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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
coe_to_add_submonoid (s : non_unital_subsemiring R) :
(s.to_add_submonoid : set R) = s | rfl | lemma | non_unital_subsemiring.coe_to_add_submonoid | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : R) : x ∈ (⊤ : non_unital_subsemiring R) | set.mem_univ x | lemma | non_unital_subsemiring.mem_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : non_unital_subsemiring R) : set R) = set.univ | rfl | lemma | non_unital_subsemiring.coe_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : F) (s : non_unital_subsemiring S) : non_unital_subsemiring R | { carrier := f ⁻¹' s,
.. s.to_subsemigroup.comap (f : mul_hom R S), .. s.to_add_submonoid.comap (f : R →+ S) } | def | non_unital_subsemiring.comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"mul_hom",
"non_unital_subsemiring"
] | The preimage of a non-unital subsemiring along a non-unital ring homomorphism is a
non-unital subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (s : non_unital_subsemiring S) (f : F) :
(s.comap f : set R) = f ⁻¹' s | rfl | lemma | non_unital_subsemiring.coe_comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {s : non_unital_subsemiring S} {f : F} {x : R} :
x ∈ s.comap f ↔ f x ∈ s | iff.rfl | lemma | non_unital_subsemiring.mem_comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (s : non_unital_subsemiring T) (g : G) (f : F) :
((s.comap g : non_unital_subsemiring S).comap f : non_unital_subsemiring R) =
s.comap ((g : S →ₙ+* T).comp (f : R →ₙ+* S)) | rfl | lemma | non_unital_subsemiring.comap_comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : F) (s : non_unital_subsemiring R) : non_unital_subsemiring S | { carrier := f '' s,
.. s.to_subsemigroup.map (f : R →ₙ* S), .. s.to_add_submonoid.map (f : R →+ S) } | def | non_unital_subsemiring.map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | The image of a non-unital subsemiring along a ring homomorphism is a non-unital subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (f : F) (s : non_unital_subsemiring R) : (s.map f : set S) = f '' s | rfl | lemma | non_unital_subsemiring.coe_map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {f : F} {s : non_unital_subsemiring R} {y : S} :
y ∈ s.map f ↔ ∃ x ∈ s, f x = y | set.mem_image_iff_bex | lemma | non_unital_subsemiring.mem_map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"mem_map",
"non_unital_subsemiring",
"set.mem_image_iff_bex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : s.map (non_unital_ring_hom.id R) = s | set_like.coe_injective $ set.image_id _ | lemma | non_unital_subsemiring.map_id | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"map_id",
"non_unital_ring_hom.id",
"set.image_id",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (g : G) (f : F) :
(s.map (f : R →ₙ+* S)).map (g : S →ₙ+* T) = s.map ((g : S →ₙ+* T).comp (f : R →ₙ+* S)) | set_like.coe_injective $ set.image_image _ _ _ | lemma | non_unital_subsemiring.map_map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"set.image_image",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap {f : F} {s : non_unital_subsemiring R} {t : non_unital_subsemiring S} :
s.map f ≤ t ↔ s ≤ t.comap f | set.image_subset_iff | lemma | non_unital_subsemiring.map_le_iff_le_comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap (f : F) :
@galois_connection (non_unital_subsemiring R) (non_unital_subsemiring S) _ _ (map f) (comap f) | λ S T, map_le_iff_le_comap | lemma | non_unital_subsemiring.gc_map_comap | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"galois_connection",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_map_of_injective
(f : F) (hf : function.injective (f : R → S)) : s ≃+* s.map f | { map_mul' := λ _ _, subtype.ext (map_mul f _ _),
map_add' := λ _ _, subtype.ext (map_add f _ _),
..equiv.set.image f s hf } | def | non_unital_subsemiring.equiv_map_of_injective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"equiv.set.image",
"map_mul",
"subtype.ext"
] | A non-unital subsemiring is isomorphic to its image under an injective function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_equiv_map_of_injective_apply
(f : F) (hf : function.injective f) (x : s) :
(equiv_map_of_injective s f hf x : S) = f x | rfl | lemma | non_unital_subsemiring.coe_equiv_map_of_injective_apply | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange : non_unital_subsemiring S | ((⊤ : non_unital_subsemiring R).map (f : R →ₙ+* S)).copy (set.range f) set.image_univ.symm | def | non_unital_ring_hom.srange | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.range"
] | The range of a non-unital ring homomorphism is a non-unital subsemiring.
