statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
closure_eq_of_le {s : set R} {t : subring R} (h₁ : s ⊆ t) (h₂ : t ≤ closure s) :
closure s = t | le_antisymm (closure_le.2 h₁) h₂ | lemma | subring.closure_eq_of_le | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_induction {s : set R} {p : R → Prop} {x} (h : x ∈ closure s)
(Hs : ∀ x ∈ s, p x) (H0 : p 0) (H1 : p 1)
(Hadd : ∀ x y, p x → p y → p (x + y))
(Hneg : ∀ (x : R), p x → p (-x))
(Hmul : ∀ x y, p x → p y → p (x * y)) : p x | (@closure_le _ _ _ ⟨p, Hmul, H1, Hadd, H0, Hneg⟩).2 Hs h | lemma | subring.closure_induction | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure"
] | An induction principle for closure membership. If `p` holds for `0`, `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_induction₂ {s : set R} {p : R → R → Prop} {a b : R}
(ha : a ∈ closure s) (hb : b ∈ closure s)
(Hs : ∀ (x ∈ s) (y ∈ s), p x y)
(H0_left : ∀ x, p 0 x)
(H0_right : ∀ x, p x 0)
(H1_left : ∀ x, p 1 x)
(H1_right : ∀ x, p x 1)
(Hneg_left : ∀ x y, p x y → p (-x) y)
(Hneg_right : ∀ x y, p x y → p x (-y))... | begin
refine closure_induction hb _ (H0_right _) (H1_right _)
(Hadd_right a) (Hneg_right a) (Hmul_right a),
refine closure_induction ha Hs (λ x _, H0_left x) (λ x _, H1_left x) _ _ _,
{ exact (λ x y H₁ H₂ z zs, Hadd_left x y z (H₁ z zs) (H₂ z zs)) },
{ exact (λ x hx z zs, Hneg_left x z (hx z zs)) },
{ exa... | lemma | subring.closure_induction₂ | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure"
] | An induction principle for closure membership, for predicates with two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff {s : set R} {x} :
x ∈ closure s ↔ x ∈ add_subgroup.closure (submonoid.closure s : set R) | ⟨λ h, closure_induction h (λ x hx, add_subgroup.subset_closure $ submonoid.subset_closure hx)
(add_subgroup.zero_mem _)
(add_subgroup.subset_closure ( submonoid.one_mem (submonoid.closure s)) )
(λ x y hx hy, add_subgroup.add_mem _ hx hy )
(λ x hx, add_subgroup.neg_mem _ hx )
(λ x y hx hy, add_subgroup.closure_in... | lemma | subring.mem_closure_iff | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"mem_closure_iff",
"mul_zero",
"submonoid.closure",
"submonoid.closure_induction",
"submonoid.one_mem",
"submonoid.subset_closure",
"subset_closure",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_comm_ring_of_comm {s : set R} (hcomm : ∀ (a ∈ s) (b ∈ s), a * b = b * a) :
comm_ring (closure s) | { mul_comm := λ x y,
begin
ext,
simp only [subring.coe_mul],
refine closure_induction₂ x.prop y.prop
hcomm
(λ x, by simp only [mul_zero, zero_mul])
(λ x, by simp only [mul_zero, zero_mul])
(λ x, by simp only [mul_one, one_mul])
(λ x, by simp only [mul_one, one_mul])
(λ x y hxy, by ... | def | subring.closure_comm_ring_of_comm | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"comm_ring",
"mul_assoc",
"mul_comm",
"mul_neg",
"mul_one",
"mul_zero",
"neg_mul",
"one_mul",
"subring.coe_mul",
"zero_mul"
] | If all elements of `s : set A` commute pairwise, then `closure s` is a commutative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_list_of_mem_closure {s : set R} {x : R} (h : x ∈ closure s) :
(∃ L : list (list R), (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = (-1:R)) ∧ (L.map list.prod).sum = x) | add_subgroup.closure_induction (mem_closure_iff.1 h)
(λ x hx, let ⟨l, hl, h⟩ :=submonoid.exists_list_of_mem_closure hx in ⟨[l], by simp [h];
clear_aux_decl; tauto!⟩)
⟨[], by simp⟩
(λ x y ⟨l, hl1, hl2⟩ ⟨m, hm1, hm2⟩, ⟨l ++ m, λ t ht, (list.mem_append.1 ht).elim (hl1 t) (hm1 t),
by simp [hl2, hm2]⟩)
(λ x ... | theorem | subring.exists_list_of_mem_closure | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"list.prod",
"submonoid.exists_list_of_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq (s : subring R) : closure (s : set R) = s | (subring.gi R).l_u_eq s | lemma | subring.closure_eq | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subring",
"subring.gi"
] | Closure of a subring `S` equals `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_empty : closure (∅ : set R) = ⊥ | (subring.gi R).gc.l_bot | lemma | subring.closure_empty | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"closure_empty",
"subring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_union (s t : set R) : closure (s ∪ t) = closure s ⊔ closure t | (subring.gi R).gc.l_sup | lemma | subring.closure_union | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"closure_union",
"subring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Union {ι} (s : ι → set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) | (subring.gi R).gc.l_supr | lemma | subring.closure_Union | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"closure_Union",
