mathlib documentation

ring_theory.subsemiring

Bundled subsemirings #

We define bundled subsemirings and some standard constructions: complete_lattice structure, subtype and inclusion ring homomorphisms, subsemiring map, comap and range (srange) of a ring_hom etc.

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def subsemiring.to_add_submonoid {R : Type u} [semiring R] (self : subsemiring R) :
add_submonoid R

Reinterpret a subsemiring as an add_submonoid.

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def subsemiring.to_submonoid {R : Type u} [semiring R] (self : subsemiring R) :
submonoid R

Reinterpret a subsemiring as a submonoid.

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structure subsemiring (R : Type u) [semiring R] :
Type u

A subsemiring of a semiring R is a subset s that is both a multiplicative and an additive submonoid.

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@[protected, instance]
def subsemiring.set_like {R : Type u} [semiring R] :
set_like (subsemiring R) R
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@[simp]
theorem subsemiring.mem_carrier {R : Type u} [semiring R] {s : subsemiring R} {x : R} :
x ∈ s.carrier ↔ x ∈ s
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@[ext]
theorem subsemiring.ext {R : Type u} [semiring R] {S T : subsemiring R} (h : ∀ (x : R), x ∈ S ↔ x ∈ T) :
S = T

Two subsemirings are equal if they have the same elements.

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@[protected]
def subsemiring.copy {R : Type u} [semiring R] (S : subsemiring R) (s : set R) (hs : s = ↑S) :
subsemiring R

Copy of a subsemiring with a new carrier equal to the old one. Useful to fix definitional equalities.

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theorem subsemiring.to_submonoid_injective {R : Type u} [semiring R] :
function.injective subsemiring.to_submonoid
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theorem subsemiring.to_submonoid_strict_mono {R : Type u} [semiring R] :
strict_mono subsemiring.to_submonoid
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theorem subsemiring.to_submonoid_mono {R : Type u} [semiring R] :
monotone subsemiring.to_submonoid
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theorem subsemiring.to_add_submonoid_injective {R : Type u} [semiring R] :
function.injective subsemiring.to_add_submonoid
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theorem subsemiring.to_add_submonoid_strict_mono {R : Type u} [semiring R] :
strict_mono subsemiring.to_add_submonoid
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theorem subsemiring.to_add_submonoid_mono {R : Type u} [semiring R] :
monotone subsemiring.to_add_submonoid
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@[protected]
def subsemiring.mk' {R : Type u} [semiring R] (s : set R) (sm : submonoid R) (hm : ↑sm = s) (sa : add_submonoid R) (ha : ↑sa = s) :
subsemiring R

Construct a subsemiring R from a set s, a submonoid sm, and an additive submonoid sa such that x ∈ s ↔ x ∈ sm ↔ x ∈ sa.

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@[simp]
theorem subsemiring.coe_mk' {R : Type u} [semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :
↑(subsemiring.mk' s sm hm sa ha) = s
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@[simp]
theorem subsemiring.mem_mk' {R : Type u} [semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) {x : R} :
x ∈ subsemiring.mk' s sm hm sa ha ↔ x ∈ s
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@[simp]
theorem subsemiring.mk'_to_submonoid {R : Type u} [semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :
(subsemiring.mk' s sm hm sa ha).to_submonoid = sm
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@[simp]
theorem subsemiring.mk'_to_add_submonoid {R : Type u} [semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :
(subsemiring.mk' s sm hm sa ha).to_add_submonoid = sa
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theorem subsemiring.one_mem {R : Type u} [semiring R] (s : subsemiring R) :
1 ∈ s

A subsemiring contains the semiring's 1.

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theorem subsemiring.zero_mem {R : Type u} [semiring R] (s : subsemiring R) :
0 ∈ s

A subsemiring contains the semiring's 0.

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theorem subsemiring.mul_mem {R : Type u} [semiring R] (s : subsemiring R) {x y : R} :
x ∈ s → y ∈ s → x * y ∈ s

A subsemiring is closed under multiplication.

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theorem subsemiring.add_mem {R : Type u} [semiring R] (s : subsemiring R) {x y : R} :
x ∈ s → y ∈ s → x + y ∈ s

A subsemiring is closed under addition.