See note [range copy pattern]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_srange : (@srange R S _ _ _ _ f : set S) = set.range f | rfl | lemma | non_unital_ring_hom.coe_srange | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_srange {f : F} {y : S} : y ∈ (@srange R S _ _ _ _ f) ↔ ∃ x, f x = y | iff.rfl | lemma | non_unital_ring_hom.mem_srange | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_eq_map : @srange R S _ _ _ _ f = (⊤ : non_unital_subsemiring R).map f | by { ext, simp } | lemma | non_unital_ring_hom.srange_eq_map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_srange_self (f : F) (x : R) : f x ∈ @srange R S _ _ _ _ f | mem_srange.mpr ⟨x, rfl⟩ | lemma | non_unital_ring_hom.mem_srange_self | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_srange (g : S →ₙ+* T) (f : R →ₙ+* S) : map g (srange f) = srange (g.comp f) | by simpa only [srange_eq_map] using (⊤ : non_unital_subsemiring R).map_map g f | lemma | non_unital_ring_hom.map_srange | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_srange [finite R] (f : F) : finite (srange f : non_unital_subsemiring S) | (set.finite_range f).to_subtype | instance | non_unital_ring_hom.finite_srange | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"finite",
"non_unital_subsemiring",
"set.finite_range"
] | The range of a morphism of non-unital semirings is finite if the domain is a finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_bot : ((⊥ : non_unital_subsemiring R) : set R) = {0} | rfl | lemma | non_unital_subsemiring.coe_bot | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : R} : x ∈ (⊥ : non_unital_subsemiring R) ↔ x = 0 | set.mem_singleton_iff | lemma | non_unital_subsemiring.mem_bot | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (p p' : non_unital_subsemiring R) :
((p ⊓ p' : non_unital_subsemiring R) : set R) = p ∩ p' | rfl | lemma | non_unital_subsemiring.coe_inf | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {p p' : non_unital_subsemiring R} {x : R} :x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' | iff.rfl | lemma | non_unital_subsemiring.mem_inf | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (non_unital_subsemiring R)) :
((Inf S : non_unital_subsemiring R) : set R) = ⋂ s ∈ S, ↑s | rfl | lemma | non_unital_subsemiring.coe_Inf | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (non_unital_subsemiring R)} {x : R} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | set.mem_Inter₂ | lemma | non_unital_subsemiring.mem_Inf | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.mem_Inter₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_subsemigroup (s : set (non_unital_subsemiring R)) :
(Inf s).to_subsemigroup = ⨅ t ∈ s, non_unital_subsemiring.to_subsemigroup t | mk'_to_subsemigroup _ _ | lemma | non_unital_subsemiring.Inf_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_add_submonoid (s : set (non_unital_subsemiring R)) :
(Inf s).to_add_submonoid = ⨅ t ∈ s, non_unital_subsemiring.to_add_submonoid t | mk'_to_add_submonoid _ _ | lemma | non_unital_subsemiring.Inf_to_add_submonoid | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff' (A : non_unital_subsemiring R) : A = ⊤ ↔ ∀ x : R, x ∈ A | eq_top_iff.trans ⟨λ h m, h $ mem_top m, λ h m _, h m⟩ | lemma | non_unital_subsemiring.eq_top_iff' | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center (R) [non_unital_semiring R] : non_unital_subsemiring R | { carrier := set.center R,
zero_mem' := set.zero_mem_center R,
add_mem' := λ a b, set.add_mem_center,
.. subsemigroup.center R } | def | non_unital_subsemiring.center | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"non_unital_subsemiring",
"set.add_mem_center",
"set.center",
"set.zero_mem_center",
"subsemigroup.center"
] | The center of a semiring `R` is the set of elements that commute with everything in `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_center (R) [non_unital_semiring R] : ↑(center R) = set.center R | rfl | lemma | non_unital_subsemiring.coe_center | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"set.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_to_subsemigroup (R) [non_unital_semiring R] :
(center R).to_subsemigroup = subsemigroup.center R | rfl | lemma | non_unital_subsemiring.center_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"subsemigroup.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_center_iff {R} [non_unital_semiring R] {z : R} : z ∈ center R ↔ ∀ g, g * z = z * g | iff.rfl | lemma | non_unital_subsemiring.mem_center_iff | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_mem_center {R} [non_unital_semiring R] [decidable_eq R] [fintype R] :