"subring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_sUnion (s : set (set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t | (subring.gi R).gc.l_Sup | lemma | subring.closure_sUnion | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup (s t : subring R) (f : R →+* S) : (s ⊔ t).map f = s.map f ⊔ t.map f | (gc_map_comap f).l_sup | lemma | subring.map_sup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr {ι : Sort*} (f : R →+* S) (s : ι → subring R) :
(supr s).map f = ⨆ i, (s i).map f | (gc_map_comap f).l_supr | lemma | subring.map_supr | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_supr",
"subring",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf (s t : subring S) (f : R →+* S) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f | (gc_map_comap f).u_inf | lemma | subring.comap_inf | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_infi {ι : Sort*} (f : R →+* S) (s : ι → subring S) :
(infi s).comap f = ⨅ i, (s i).comap f | (gc_map_comap f).u_infi | lemma | subring.comap_infi | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"infi",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot (f : R →+* S) : (⊥ : subring R).map f = ⊥ | (gc_map_comap f).l_bot | lemma | subring.map_bot | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top (f : R →+* S) : (⊤ : subring S).comap f = ⊤ | (gc_map_comap f).u_top | lemma | subring.comap_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (s : subring R) (t : subring S) : subring (R × S) | { carrier := s ×ˢ t,
.. s.to_submonoid.prod t.to_submonoid, .. s.to_add_subgroup.prod t.to_add_subgroup} | def | subring.prod | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | Given `subring`s `s`, `t` of rings `R`, `S` respectively, `s.prod t` is `s ×̂ t`
as a subring of `R × S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (s : subring R) (t : subring S) : (s.prod t : set (R × S)) = s ×ˢ t | rfl | lemma | subring.coe_prod | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod {s : subring R} {t : subring S} {p : R × S} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t | iff.rfl | lemma | subring.mem_prod | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono ⦃s₁ s₂ : subring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : subring S⦄
(ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ | set.prod_mono hs ht | lemma | subring.prod_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.prod_mono",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_right (s : subring R) : monotone (λ t : subring S, s.prod t) | prod_mono (le_refl s) | lemma | subring.prod_mono_right | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monotone",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_left (t : subring S) : monotone (λ s : subring R, s.prod t) | λ s₁ s₂ hs, prod_mono hs (le_refl t) | lemma | subring.prod_mono_left | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monotone",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top (s : subring R) :
s.prod (⊤ : subring S) = s.comap (ring_hom.fst R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_fst] | lemma | subring.prod_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monoid_hom.coe_fst",
"ring_hom.fst",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod (s : subring S) :
(⊤ : subring R).prod s = s.comap (ring_hom.snd R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_snd] | lemma | subring.top_prod | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"monoid_hom.coe_snd",
"ring_hom.snd",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod_top : (⊤ : subring R).prod (⊤ : subring S) = ⊤ | (top_prod _).trans $ comap_top _ | lemma | subring.top_prod_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv (s : subring R) (t : subring S) : s.prod t ≃+* s × t | { map_mul' := λ x y, rfl, map_add' := λ x y, rfl, .. equiv.set.prod ↑s ↑t } | def | subring.prod_equiv | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"equiv.set.prod",
"subring"
] | Product of subrings is isomorphic to their product as rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → subring R} (hS : directed (≤) S)
{x : R} :
x ∈ (⨆ i, S i) ↔ ∃ i, x ∈ S i | begin
refine ⟨_, λ ⟨i, hi⟩, (set_like.le_def.1 $ le_supr S i) hi⟩,
let U : subring R := subring.mk' (⋃ i, (S i : set R))
(⨆ i, (S i).to_submonoid) (⨆ i, (S i).to_add_subgroup)
(submonoid.coe_supr_of_directed $ hS.mono_comp _ (λ _ _, id))
(add_subgroup.coe_supr_of_directed $ hS.mono_comp _ (λ _ _, id)),
... | lemma | subring.mem_supr_of_directed | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"directed",
"le_supr",
"submonoid.coe_supr_of_directed",
"subring",
"subring.mk'",
"supr_le"
] | The underlying set of a non-empty directed Sup of subrings is just a union of the subrings.