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theorem subsemiring.list_prod_mem {R : Type u} [semiring R] (s : subsemiring R) {l : list R} :
(∀ (x : R), x ∈ l → x ∈ s) → l.prod ∈ s

Product of a list of elements in a subsemiring is in the subsemiring.

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theorem subsemiring.list_sum_mem {R : Type u} [semiring R] (s : subsemiring R) {l : list R} :
(∀ (x : R), x ∈ l → x ∈ s) → l.sum ∈ s

Sum of a list of elements in a subsemiring is in the subsemiring.

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theorem subsemiring.multiset_prod_mem {R : Type u_1} [comm_semiring R] (s : subsemiring R) (m : multiset R) :
(∀ (a : R), a ∈ m → a ∈ s) → m.prod ∈ s

Product of a multiset of elements in a subsemiring of a comm_semiring is in the subsemiring.

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theorem subsemiring.multiset_sum_mem {R : Type u_1} [semiring R] (s : subsemiring R) (m : multiset R) :
(∀ (a : R), a ∈ m → a ∈ s) → m.sum ∈ s

Sum of a multiset of elements in a subsemiring of a semiring is in the add_subsemiring.

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theorem subsemiring.prod_mem {R : Type u_1} [comm_semiring R] (s : subsemiring R) {ι : Type u_2} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :
∏ (i : ι) in t, f i ∈ s

Product of elements of a subsemiring of a comm_semiring indexed by a finset is in the subsemiring.

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theorem subsemiring.sum_mem {R : Type u_1} [semiring R] (s : subsemiring R) {ι : Type u_2} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :
∑ (i : ι) in t, f i ∈ s

Sum of elements in an subsemiring of an semiring indexed by a finset is in the add_subsemiring.

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theorem subsemiring.pow_mem {R : Type u} [semiring R] (s : subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :
x ^ n ∈ s
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theorem subsemiring.nsmul_mem {R : Type u} [semiring R] (s : subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :
n • x ∈ s
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theorem subsemiring.coe_nat_mem {R : Type u} [semiring R] (s : subsemiring R) (n : ℕ) :
↑n ∈ s
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@[protected, instance]
def subsemiring.to_semiring {R : Type u} [semiring R] (s : subsemiring R) :
semiring ↥s

A subsemiring of a semiring inherits a semiring structure

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@[simp, norm_cast]
theorem subsemiring.coe_one {R : Type u} [semiring R] (s : subsemiring R) :
↑1 = 1
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@[simp, norm_cast]
theorem subsemiring.coe_zero {R : Type u} [semiring R] (s : subsemiring R) :
↑0 = 0
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@[simp, norm_cast]
theorem subsemiring.coe_add {R : Type u} [semiring R] (s : subsemiring R) (x y : ↥s) :
↑(x + y) = ↑x + ↑y
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@[simp, norm_cast]
theorem subsemiring.coe_mul {R : Type u} [semiring R] (s : subsemiring R) (x y : ↥s) :
↑x * y = (↑x) * ↑y
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@[simp, norm_cast]
theorem subsemiring.coe_pow {R : Type u} [semiring R] (s : subsemiring R) (x : ↥s) (n : ℕ) :
↑(x ^ n) = ↑x ^ n
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@[protected, instance]
def subsemiring.nontrivial {R : Type u} [semiring R] (s : subsemiring R) [nontrivial R] :
nontrivial ↥s
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@[protected, instance]
def subsemiring.no_zero_divisors {R : Type u} [semiring R] (s : subsemiring R) [no_zero_divisors R] :
no_zero_divisors ↥s
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@[protected, instance]
def subsemiring.to_comm_semiring {R : Type u_1} [comm_semiring R] (s : subsemiring R) :
comm_semiring ↥s

A subsemiring of a comm_semiring is a comm_semiring.

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def subsemiring.subtype {R : Type u} [semiring R] (s : subsemiring R) :
↥s →+* R

The natural ring hom from a subsemiring of semiring R to R.