decidable_pred (∈ center R) | λ _, decidable_of_iff' _ mem_center_iff | instance | non_unital_subsemiring.decidable_mem_center | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"decidable_of_iff'",
"fintype",
"non_unital_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_eq_top (R) [non_unital_comm_semiring R] : center R = ⊤ | set_like.coe_injective (set.center_eq_univ R) | lemma | non_unital_subsemiring.center_eq_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_comm_semiring",
"set.center_eq_univ",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer {R} [non_unital_semiring R] (s : set R) : non_unital_subsemiring R | { carrier := s.centralizer,
zero_mem' := set.zero_mem_centralizer _,
add_mem' := λ x y hx hy, set.add_mem_centralizer hx hy,
..subsemigroup.centralizer s } | def | non_unital_subsemiring.centralizer | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"non_unital_subsemiring",
"set.add_mem_centralizer",
"set.zero_mem_centralizer",
"subsemigroup.centralizer"
] | The centralizer of a set as non-unital subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_centralizer {R} [non_unital_semiring R] (s : set R) :
(centralizer s : set R) = s.centralizer | rfl | lemma | non_unital_subsemiring.coe_centralizer | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_to_subsemigroup {R} [non_unital_semiring R] (s : set R) :
(centralizer s).to_subsemigroup = subsemigroup.centralizer s | rfl | lemma | non_unital_subsemiring.centralizer_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"subsemigroup.centralizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_centralizer_iff {R} [non_unital_semiring R] {s : set R} {z : R} :
z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g | iff.rfl | lemma | non_unital_subsemiring.mem_centralizer_iff | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_le_centralizer {R} [non_unital_semiring R] (s) : center R ≤ centralizer s | s.center_subset_centralizer | lemma | non_unital_subsemiring.center_le_centralizer | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_le {R} [non_unital_semiring R] (s t : set R) (h : s ⊆ t) :
centralizer t ≤ centralizer s | set.centralizer_subset h | lemma | non_unital_subsemiring.centralizer_le | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"set.centralizer_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_eq_top_iff_subset {R} [non_unital_semiring R] {s : set R} :
centralizer s = ⊤ ↔ s ⊆ center R | set_like.ext'_iff.trans set.centralizer_eq_top_iff_subset | lemma | non_unital_subsemiring.centralizer_eq_top_iff_subset | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"set.centralizer_eq_top_iff_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_univ {R} [non_unital_semiring R] : centralizer set.univ = center R | set_like.ext' (set.centralizer_univ R) | lemma | non_unital_subsemiring.centralizer_univ | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"set.centralizer_univ",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (s : set R) : non_unital_subsemiring R | Inf {S | s ⊆ S} | def | non_unital_subsemiring.closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"non_unital_subsemiring"
] | The `non_unital_subsemiring` generated by a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure {x : R} {s : set R} : x ∈ closure s ↔ ∀ S :
non_unital_subsemiring R, s ⊆ S → x ∈ S | mem_Inf | lemma | non_unital_subsemiring.mem_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_closure {s : set R} : s ⊆ closure s | λ x hx, mem_closure.2 $ λ S hS, hS hx | lemma | non_unital_subsemiring.subset_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"subset_closure"
] | The non-unital subsemiring generated by a set includes the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_of_not_mem_closure {s : set R} {P : R} (hP : P ∉ closure s) : P ∉ s | λ h, hP (subset_closure h) | lemma | non_unital_subsemiring.not_mem_of_not_mem_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"not_mem_of_not_mem_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_le {s : set R} {t : non_unital_subsemiring R} : closure s ≤ t ↔ s ⊆ t | ⟨set.subset.trans subset_closure, λ h, Inf_le h⟩ | lemma | non_unital_subsemiring.closure_le | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"Inf_le",