Note that this fails without the directedness assumption (the union of two subrings is
typically not a subring) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_supr_of_directed {ι} [hι : nonempty ι] {S : ι → subring R} (hS : directed (≤) S) :
((⨆ i, S i : subring R) : set R) = ⋃ i, ↑(S i) | set.ext $ λ x, by simp [mem_supr_of_directed hS] | lemma | subring.coe_supr_of_directed | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"directed",
"set.ext",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup_of_directed_on {S : set (subring R)} (Sne : S.nonempty)
(hS : directed_on (≤) S) {x : R} :
x ∈ Sup S ↔ ∃ s ∈ S, x ∈ s | begin
haveI : nonempty S := Sne.to_subtype,
simp only [Sup_eq_supr', mem_supr_of_directed hS.directed_coe, set_coe.exists, subtype.coe_mk]
end | lemma | subring.mem_Sup_of_directed_on | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"Sup_eq_supr'",
"directed_on",
"set_coe.exists",
"subring",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup_of_directed_on {S : set (subring R)} (Sne : S.nonempty) (hS : directed_on (≤) S) :
(↑(Sup S) : set R) = ⋃ s ∈ S, ↑s | set.ext $ λ x, by simp [mem_Sup_of_directed_on Sne hS] | lemma | subring.coe_Sup_of_directed_on | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"directed_on",
"set.ext",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_equiv {f : R ≃+* S} {K : subring R} {x : S} :
x ∈ K.map (f : R →+* S) ↔ f.symm x ∈ K | @set.mem_image_equiv _ _ ↑K f.to_equiv x | lemma | subring.mem_map_equiv | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.mem_image_equiv",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_eq_comap_symm (f : R ≃+* S) (K : subring R) :
K.map (f : R →+* S) = K.comap f.symm | set_like.coe_injective (f.to_equiv.image_eq_preimage K) | lemma | subring.map_equiv_eq_comap_symm | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set_like.coe_injective",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_equiv_eq_map_symm (f : R ≃+* S) (K : subring S) :
K.comap (f : R →+* S) = K.map f.symm | (map_equiv_eq_comap_symm f.symm K).symm | lemma | subring.comap_equiv_eq_map_symm | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_restrict (f : R →+* S) : R →+* f.range | f.cod_restrict f.range $ λ x, ⟨x, rfl⟩ | def | ring_hom.range_restrict | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | Restriction of a ring homomorphism to its range interpreted as a subsemiring.
This is the bundled version of `set.range_factorization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_range_restrict (f : R →+* S) (x : R) : (f.range_restrict x : S) = f x | rfl | lemma | ring_hom.coe_range_restrict | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_restrict_surjective (f : R →+* S) : function.surjective f.range_restrict | λ ⟨y, hy⟩, let ⟨x, hx⟩ := mem_range.mp hy in ⟨x, subtype.ext hx⟩ | lemma | ring_hom.range_restrict_surjective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_top_iff_surjective {f : R →+* S} :
f.range = (⊤ : subring S) ↔ function.surjective f | set_like.ext'_iff.trans $ iff.trans (by rw [coe_range, coe_top]) set.range_iff_surjective | lemma | ring_hom.range_top_iff_surjective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.range_iff_surjective",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_top_of_surjective (f : R →+* S) (hf : function.surjective f) :
f.range = (⊤ : subring S) | range_top_iff_surjective.2 hf | lemma | ring_hom.range_top_of_surjective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | The range of a surjective ring homomorphism is the whole of the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_locus (f g : R →+* S) : subring R | { carrier := {x | f x = g x}, .. (f : R →* S).eq_mlocus g, .. (f : R →+ S).eq_locus g } | def | ring_hom.eq_locus | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | The subring of elements `x : R` such that `f x = g x`, i.e.,
the equalizer of f and g as a subring of R | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_locus_same (f : R →+* S) : f.eq_locus f = ⊤ | set_like.ext $ λ _, eq_self_iff_true _ | lemma | ring_hom.eq_locus_same | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_set_closure {f g : R →+* S} {s : set R} (h : set.eq_on f g s) :
set.eq_on f g (closure s) | show closure s ≤ f.eq_locus g, from closure_le.2 h | lemma | ring_hom.eq_on_set_closure | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"set.eq_on"