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@[simp]
theorem subsemiring.coe_subtype {R : Type u} [semiring R] (s : subsemiring R) :
⇑(s.subtype) = coe
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@[protected, instance]
def subsemiring.to_ordered_semiring {R : Type u_1} [ordered_semiring R] (s : subsemiring R) :
ordered_semiring ↥s

A subsemiring of an ordered_semiring is an ordered_semiring.

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@[protected, instance]
def subsemiring.to_ordered_comm_semiring {R : Type u_1} [ordered_comm_semiring R] (s : subsemiring R) :
ordered_comm_semiring ↥s

A subsemiring of an ordered_comm_semiring is an ordered_comm_semiring.

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@[protected, instance]
def subsemiring.to_linear_ordered_semiring {R : Type u_1} [linear_ordered_semiring R] (s : subsemiring R) :
linear_ordered_semiring ↥s

A subsemiring of a linear_ordered_semiring is a linear_ordered_semiring.

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Note: currently, there is no linear_ordered_comm_semiring.

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@[simp]
theorem subsemiring.mem_to_submonoid {R : Type u} [semiring R] {s : subsemiring R} {x : R} :
x ∈ s.to_submonoid ↔ x ∈ s
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@[simp]
theorem subsemiring.coe_to_submonoid {R : Type u} [semiring R] (s : subsemiring R) :
↑(s.to_submonoid) = ↑s
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@[simp]
theorem subsemiring.mem_to_add_submonoid {R : Type u} [semiring R] {s : subsemiring R} {x : R} :
x ∈ s.to_add_submonoid ↔ x ∈ s
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@[simp]
theorem subsemiring.coe_to_add_submonoid {R : Type u} [semiring R] (s : subsemiring R) :
↑(s.to_add_submonoid) = ↑s
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@[protected, instance]
def subsemiring.has_top {R : Type u} [semiring R] :
has_top (subsemiring R)

The subsemiring R of the semiring R.

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@[simp]
theorem subsemiring.mem_top {R : Type u} [semiring R] (x : R) :
x ∈ ⊤
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@[simp]
theorem subsemiring.coe_top {R : Type u} [semiring R] :
↑⊤ = set.univ
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def subsemiring.comap {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : subsemiring S) :
subsemiring R

The preimage of a subsemiring along a ring homomorphism is a subsemiring.

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@[simp]
theorem subsemiring.coe_comap {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring S) (f : R →+* S) :
↑(subsemiring.comap f s) = ⇑f ⁻¹' ↑s
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@[simp]
theorem subsemiring.mem_comap {R : Type u} {S : Type v} [semiring R] [semiring S] {s : subsemiring S} {f : R →+* S} {x : R} :
x ∈ subsemiring.comap f s ↔ ⇑f x ∈ s
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theorem subsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [semiring R] [semiring S] [semiring T] (s : subsemiring T) (g : S →+* T) (f : R →+* S) :
subsemiring.comap f (subsemiring.comap g s) = subsemiring.comap (g.comp f) s
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def subsemiring.map {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : subsemiring R) :
subsemiring S

The image of a subsemiring along a ring homomorphism is a subsemiring.

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@[simp]
theorem subsemiring.coe_map {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : subsemiring R) :
↑(subsemiring.map f s) = ⇑f '' ↑s
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@[simp]
theorem subsemiring.mem_map {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} {s : subsemiring R} {y : S} :
y ∈ subsemiring.map f s ↔ ∃ (x : R) (H : x ∈ s), ⇑f x = y
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theorem subsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [semiring R] [semiring S] [semiring T] (s : subsemiring R) (g : S →+* T) (f : R →+* S) :
subsemiring.map g (subsemiring.map f s) = subsemiring.map (g.comp f) s
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theorem subsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} {s : subsemiring R} {t : subsemiring S} :
subsemiring.map f s ≤ t ↔ s ≤ subsemiring.comap f t
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theorem subsemiring.gc_map_comap {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
galois_connection (subsemiring.map f) (subsemiring.comap f)
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def ring_hom.srange {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
subsemiring S

The range of a ring homomorphism is a subsemiring. See Note [range copy pattern].

