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
value |
|---|---|---|---|---|---|---|---|---|---|---|
subsemiring_closure_eq_closure : M.subsemiring_closure = subsemiring.closure (M : set R) | begin
ext,
refine ⟨λ hx, _, λ hx, (subsemiring.mem_closure.mp hx) M.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 | submonoid.subsemiring_closure_eq_closure | 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"
] | [
"subsemiring.closure"
] | The `subsemiring` generated by a multiplicative submonoid coincides with the
`subsemiring.closure` of the submonoid itself . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_submonoid_closure (s : set R) : closure ↑(submonoid.closure s) = closure s | le_antisymm
(closure_le.mpr (λ y hy, (submonoid.mem_closure.mp hy) (closure s).to_submonoid subset_closure))
(closure_mono (submonoid.subset_closure)) | lemma | subsemiring.closure_submonoid_closure | 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"
] | [
"closure",
"closure_mono",
"submonoid.closure",
"submonoid.subset_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_closure_eq (s : set R) :
(closure s : set R) = add_submonoid.closure (submonoid.closure s : set R) | by simp [← submonoid.subsemiring_closure_to_add_submonoid, submonoid.subsemiring_closure_eq_closure] | lemma | subsemiring.coe_closure_eq | 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"
] | [
"closure",
"submonoid.closure",
"submonoid.subsemiring_closure_eq_closure",
"submonoid.subsemiring_closure_to_add_submonoid"
] | The elements of the subsemiring closure of `M` are exactly the elements of the additive closure
of a multiplicative submonoid `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff {s : set R} {x} :
x ∈ closure s ↔ x ∈ add_submonoid.closure (submonoid.closure s : set R) | set.ext_iff.mp (coe_closure_eq s) x | lemma | subsemiring.mem_closure_iff | 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"
] | [
"closure",
"mem_closure_iff",
"submonoid.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 (submonoid.mem_closure.mp hy) H.to_submonoid (λ z hz, _),
exact (add_submonoid.mem... | lemma | subsemiring.closure_add_submonoid_closure | 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"
] | [
"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) (H1 : p 1)
(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, Hmul, H1, Hadd, H0⟩).2 Hs h | lemma | subsemiring.closure_induction | 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"
] | [
"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)
(H1_left : ∀ x, p 1 x)
(H1_right : ∀ x, p x 1)
(Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)
(Hadd_right : ∀ x... | closure_induction hx
(λ x₁ x₁s, closure_induction hy (Hs x₁ x₁s) (H0_right x₁) (H1_right x₁) (Hadd_right x₁)
(Hmul_right x₁))
(H0_left y) (H1_left y) (λ z z', Hadd_left z z' y) (λ z z', Hmul_left z z' y) | lemma | subsemiring.closure_induction₂ | 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"
] | [
"closure"
] | An induction principle for closure membership for predicates with two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff_exists_list {R} [semiring R] {s : set R} {x} : x ∈ closure s ↔
∃ L : list (list R), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (L.map list.prod).sum = x | ⟨λ hx, add_submonoid.closure_induction (mem_closure_iff.1 hx)
(λ x hx, suffices ∃ t : list R, (∀ y ∈ t, y ∈ s) ∧ t.prod = x,
from let ⟨t, ht1, ht2⟩ := this in ⟨[t], list.forall_mem_singleton.2 ht1,
by rw [list.map_singleton, list.sum_singleton, ht2]⟩,
submonoid.closure_induction hx
(λ x hx, ⟨[x], ... | lemma | subsemiring.mem_closure_iff_exists_list | 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"
] | [
"closure",
"list.forall_mem_nil",
"list.prod",
"list.prod_append",
"list_prod_mem",
"one_mul",
"semiring",
"submonoid.closure_induction",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq (s : subsemiring R) : closure (s : set R) = s | (subsemiring.gi R).l_u_eq s | lemma | subsemiring.closure_eq | 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"
] | [
"closure",
"subsemiring",
"subsemiring.gi"
] | Closure of a subsemiring `S` equals `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_empty : closure (∅ : set R) = ⊥ | (subsemiring.gi R).gc.l_bot | lemma | subsemiring.closure_empty | 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"
] | [
"closure",
"closure_empty",
"subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_union (s t : set R) : closure (s ∪ t) = closure s ⊔ closure t | (subsemiring.gi R).gc.l_sup | lemma | subsemiring.closure_union | 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"
] | [
"closure",
"closure_union",
"subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Union {ι} (s : ι → set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) | (subsemiring.gi R).gc.l_supr | lemma | subsemiring.closure_Union | 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"
] | [
"closure",
"closure_Union",
"subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_sUnion (s : set (set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t | (subsemiring.gi R).gc.l_Sup | lemma | subsemiring.closure_sUnion | 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"
] | [
"closure",
"subsemiring.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup (s t : subsemiring R) (f : R →+* S) : (s ⊔ t).map f = s.map f ⊔ t.map f | (gc_map_comap f).l_sup | lemma | subsemiring.map_sup | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr {ι : Sort*} (f : R →+* S) (s : ι → subsemiring R) :