] | If two ring homomorphisms are equal on a set, then they are equal on its subring closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_eq_on_set_top {f g : R →+* S} (h : set.eq_on f g (⊤ : subring R)) :
f = g | ext $ λ x, h trivial | lemma | ring_hom.eq_of_eq_on_set_top | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set.eq_on",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq_on_set_dense {s : set R} (hs : closure s = ⊤) {f g : R →+* S} (h : s.eq_on f g) :
f = g | eq_of_eq_on_set_top $ hs ▸ eq_on_set_closure h | lemma | ring_hom.eq_of_eq_on_set_dense | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_preimage_le (f : R →+* S) (s : set S) :
closure (f ⁻¹' s) ≤ (closure s).comap f | closure_le.2 $ λ x hx, set_like.mem_coe.2 $ mem_comap.2 $ subset_closure hx | lemma | ring_hom.closure_preimage_le | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_closure (f : R →+* S) (s : set R) :
(closure s).map f = closure (f '' s) | le_antisymm
(map_le_iff_le_comap.2 $ le_trans (closure_mono $ set.subset_preimage_image _ _)
(closure_preimage_le _ _))
(closure_le.2 $ set.image_subset _ subset_closure) | lemma | ring_hom.map_closure | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"closure_mono",
"set.image_subset",
"set.subset_preimage_image",
"subset_closure"
] | The image under a ring homomorphism of the subring generated by a set equals
the subring generated by the image of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion {S T : subring R} (h : S ≤ T) : S →+* T | S.subtype.cod_restrict _ (λ x, h x.2) | def | subring.inclusion | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | The ring homomorphism associated to an inclusion of subrings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_subtype (s : subring R) : s.subtype.range = s | set_like.coe_injective $ (coe_srange _).trans subtype.range_coe | lemma | subring.range_subtype | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set_like.coe_injective",
"subring",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_fst : (fst R S).srange = ⊤ | (fst R S).srange_top_of_surjective $ prod.fst_surjective | lemma | subring.range_fst | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"prod.fst_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_snd : (snd R S).srange = ⊤ | (snd R S).srange_top_of_surjective $ prod.snd_surjective | lemma | subring.range_snd | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"prod.snd_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_bot_sup_bot_prod (s : subring R) (t : subring S) :
(s.prod ⊥) ⊔ (prod ⊥ t) = s.prod t | le_antisymm (sup_le (prod_mono_right s bot_le) (prod_mono_left t bot_le)) $
assume p hp, prod.fst_mul_snd p ▸ mul_mem
((le_sup_left : s.prod ⊥ ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨hp.1, set_like.mem_coe.2 $ one_mem ⊥⟩)
((le_sup_right : prod ⊥ t ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨set_like.mem_coe.2 $ one_mem ⊥, hp.2⟩) | lemma | subring.prod_bot_sup_bot_prod | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"bot_le",
"le_sup_left",
"le_sup_right",
"prod.fst_mul_snd",
"subring",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring_congr (h : s = t) : s ≃+* t | { map_mul' := λ _ _, rfl, map_add' := λ _ _, rfl, ..equiv.set_congr $ congr_arg _ h } | def | ring_equiv.subring_congr | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"equiv.set_congr"
] | Makes the identity isomorphism from a proof two subrings of a multiplicative
monoid are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_inverse {g : S → R} {f : R →+* S} (h : function.left_inverse g f) :
R ≃+* f.range | { to_fun := λ x, f.range_restrict x,
inv_fun := λ x, (g ∘ f.range.subtype) x,
left_inv := h,
right_inv := λ x, subtype.ext $
let ⟨x', hx'⟩ := ring_hom.mem_range.mp x.prop in
show f (g x) = x, by rw [←hx', h x'],
..f.range_restrict } | def | ring_equiv.of_left_inverse | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"inv_fun",
"subtype.ext"
] | Restrict a ring homomorphism with a left inverse to a ring isomorphism to its
`ring_hom.range`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_inverse_apply
{g : S → R} {f : R →+* S} (h : function.left_inverse g f) (x : R) :
↑(of_left_inverse h x) = f x | rfl | lemma | ring_equiv.of_left_inverse_apply | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_left_inverse_symm_apply
{g : S → R} {f : R →+* S} (h : function.left_inverse g f) (x : f.range) :
(of_left_inverse h).symm x = g x | rfl | lemma | ring_equiv.of_left_inverse_symm_apply | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring_map (e : R ≃+* S) :