"closure",
"non_unital_subsemiring",
"subset_closure"
] | A non-unital subsemiring `S` includes `closure s` if and only if it includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_mono ⦃s t : set R⦄ (h : s ⊆ t) : closure s ≤ closure t | closure_le.2 $ set.subset.trans h subset_closure | lemma | non_unital_subsemiring.closure_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_mono",
"set.subset.trans",
"subset_closure"
] | Subsemiring closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_of_le {s : set R} {t : non_unital_subsemiring R} (h₁ : s ⊆ t)
(h₂ : t ≤ closure s) : closure s = t | le_antisymm (closure_le.2 h₁) h₂ | lemma | non_unital_subsemiring.closure_eq_of_le | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_equiv {f : R ≃+* S} {K : non_unital_subsemiring R} {x : S} :
x ∈ K.map (f : R →ₙ+* S) ↔ f.symm x ∈ K | @set.mem_image_equiv _ _ ↑K f.to_equiv x | lemma | non_unital_subsemiring.mem_map_equiv | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.mem_image_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_eq_comap_symm (f : R ≃+* S) (K : non_unital_subsemiring R) :
K.map (f : R →ₙ+* S) = K.comap f.symm | set_like.coe_injective (f.to_equiv.image_eq_preimage K) | lemma | non_unital_subsemiring.map_equiv_eq_comap_symm | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_equiv_eq_map_symm (f : R ≃+* S) (K : non_unital_subsemiring S) :
K.comap (f : R →ₙ+* S) = K.map f.symm | (map_equiv_eq_comap_symm f.symm K).symm | lemma | non_unital_subsemiring.comap_equiv_eq_map_symm | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_subsemiring_closure (M : subsemigroup R) : non_unital_subsemiring R | { mul_mem' := λ x y, mul_mem_class.mul_mem_add_closure,
..add_submonoid.closure (M : set R)} | def | subsemigroup.non_unital_subsemiring_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"mul_mem_class.mul_mem_add_closure",
"non_unital_subsemiring",
"subsemigroup"
] | The additive closure of a non-unital subsemigroup is a non-unital subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_subsemiring_closure_coe :
(M.non_unital_subsemiring_closure : set R) = add_submonoid.closure (M : set R) | rfl | lemma | subsemigroup.non_unital_subsemiring_closure_coe | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_subsemiring_closure_to_add_submonoid :
M.non_unital_subsemiring_closure.to_add_submonoid = add_submonoid.closure (M : set R) | rfl | lemma | subsemigroup.non_unital_subsemiring_closure_to_add_submonoid | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_subsemiring_closure_eq_closure :
M.non_unital_subsemiring_closure = non_unital_subsemiring.closure (M : set R) | begin
ext,
refine ⟨λ hx, _,
λ hx, (non_unital_subsemiring.mem_closure.mp hx) M.non_unital_subsemiring_closure (λ s sM, _)⟩;
rintros - ⟨H1, rfl⟩;
rintros - ⟨H2, rfl⟩,
{ exact add_submonoid.mem_closure.mp hx H1.to_add_submonoid H2 },
{ exact H2 sM }
end | lemma | subsemigroup.non_unital_subsemiring_closure_eq_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring.closure"
] | The `non_unital_subsemiring` generated by a multiplicative subsemigroup coincides with the
`non_unital_subsemiring.closure` of the subsemigroup itself . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_subsemigroup_closure (s : set R) : closure ↑(subsemigroup.closure s) = closure s | le_antisymm
(closure_le.mpr (λ y hy, (subsemigroup.mem_closure.mp hy)
(closure s).to_subsemigroup subset_closure))
(closure_mono (subsemigroup.subset_closure)) | lemma | non_unital_subsemiring.closure_subsemigroup_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_mono",
"subsemigroup.closure",
"subsemigroup.subset_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_closure_eq (s : set R) :
(closure s : set R) = add_submonoid.closure (subsemigroup.closure s : set R) | by simp [← subsemigroup.non_unital_subsemiring_closure_to_add_submonoid,
subsemigroup.non_unital_subsemiring_closure_eq_closure] | lemma | non_unital_subsemiring.coe_closure_eq | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"subsemigroup.closure",
"subsemigroup.non_unital_subsemiring_closure_eq_closure",
"subsemigroup.non_unital_subsemiring_closure_to_add_submonoid"
] | The elements of the non-unital subsemiring closure of `M` are exactly the elements of the