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@[simp]
theorem ring_hom.coe_srange {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
↑(f.srange) = set.range ⇑f
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@[simp]
theorem ring_hom.mem_srange {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} {y : S} :
y ∈ f.srange ↔ ∃ (x : R), ⇑f x = y
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theorem ring_hom.srange_eq_map {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
f.srange = subsemiring.map f ⊤
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theorem ring_hom.mem_srange_self {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (x : R) :
⇑f x ∈ f.srange
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theorem ring_hom.map_srange {R : Type u} {S : Type v} {T : Type w} [semiring R] [semiring S] [semiring T] (g : S →+* T) (f : R →+* S) :
subsemiring.map g f.srange = (g.comp f).srange
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@[protected, instance]
def subsemiring.has_bot {R : Type u} [semiring R] :
has_bot (subsemiring R)
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@[protected, instance]
def subsemiring.inhabited {R : Type u} [semiring R] :
inhabited (subsemiring R)
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theorem subsemiring.coe_bot {R : Type u} [semiring R] :
↑⊥ = set.range coe
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theorem subsemiring.mem_bot {R : Type u} [semiring R] {x : R} :
x ∈ ⊥ ↔ ∃ (n : ℕ), ↑n = x
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@[protected, instance]
def subsemiring.has_inf {R : Type u} [semiring R] :
has_inf (subsemiring R)

The inf of two subsemirings is their intersection.

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@[simp]
theorem subsemiring.coe_inf {R : Type u} [semiring R] (p p' : subsemiring R) :
↑(p ⊓ p') = ↑p ∩ ↑p'
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@[simp]
theorem subsemiring.mem_inf {R : Type u} [semiring R] {p p' : subsemiring R} {x : R} :
x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'
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@[protected, instance]
def subsemiring.has_Inf {R : Type u} [semiring R] :
has_Inf (subsemiring R)
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@[simp, norm_cast]
theorem subsemiring.coe_Inf {R : Type u} [semiring R] (S : set (subsemiring R)) :
↑(Inf S) = ⋂ (s : subsemiring R) (H : s ∈ S), ↑s
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theorem subsemiring.mem_Inf {R : Type u} [semiring R] {S : set (subsemiring R)} {x : R} :
x ∈ Inf S ↔ ∀ (p : subsemiring R), p ∈ S → x ∈ p
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@[simp]
theorem subsemiring.Inf_to_submonoid {R : Type u} [semiring R] (s : set (subsemiring R)) :
(Inf s).to_submonoid = ⨅ (t : subsemiring R) (H : t ∈ s), t.to_submonoid
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@[simp]
theorem subsemiring.Inf_to_add_submonoid {R : Type u} [semiring R] (s : set (subsemiring R)) :
(Inf s).to_add_submonoid = ⨅ (t : subsemiring R) (H : t ∈ s), t.to_add_submonoid
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@[protected, instance]
def subsemiring.complete_lattice {R : Type u} [semiring R] :
complete_lattice (subsemiring R)

Subsemirings of a semiring form a complete lattice.

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theorem subsemiring.eq_top_iff' {R : Type u} [semiring R] (A : subsemiring R) :
A = ⊤ ↔ ∀ (x : R), x ∈ A
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def subsemiring.closure {R : Type u} [semiring R] (s : set R) :
subsemiring R

The subsemiring generated by a set.

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theorem subsemiring.mem_closure {R : Type u} [semiring R] {x : R} {s : set R} :
x ∈ subsemiring.closure s ↔ ∀ (S : subsemiring R), s ⊆ ↑S → x ∈ S
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@[simp]
theorem subsemiring.subset_closure {R : Type u} [semiring R] {s : set R} :
s ⊆ ↑(subsemiring.closure s)

The subsemiring generated by a set includes the set.

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@[simp]
theorem subsemiring.closure_le {R : Type u} [semiring R] {s : set R} {t : subsemiring R} :
subsemiring.closure s ≤ t ↔ s ⊆ ↑t

A subsemiring S includes closure s if and only if it includes s.

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theorem subsemiring.closure_mono {R : Type u} [semiring R] ⦃s t : set R⦄ (h : s ⊆ t) :
subsemiring.closure s ≤ subsemiring.closure t

Subsemiring closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

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theorem subsemiring.closure_eq_of_le {R : Type u} [semiring R] {s : set R} {t : subsemiring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ subsemiring.closure s) :
subsemiring.closure s = t
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def submonoid.subsemiring_closure {R : Type u} [semiring R] (M : submonoid R) :
subsemiring R

The additive closure of a submonoid is a subsemiring.