s ≃+* s.map e.to_ring_hom | e.subsemiring_map s.to_subsemiring | def | ring_equiv.subring_map | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [] | Given an equivalence `e : R ≃+* S` of rings and a subring `s` of `R`,
`subring_equiv_map e s` is the induced equivalence between `s` and `s.map e` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
in_closure.rec_on {C : R → Prop} {x : R} (hx : x ∈ closure s)
(h1 : C 1) (hneg1 : C (-1)) (hs : ∀ z ∈ s, ∀ n, C n → C (z * n))
(ha : ∀ {x y}, C x → C y → C (x + y)) : C x | begin
have h0 : C 0 := add_neg_self (1:R) ▸ ha h1 hneg1,
rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩, clear hx,
induction L with hd tl ih, { exact h0 },
rw list.forall_mem_cons at HL,
suffices : C (list.prod hd),
{ rw [list.map_cons, list.sum_cons],
exact ha this (ih HL.2) },
replace HL := ... | theorem | subring.in_closure.rec_on | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"closure",
"ih",
"list.forall_mem_cons",
"list.forall_mem_nil",
"list.prod",
"list.prod_cons",
"neg_mul_eq_mul_neg",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_subgroup.int_mul_mem {G : add_subgroup R} (k : ℤ) {g : R} (h : g ∈ G) :
(k : R) * g ∈ G | by { convert add_subgroup.zsmul_mem G h k, simp } | lemma | add_subgroup.int_mul_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def [has_smul R α] {S : subring R} (g : S) (m : α) : g • m = (g : R) • m | rfl | lemma | subring.smul_def | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"has_smul",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_left
[has_smul R β] [has_smul α β] [smul_comm_class R α β] (S : subring R) :
smul_comm_class S α β | S.to_subsemiring.smul_comm_class_left | instance | subring.smul_comm_class_left | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"has_smul",
"smul_comm_class",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_right
[has_smul α β] [has_smul R β] [smul_comm_class α R β] (S : subring R) :
smul_comm_class α S β | S.to_subsemiring.smul_comm_class_right | instance | subring.smul_comm_class_right | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"has_smul",
"smul_comm_class",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center.smul_comm_class_left : smul_comm_class (center R) R R | subsemiring.center.smul_comm_class_left | instance | subring.center.smul_comm_class_left | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"smul_comm_class",
"subsemiring.center.smul_comm_class_left"
] | The center of a semiring acts commutatively on that semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
center.smul_comm_class_right : smul_comm_class R (center R) R | subsemiring.center.smul_comm_class_right | instance | subring.center.smul_comm_class_right | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"smul_comm_class",
"subsemiring.center.smul_comm_class_right"
] | The center of a semiring acts commutatively on that semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.pos_subgroup (R : Type*) [linear_ordered_semiring R] :
subgroup Rˣ | { carrier := {x | (0 : R) < x},
inv_mem' := λ x, units.inv_pos.mpr,
..(pos_submonoid R).comap (units.coe_hom R)} | def | units.pos_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_semiring",
"pos_submonoid",
"subgroup",
"units.coe_hom"
] | The subgroup of positive units of a linear ordered semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.mem_pos_subgroup {R : Type*} [linear_ordered_semiring R]
(u : Rˣ) : u ∈ units.pos_subgroup R ↔ (0 : R) < u | iff.rfl | lemma | units.mem_pos_subgroup | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_semiring",
"units.pos_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_mul_action : mul_action M (subring R) | { smul := λ a S, S.map (mul_semiring_action.to_ring_hom _ _ a),
one_smul := λ S,
(congr_arg (λ f, S.map f) (ring_hom.ext $ by exact one_smul M)).trans S.map_id,
mul_smul := λ a₁ a₂ S,
(congr_arg (λ f, S.map f) (ring_hom.ext $ by exact mul_smul _ _)).trans (S.map_map _ _).symm } | def | subring.pointwise_mul_action | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"mul_action",
"mul_semiring_action.to_ring_hom",
"one_smul",
"ring_hom.ext",
"subring"
] | The action on a subring corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointwise_smul_def {a : M} (S : subring R) :
a • S = S.map (mul_semiring_action.to_ring_hom _ _ a) | rfl | lemma | subring.pointwise_smul_def | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"mul_semiring_action.to_ring_hom",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pointwise_smul (m : M) (S : subring R) : ↑(m • S) = m • (S : set R) | rfl | lemma | subring.coe_pointwise_smul | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_add_subgroup (m : M) (S : subring R) :