additive closure of a multiplicative subsemigroup `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff {s : set R} {x} :
x ∈ closure s ↔ x ∈ add_submonoid.closure (subsemigroup.closure s : set R) | set.ext_iff.mp (coe_closure_eq s) x | lemma | non_unital_subsemiring.mem_closure_iff | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"mem_closure_iff",
"subsemigroup.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_add_submonoid_closure {s : set R} : closure ↑(add_submonoid.closure s) = closure s | begin
ext x,
refine ⟨λ hx, _, λ hx, closure_mono add_submonoid.subset_closure hx⟩,
rintros - ⟨H, rfl⟩,
rintros - ⟨J, rfl⟩,
refine (add_submonoid.mem_closure.mp (mem_closure_iff.mp hx)) H.to_add_submonoid (λ y hy, _),
refine (subsemigroup.mem_closure.mp hy) H.to_subsemigroup (λ z hz, _),
exact (add_submono... | lemma | non_unital_subsemiring.closure_add_submonoid_closure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_mono"
] | 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)
(Hadd : ∀ x y, p x → p y → p (x + y)) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x | (@closure_le _ _ _ ⟨p, Hadd, H0, Hmul⟩).2 Hs h | lemma | non_unital_subsemiring.closure_induction | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_induction₂ {s : set R} {p : R → R → Prop} {x} {y : R} (hx : x ∈ closure s)
(hy : y ∈ closure s)
(Hs : ∀ (x ∈ s) (y ∈ s), p x y)
(H0_left : ∀ x, p 0 x)
(H0_right : ∀ x, p x 0)
(Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)
(Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))
(Hmul_le... | closure_induction hx
(λ x₁ x₁s, closure_induction hy (Hs x₁ x₁s) (H0_right x₁) (Hadd_right x₁) (Hmul_right x₁))
(H0_left y) (λ z z', Hadd_left z z' y) (λ z z', Hmul_left z z' y) | lemma | non_unital_subsemiring.closure_induction₂ | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure"
] | An induction principle for closure membership for predicates with two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gi : galois_insertion (@closure R _) coe | { choice := λ s _, closure s,
gc := λ s t, closure_le,
le_l_u := λ s, subset_closure,
choice_eq := λ s h, rfl } | def | non_unital_subsemiring.gi | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"galois_insertion",
"subset_closure"
] | `closure` forms a Galois insertion with the coercion to set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq (s : non_unital_subsemiring R) : closure (s : set R) = s | (non_unital_subsemiring.gi R).l_u_eq s | lemma | non_unital_subsemiring.closure_eq | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"non_unital_subsemiring",
"non_unital_subsemiring.gi"
] | Closure of a non-unital subsemiring `S` equals `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_empty : closure (∅ : set R) = ⊥ | (non_unital_subsemiring.gi R).gc.l_bot | lemma | non_unital_subsemiring.closure_empty | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_empty",
"non_unital_subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_univ : closure (set.univ : set R) = ⊤ | @coe_top R _ ▸ closure_eq ⊤ | lemma | non_unital_subsemiring.closure_univ | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_union (s t : set R) : closure (s ∪ t) = closure s ⊔ closure t | (non_unital_subsemiring.gi R).gc.l_sup | lemma | non_unital_subsemiring.closure_union | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_union",
"non_unital_subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Union {ι} (s : ι → set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) | (non_unital_subsemiring.gi R).gc.l_supr | lemma | non_unital_subsemiring.closure_Union | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_Union",
"non_unital_subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_sUnion (s : set (set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t | (non_unital_subsemiring.gi R).gc.l_Sup | lemma | non_unital_subsemiring.closure_sUnion | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"non_unital_subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup (s t : non_unital_subsemiring R) (f : F) :
(map f (s ⊔ t) : non_unital_subsemiring S) = map f s ⊔ map f t | @galois_connection.l_sup _ _ s t _ _ _ _ (gc_map_comap f) | lemma | non_unital_subsemiring.map_sup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"galois_connection.l_sup",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr {ι : Sort*} (f : F) (s : ι → non_unital_subsemiring R) :