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theorem submonoid.subsemiring_closure_coe {R : Type u} [semiring R] (M : submonoid R) :
↑(M.subsemiring_closure) = ↑(add_submonoid.closure ↑M)
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theorem submonoid.subsemiring_closure_to_add_submonoid {R : Type u} [semiring R] (M : submonoid R) :
M.subsemiring_closure.to_add_submonoid = add_submonoid.closure ↑M
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theorem submonoid.subsemiring_closure_eq_closure {R : Type u} [semiring R] (M : submonoid R) :
M.subsemiring_closure = subsemiring.closure ↑M

The subsemiring generated by a multiplicative submonoid coincides with the subsemiring.closure of the submonoid itself .

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@[simp]
theorem subsemiring.closure_submonoid_closure {R : Type u} [semiring R] (s : set R) :
subsemiring.closure ↑(submonoid.closure s) = subsemiring.closure s
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theorem subsemiring.coe_closure_eq {R : Type u} [semiring R] (s : set R) :
↑(subsemiring.closure s) = ↑(add_submonoid.closure ↑(submonoid.closure s))

The elements of the subsemiring closure of M are exactly the elements of the additive closure of a multiplicative submonoid M.

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theorem subsemiring.mem_closure_iff {R : Type u} [semiring R] {s : set R} {x : R} :
x ∈ subsemiring.closure s ↔ x ∈ add_submonoid.closure ↑(submonoid.closure s)
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@[simp]
theorem subsemiring.closure_add_submonoid_closure {R : Type u} [semiring R] {s : set R} :
subsemiring.closure ↑(add_submonoid.closure s) = subsemiring.closure s
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theorem subsemiring.closure_induction {R : Type u} [semiring R] {s : set R} {p : R → Prop} {x : R} (h : x ∈ subsemiring.closure s) (Hs : ∀ (x : R), x ∈ s → p x) (H0 : p 0) (H1 : p 1) (Hadd : ∀ (x y : R), p x → p y → p (x + y)) (Hmul : ∀ (x y : R), p x → p y → p (x * y)) :
p x

An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

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theorem subsemiring.mem_closure_iff_exists_list {R : Type u} [semiring R] {s : set R} {x : R} :
x ∈ subsemiring.closure s ↔ ∃ (L : list (list R)), (∀ (t : list R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s) ∧ (list.map list.prod L).sum = x
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@[protected]
def subsemiring.gi (R : Type u) [semiring R] :
galois_insertion subsemiring.closure coe

closure forms a Galois insertion with the coercion to set.

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theorem subsemiring.closure_eq {R : Type u} [semiring R] (s : subsemiring R) :
subsemiring.closure ↑s = s

Closure of a subsemiring S equals S.

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@[simp]
theorem subsemiring.closure_empty {R : Type u} [semiring R] :
subsemiring.closure ∅ = ⊥
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@[simp]
theorem subsemiring.closure_univ {R : Type u} [semiring R] :
subsemiring.closure set.univ = ⊤
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theorem subsemiring.closure_union {R : Type u} [semiring R] (s t : set R) :
subsemiring.closure (s ∪ t) = subsemiring.closure s ⊔ subsemiring.closure t
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theorem subsemiring.closure_Union {R : Type u} [semiring R] {ι : Sort u_1} (s : ι → set R) :
subsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), subsemiring.closure (s i)
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theorem subsemiring.closure_sUnion {R : Type u} [semiring R] (s : set (set R)) :
subsemiring.closure (⋃₀s) = ⨆ (t : set R) (H : t ∈ s), subsemiring.closure t
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theorem subsemiring.map_sup {R : Type u} {S : Type v} [semiring R] [semiring S] (s t : subsemiring R) (f : R →+* S) :
subsemiring.map f (s ⊔ t) = subsemiring.map f s ⊔ subsemiring.map f t
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theorem subsemiring.map_supr {R : Type u} {S : Type v} [semiring R] [semiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subsemiring R) :
subsemiring.map f (supr s) = ⨆ (i : ι), subsemiring.map f (s i)
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theorem subsemiring.comap_inf {R : Type u} {S : Type v} [semiring R] [semiring S] (s t : subsemiring S) (f : R →+* S) :
subsemiring.comap f (s ⊓ t) = subsemiring.comap f s ⊓ subsemiring.comap f t
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theorem subsemiring.comap_infi {R : Type u} {S : Type v} [semiring R] [semiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subsemiring S) :
subsemiring.comap f (infi s) = ⨅ (i : ι), subsemiring.comap f (s i)
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@[simp]
theorem subsemiring.map_bot {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
subsemiring.map f ⊥ = ⊥
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@[simp]
theorem subsemiring.comap_top {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
subsemiring.comap f ⊤ = ⊤
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def subsemiring.prod {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) (t : subsemiring S) :
subsemiring (R × S)