(m • S).to_add_subgroup = m • S.to_add_subgroup | rfl | lemma | subring.pointwise_smul_to_add_subgroup | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_subsemiring (m : M) (S : subring R) :
(m • S).to_subsemiring = m • S.to_subsemiring | rfl | lemma | subring.pointwise_smul_to_subsemiring | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul (m : M) (r : R) (S : subring R) : r ∈ S → m • r ∈ m • S | (set.smul_mem_smul_set : _ → _ ∈ m • (S : set R)) | lemma | subring.smul_mem_pointwise_smul | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"set.smul_mem_smul_set",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_pointwise_iff_exists (m : M) (r : R) (S : subring R) :
r ∈ m • S ↔ ∃ (s : R), s ∈ S ∧ m • s = r | (set.mem_smul_set : r ∈ m • (S : set R) ↔ _) | lemma | subring.mem_smul_pointwise_iff_exists | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"set.mem_smul_set",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_bot (a : M) : a • (⊥ : subring R) = ⊥ | map_bot _ | lemma | subring.smul_bot | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup (a : M) (S T : subring R) : a • (S ⊔ T) = a • S ⊔ a • T | map_sup _ _ _ | lemma | subring.smul_sup | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closure (a : M) (s : set R) : a • closure s = closure (a • s) | ring_hom.map_closure _ _ | lemma | subring.smul_closure | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"closure",
"ring_hom.map_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_central_scalar [mul_semiring_action Mᵐᵒᵖ R] [is_central_scalar M R] :
is_central_scalar M (subring R) | ⟨λ a S, congr_arg (λ f, S.map f) $ ring_hom.ext $ by exact op_smul_eq_smul _⟩ | instance | subring.pointwise_central_scalar | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"is_central_scalar",
"mul_semiring_action",
"ring_hom.ext",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul_iff {a : M} {S : subring R} {x : R} :
a • x ∈ a • S ↔ x ∈ S | smul_mem_smul_set_iff | lemma | subring.smul_mem_pointwise_smul_iff | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : subring R} {x : R} :
x ∈ a • S ↔ a⁻¹ • x ∈ S | mem_smul_set_iff_inv_smul_mem | lemma | subring.mem_pointwise_smul_iff_inv_smul_mem | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_pointwise_smul_iff {a : M} {S : subring R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S | mem_inv_smul_set_iff | lemma | subring.mem_inv_pointwise_smul_iff | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_pointwise_smul_iff {a : M} {S T : subring R} :
a • S ≤ a • T ↔ S ≤ T | set_smul_subset_set_smul_iff | lemma | subring.pointwise_smul_le_pointwise_smul_iff | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_subset_iff {a : M} {S T : subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T | set_smul_subset_iff | lemma | subring.pointwise_smul_subset_iff | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_pointwise_smul_iff {a : M} {S T : subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T | subset_set_smul_iff | lemma | subring.subset_pointwise_smul_iff | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subring R)
(x : R) : a • x ∈ a • S ↔ x ∈ S | smul_mem_smul_set_iff₀ ha (S : set R) x | lemma | subring.smul_mem_pointwise_smul_iff₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : subring R) (x : R) :
x ∈ a • S ↔ a⁻¹ • x ∈ S | mem_smul_set_iff_inv_smul_mem₀ ha (S : set R) x | lemma | subring.mem_pointwise_smul_iff_inv_smul_mem₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subring R) (x : R) :
x ∈ a⁻¹ • S ↔ a • x ∈ S | mem_inv_smul_set_iff₀ ha (S : set R) x | lemma | subring.mem_inv_pointwise_smul_iff₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} :
a • S ≤ a • T ↔ S ≤ T | set_smul_subset_set_smul_iff₀ ha | lemma | subring.pointwise_smul_le_pointwise_smul_iff₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T | set_smul_subset_iff₀ ha | lemma | subring.pointwise_smul_le_iff₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T | subset_set_smul_iff₀ ha | lemma | subring.le_pointwise_smul_iff₀ | ring_theory.subring | src/ring_theory/subring/pointwise.lean | [
"ring_theory.subring.basic",
"group_theory.subgroup.pointwise",
"ring_theory.subsemiring.pointwise",
"data.set.pointwise.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_submonoid_with_one_class (S : Type*) (R : Type*)