(map f (supr s) : non_unital_subsemiring S) = ⨆ i, map f (s i) | @galois_connection.l_supr _ _ _ _ _ _ _ (gc_map_comap f) s | lemma | non_unital_subsemiring.map_supr | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"galois_connection.l_supr",
"map_supr",
"non_unital_subsemiring",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf (s t : non_unital_subsemiring S) (f : F) :
(comap f (s ⊓ t) : non_unital_subsemiring R) = comap f s ⊓ comap f t | @galois_connection.u_inf _ _ s t _ _ _ _ (gc_map_comap f) | lemma | non_unital_subsemiring.comap_inf | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"galois_connection.u_inf",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_infi {ι : Sort*} (f : F) (s : ι → non_unital_subsemiring S) :
(comap f (infi s) : non_unital_subsemiring R) = ⨅ i, comap f (s i) | @galois_connection.u_infi _ _ _ _ _ _ _ (gc_map_comap f) s | lemma | non_unital_subsemiring.comap_infi | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"galois_connection.u_infi",
"infi",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot (f : F) :
map f (⊥ : non_unital_subsemiring R) = (⊥ : non_unital_subsemiring S) | (gc_map_comap f).l_bot | lemma | non_unital_subsemiring.map_bot | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top (f : F) :
comap f (⊤ : non_unital_subsemiring S) = (⊤ : non_unital_subsemiring R) | (gc_map_comap f).u_top | lemma | non_unital_subsemiring.comap_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (s : non_unital_subsemiring R) (t : non_unital_subsemiring S) :
non_unital_subsemiring (R × S) | { carrier := (s : set R) ×ˢ (t : set S),
.. s.to_subsemigroup.prod t.to_subsemigroup, .. s.to_add_submonoid.prod t.to_add_submonoid} | def | non_unital_subsemiring.prod | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | Given `non_unital_subsemiring`s `s`, `t` of semirings `R`, `S` respectively, `s.prod t` is
`s × t` as a non-unital subsemiring of `R × S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (s : non_unital_subsemiring R) (t : non_unital_subsemiring S) :
(s.prod t : set (R × S)) = (s : set R) ×ˢ (t : set S) | rfl | lemma | non_unital_subsemiring.coe_prod | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod {s : non_unital_subsemiring R} {t : non_unital_subsemiring S} {p : R × S} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t | iff.rfl | lemma | non_unital_subsemiring.mem_prod | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono ⦃s₁ s₂ : non_unital_subsemiring R⦄ (hs : s₁ ≤ s₂)
⦃t₁ t₂ : non_unital_subsemiring S⦄ (ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ | set.prod_mono hs ht | lemma | non_unital_subsemiring.prod_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_right (s : non_unital_subsemiring R) :
monotone (λ t : non_unital_subsemiring S, s.prod t) | prod_mono (le_refl s) | lemma | non_unital_subsemiring.prod_mono_right | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"monotone",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_left (t : non_unital_subsemiring S) :
monotone (λ s : non_unital_subsemiring R, s.prod t) | λ s₁ s₂ hs, prod_mono hs (le_refl t) | lemma | non_unital_subsemiring.prod_mono_left | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"monotone",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top (s : non_unital_subsemiring R) :
s.prod (⊤ : non_unital_subsemiring S) = s.comap (non_unital_ring_hom.fst R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_fst] | lemma | non_unital_subsemiring.prod_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"monoid_hom.coe_fst",
"non_unital_ring_hom.fst",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod (s : non_unital_subsemiring S) :
(⊤ : non_unital_subsemiring R).prod s = s.comap (non_unital_ring_hom.snd R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_snd] | lemma | non_unital_subsemiring.top_prod | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"monoid_hom.coe_snd",
"non_unital_ring_hom.snd",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod_top : (⊤ : non_unital_subsemiring R).prod (⊤ : non_unital_subsemiring S) = ⊤ | (top_prod _).trans $ comap_top _ | lemma | non_unital_subsemiring.top_prod_top | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv (s : non_unital_subsemiring R) (t : non_unital_subsemiring S) : s.prod t ≃+* s × t | { map_mul' := λ x y, rfl, map_add' := λ x y, rfl, .. equiv.set.prod ↑s ↑t } | def | non_unital_subsemiring.prod_equiv | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"equiv.set.prod",