Given subsemirings s, t of semirings R, S respectively, s.prod t is s × t as a subsemiring of R × S.

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@[norm_cast]
theorem subsemiring.coe_prod {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) (t : subsemiring S) :
↑(s.prod t) = ↑s.prod ↑t
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theorem subsemiring.mem_prod {R : Type u} {S : Type v} [semiring R] [semiring S] {s : subsemiring R} {t : subsemiring S} {p : R × S} :
p ∈ s.prod t ↔ p.fst ∈ s ∧ p.snd ∈ t
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theorem subsemiring.prod_mono {R : Type u} {S : Type v} [semiring R] [semiring S] ⦃s₁ s₂ : subsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : subsemiring S⦄ (ht : t₁ ≤ t₂) :
s₁.prod t₁ ≤ s₂.prod t₂
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theorem subsemiring.prod_mono_right {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) :
monotone (λ (t : subsemiring S), s.prod t)
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theorem subsemiring.prod_mono_left {R : Type u} {S : Type v} [semiring R] [semiring S] (t : subsemiring S) :
monotone (λ (s : subsemiring R), s.prod t)
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theorem subsemiring.prod_top {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) :
s.prod ⊤ = subsemiring.comap (ring_hom.fst R S) s
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theorem subsemiring.top_prod {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring S) :
⊤.prod s = subsemiring.comap (ring_hom.snd R S) s
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@[simp]
theorem subsemiring.top_prod_top {R : Type u} {S : Type v} [semiring R] [semiring S] :
⊤.prod ⊤ = ⊤
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def subsemiring.prod_equiv {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) (t : subsemiring S) :
↥(s.prod t) ≃+* ↥s × ↥t

Product of subsemirings is isomorphic to their product as monoids.

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theorem subsemiring.mem_supr_of_directed {R : Type u} [semiring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subsemiring R} (hS : directed has_le.le S) {x : R} :
(x ∈ ⨆ (i : ι), S i) ↔ ∃ (i : ι), x ∈ S i
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theorem subsemiring.coe_supr_of_directed {R : Type u} [semiring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subsemiring R} (hS : directed has_le.le S) :
(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)
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theorem subsemiring.mem_Sup_of_directed_on {R : Type u} [semiring R] {S : set (subsemiring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : R} :
x ∈ Sup S ↔ ∃ (s : subsemiring R) (H : s ∈ S), x ∈ s
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theorem subsemiring.coe_Sup_of_directed_on {R : Type u} [semiring R] {S : set (subsemiring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :
↑(Sup S) = ⋃ (s : subsemiring R) (H : s ∈ S), ↑s
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def ring_hom.srestrict {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : subsemiring R) :
↥s →+* S

Restriction of a ring homomorphism to a subsemiring of the domain.

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@[simp]
theorem ring_hom.srestrict_apply {R : Type u} {S : Type v} [semiring R] [semiring S] {s : subsemiring R} (f : R →+* S) (x : ↥s) :
⇑(f.srestrict s) x = ⇑f ↑x
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def ring_hom.cod_srestrict {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : subsemiring S) (h : ∀ (x : R), ⇑f x ∈ s) :
R →+* ↥s

Restriction of a ring homomorphism to a subsemiring of the codomain.

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def ring_hom.srange_restrict {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
R →+* ↥(f.srange)

Restriction of a ring homomorphism to its range interpreted as a subsemiring.