[add_monoid_with_one R] [set_like S R]
extends add_submonoid_class S R, one_mem_class S R : Prop | class | add_submonoid_with_one_class | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_monoid_with_one",
"add_submonoid_class",
"one_mem_class",
"set_like"
] | `add_submonoid_with_one_class S R` says `S` is a type of subsets `s ≤ R` that contain `0`, `1`,
and are closed under `(+)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_mem [add_submonoid_with_one_class S R] (n : ℕ) : (n : R) ∈ s | by induction n; simp [zero_mem, add_mem, one_mem, *] | lemma | nat_cast_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid_with_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_submonoid_with_one_class.to_add_monoid_with_one [add_submonoid_with_one_class S R] :
add_monoid_with_one s | { one := ⟨_, one_mem s⟩,
nat_cast := λ n, ⟨n, nat_cast_mem s n⟩,
nat_cast_zero := subtype.ext nat.cast_zero,
nat_cast_succ := λ n, subtype.ext (nat.cast_succ _),
.. add_submonoid_class.to_add_monoid s } | instance | add_submonoid_with_one_class.to_add_monoid_with_one | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_monoid_with_one",
"add_submonoid_with_one_class",
"nat.cast_succ",
"nat.cast_zero",
"nat_cast_mem",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring_class (S : Type*) (R : Type u) [non_assoc_semiring R] [set_like S R]
extends submonoid_class S R, add_submonoid_class S R : Prop | class | subsemiring_class | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid_class",
"non_assoc_semiring",
"set_like",
"submonoid_class"
] | `subsemiring_class S R` states that `S` is a type of subsets `s ⊆ R` that
are both a multiplicative and an additive submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring_class.add_submonoid_with_one_class (S : Type*) (R : Type u)
[non_assoc_semiring R] [set_like S R] [h : subsemiring_class S R] :
add_submonoid_with_one_class S R | { .. h } | instance | subsemiring_class.add_submonoid_with_one_class | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid_with_one_class",
"non_assoc_semiring",
"set_like",
"subsemiring_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_mem (n : ℕ) : (n : R) ∈ s | by { rw ← nsmul_one, exact nsmul_mem (one_mem _) _ } | lemma | coe_nat_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"nsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_assoc_semiring : non_assoc_semiring s | subtype.coe_injective.non_assoc_semiring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl)
(λ _, rfl) | instance | subsemiring_class.to_non_assoc_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"non_assoc_semiring"
] | A subsemiring of a `non_assoc_semiring` inherits a `non_assoc_semiring` structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nontrivial [nontrivial R] : nontrivial s | nontrivial_of_ne 0 1 $ λ H, zero_ne_one (congr_arg subtype.val H) | instance | subsemiring_class.nontrivial | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"nontrivial",
"nontrivial_of_ne",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors [no_zero_divisors R] : no_zero_divisors s | { eq_zero_or_eq_zero_of_mul_eq_zero := λ x y h,
or.cases_on (eq_zero_or_eq_zero_of_mul_eq_zero $ subtype.ext_iff.mp h)
(λ h, or.inl $ subtype.eq h) (λ h, or.inr $ subtype.eq h) } | instance | subsemiring_class.no_zero_divisors | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype : s →+* R | { to_fun := coe, .. submonoid_class.subtype s, .. add_submonoid_class.subtype s } | def | subsemiring_class.subtype | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"submonoid_class.subtype"
] | The natural ring hom from a subsemiring of semiring `R` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_semiring {R} [semiring R] [set_like S R] [subsemiring_class S R] : semiring s | subtype.coe_injective.semiring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl)
(λ _, rfl) | instance | subsemiring_class.to_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of a `semiring` is a `semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pow {R} [semiring R] [set_like S R] [subsemiring_class S R]
(x : s) (n : ℕ) :
((x^n : s) : R) = (x^n : R) | begin
induction n with n ih,
{ simp, },
{ simp [pow_succ, ih], },
end | lemma | subsemiring_class.coe_pow | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ih",
"pow_succ",
"semiring",
"set_like",
"subsemiring_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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