"non_unital_subsemiring"
] | Product of non-unital subsemirings is isomorphic to their product as semigroups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → non_unital_subsemiring 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 : non_unital_subsemiring R := non_unital_subsemiring.mk' (⋃ i, (S i : set R))
(⨆ i, (S i).to_subsemigroup) (subsemigroup.coe_supr_of_directed $ hS.mono_comp _ (λ _ _, id))
(⨆ i, (S i).to_add_submonoid) (add_submonoid.coe_supr_of_direct... | lemma | non_unital_subsemiring.mem_supr_of_directed | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"directed",
"le_supr",
"non_unital_subsemiring",
"non_unital_subsemiring.mk'",
"subsemigroup.coe_supr_of_directed",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr_of_directed {ι} [hι : nonempty ι] {S : ι → non_unital_subsemiring R}
(hS : directed (≤) S) : ((⨆ i, S i : non_unital_subsemiring R) : set R) = ⋃ i, ↑(S i) | set.ext $ λ x, by simp [mem_supr_of_directed hS] | lemma | non_unital_subsemiring.coe_supr_of_directed | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"directed",
"non_unital_subsemiring",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup_of_directed_on {S : set (non_unital_subsemiring 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 | non_unital_subsemiring.mem_Sup_of_directed_on | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"Sup_eq_supr'",
"directed_on",
"non_unital_subsemiring",
"set_coe.exists",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup_of_directed_on {S : set (non_unital_subsemiring 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 | non_unital_subsemiring.coe_Sup_of_directed_on | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"directed_on",
"non_unital_subsemiring",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (f : F) (s : non_unital_subsemiring S) (h : ∀ x, f x ∈ s) : R →ₙ+* s | { to_fun := λ n, ⟨f n, h n⟩,
.. (f : R →ₙ* S).cod_restrict s.to_subsemigroup h,
.. (f : R →+ S).cod_restrict s.to_add_submonoid h } | def | non_unital_ring_hom.cod_restrict | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | Restriction of a non-unital ring homomorphism to a non-unital subsemiring of the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
srange_restrict (f : F) : R →ₙ+* (srange f : non_unital_subsemiring S) | cod_restrict f (srange f) (mem_srange_self f) | def | non_unital_ring_hom.srange_restrict | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | Restriction of a non-unital ring homomorphism to its range interpreted as a
non-unital subsemiring.
This is the bundled version of `set.range_factorization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_srange_restrict (f : F) (x : R) :
(srange_restrict f x : S) = f x | rfl | lemma | non_unital_ring_hom.coe_srange_restrict | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_restrict_surjective (f : F) :
function.surjective (srange_restrict f : R → (srange f : non_unital_subsemiring S)) | λ ⟨y, hy⟩, let ⟨x, hx⟩ := mem_srange.mp hy in ⟨x, subtype.ext hx⟩ | lemma | non_unital_ring_hom.srange_restrict_surjective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_top_iff_surjective {f : F} :
srange f = (⊤ : non_unital_subsemiring S) ↔ function.surjective (f : R → S) | set_like.ext'_iff.trans $ iff.trans (by rw [coe_srange, coe_top]) set.range_iff_surjective | lemma | non_unital_ring_hom.srange_top_iff_surjective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set.range_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_top_of_surjective (f : F) (hf : function.surjective (f : R → S)) :
srange f = (⊤ : non_unital_subsemiring S) | srange_top_iff_surjective.2 hf | lemma | non_unital_ring_hom.srange_top_of_surjective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | The range of a surjective non-unital ring homomorphism is the whole of the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_slocus (f g : F) : non_unital_subsemiring R | { carrier := {x | f x = g x},
.. (f : R →ₙ* S).eq_mlocus (g : R →ₙ* S),
.. (f : R →+ S).eq_mlocus g } | def | non_unital_ring_hom.eq_slocus | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | The non-unital subsemiring of elements `x : R` such that `f x = g x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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