This is the bundled version of set.range_factorization.

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@[simp]
theorem ring_hom.coe_srange_restrict {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (x : R) :
↑(⇑(f.srange_restrict) x) = ⇑f x
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theorem ring_hom.srange_restrict_surjective {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) :
function.surjective ⇑(f.srange_restrict)
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theorem ring_hom.srange_top_iff_surjective {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} :
f.srange = ⊤ ↔ function.surjective ⇑f
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theorem ring_hom.srange_top_of_surjective {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (hf : function.surjective ⇑f) :
f.srange = ⊤

The range of a surjective ring homomorphism is the whole of the codomain.

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def ring_hom.eq_slocus {R : Type u} {S : Type v} [semiring R] [semiring S] (f g : R →+* S) :
subsemiring R

The subsemiring of elements x : R such that f x = g x

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theorem ring_hom.eq_on_sclosure {R : Type u} {S : Type v} [semiring R] [semiring S] {f g : R →+* S} {s : set R} (h : set.eq_on ⇑f ⇑g s) :
set.eq_on ⇑f ⇑g ↑(subsemiring.closure s)

If two ring homomorphisms are equal on a set, then they are equal on its subsemiring closure.

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theorem ring_hom.eq_of_eq_on_stop {R : Type u} {S : Type v} [semiring R] [semiring S] {f g : R →+* S} (h : set.eq_on ⇑f ⇑g ↑⊤) :
f = g
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theorem ring_hom.eq_of_eq_on_sdense {R : Type u} {S : Type v} [semiring R] [semiring S] {s : set R} (hs : subsemiring.closure s = ⊤) {f g : R →+* S} (h : set.eq_on ⇑f ⇑g s) :
f = g
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theorem ring_hom.sclosure_preimage_le {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : set S) :
subsemiring.closure (⇑f ⁻¹' s) ≤ subsemiring.comap f (subsemiring.closure s)
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theorem ring_hom.map_sclosure {R : Type u} {S : Type v} [semiring R] [semiring S] (f : R →+* S) (s : set R) :
subsemiring.map f (subsemiring.closure s) = subsemiring.closure (⇑f '' s)

The image under a ring homomorphism of the subsemiring generated by a set equals the subsemiring generated by the image of the set.

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def subsemiring.inclusion {R : Type u} [semiring R] {S T : subsemiring R} (h : S ≤ T) :
↥S →* ↥T

The ring homomorphism associated to an inclusion of subsemirings.

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@[simp]
theorem subsemiring.srange_subtype {R : Type u} [semiring R] (s : subsemiring R) :
s.subtype.srange = s
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@[simp]
theorem subsemiring.range_fst {R : Type u} {S : Type v} [semiring R] [semiring S] :
(ring_hom.fst R S).srange = ⊤
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@[simp]
theorem subsemiring.range_snd {R : Type u} {S : Type v} [semiring R] [semiring S] :
(ring_hom.snd R S).srange = ⊤
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@[simp]
theorem subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [semiring R] [semiring S] (s : subsemiring R) (t : subsemiring S) :
s.prod ⊥ ⊔ ⊥.prod t = s.prod t
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def ring_equiv.subsemiring_congr {R : Type u} [semiring R] {s t : subsemiring R} (h : s = t) :
↥s ≃+* ↥t

Makes the identity isomorphism from a proof two subsemirings of a multiplicative monoid are equal.

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def ring_equiv.sof_left_inverse {R : Type u} {S : Type v} [semiring R] [semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) :
R ≃+* ↥(f.srange)

Restrict a ring homomorphism with a left inverse to a ring isomorphism to its ring_hom.srange.

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@[simp]
theorem ring_equiv.sof_left_inverse_apply {R : Type u} {S : Type v} [semiring R] [semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : R) :
↑(⇑(ring_equiv.sof_left_inverse h) x) = ⇑f x
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@[simp]
theorem ring_equiv.sof_left_inverse_symm_apply {R : Type u} {S : Type v} [semiring R] [semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : ↥(f.srange)) :
⇑((ring_equiv.sof_left_inverse h).symm) x = g ↑x