(supr s).map f = ⨆ i, (s i).map f | (gc_map_comap f).l_supr | lemma | subsemiring.map_supr | 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"
] | [
"map_supr",
"subsemiring",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf (s t : subsemiring S) (f : R →+* S) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f | (gc_map_comap f).u_inf | lemma | subsemiring.comap_inf | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_infi {ι : Sort*} (f : R →+* S) (s : ι → subsemiring S) :
(infi s).comap f = ⨅ i, (s i).comap f | (gc_map_comap f).u_infi | lemma | subsemiring.comap_infi | 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"
] | [
"infi",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot (f : R →+* S) : (⊥ : subsemiring R).map f = ⊥ | (gc_map_comap f).l_bot | lemma | subsemiring.map_bot | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top (f : R →+* S) : (⊤ : subsemiring S).comap f = ⊤ | (gc_map_comap f).u_top | lemma | subsemiring.comap_top | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (s : subsemiring R) (t : subsemiring S) : subsemiring (R × S) | { carrier := s ×ˢ t,
.. s.to_submonoid.prod t.to_submonoid, .. s.to_add_submonoid.prod t.to_add_submonoid} | def | subsemiring.prod | 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"
] | [
"subsemiring"
] | Given `subsemiring`s `s`, `t` of semirings `R`, `S` respectively, `s.prod t` is `s × t`
as a subsemiring of `R × S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (s : subsemiring R) (t : subsemiring S) : (s.prod t : set (R × S)) = s ×ˢ t | rfl | lemma | subsemiring.coe_prod | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod {s : subsemiring R} {t : subsemiring S} {p : R × S} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t | iff.rfl | lemma | subsemiring.mem_prod | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono ⦃s₁ s₂ : subsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : subsemiring S⦄
(ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ | set.prod_mono hs ht | lemma | subsemiring.prod_mono | 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"
] | [
"set.prod_mono",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_right (s : subsemiring R) : monotone (λ t : subsemiring S, s.prod t) | prod_mono (le_refl s) | lemma | subsemiring.prod_mono_right | 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"
] | [
"monotone",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_left (t : subsemiring S) : monotone (λ s : subsemiring R, s.prod t) | λ s₁ s₂ hs, prod_mono hs (le_refl t) | lemma | subsemiring.prod_mono_left | 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"
] | [
"monotone",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top (s : subsemiring R) :
s.prod (⊤ : subsemiring S) = s.comap (ring_hom.fst R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_fst] | lemma | subsemiring.prod_top | 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"
] | [
"monoid_hom.coe_fst",
"ring_hom.fst",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod (s : subsemiring S) :
(⊤ : subsemiring R).prod s = s.comap (ring_hom.snd R S) | ext $ λ x, by simp [mem_prod, monoid_hom.coe_snd] | lemma | subsemiring.top_prod | 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"
] | [
"monoid_hom.coe_snd",
"ring_hom.snd",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod_top : (⊤ : subsemiring R).prod (⊤ : subsemiring S) = ⊤ | (top_prod _).trans $ comap_top _ | lemma | subsemiring.top_prod_top | 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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv (s : subsemiring R) (t : 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 | subsemiring.prod_equiv | 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"
] | [
"equiv.set.prod",
"subsemiring"
] | Product of subsemirings is isomorphic to their product as monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → 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 : subsemiring R := subsemiring.mk' (⋃ i, (S i : set R))
(⨆ i, (S i).to_submonoid) (submonoid.coe_supr_of_directed $ hS.mono_comp _ (λ _ _, id))
(⨆ i, (S i).to_add_submonoid) (add_submonoid.coe_supr_of_directed $ hS.mono_comp _ (λ _ _, ... | lemma | subsemiring.mem_supr_of_directed | 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"
] | [
"directed",
"le_supr",
"submonoid.coe_supr_of_directed",
"subsemiring",
"subsemiring.mk'",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr_of_directed {ι} [hι : nonempty ι] {S : ι → subsemiring R} (hS : directed (≤) S) :
((⨆ i, S i : subsemiring R) : set R) = ⋃ i, ↑(S i) | set.ext $ λ x, by simp [mem_supr_of_directed hS] | lemma | subsemiring.coe_supr_of_directed | 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"
] | [
"directed",
"set.ext",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup_of_directed_on {S : set (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 | subsemiring.mem_Sup_of_directed_on | 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"
] | [
"Sup_eq_supr'",
"directed_on",
"set_coe.exists",
"subsemiring",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup_of_directed_on {S : set (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 | subsemiring.coe_Sup_of_directed_on | 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"
] | [
"directed_on",
"set.ext",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dom_restrict (f : R →+* S) (s : σR) : s →+* S | f.comp $ subsemiring_class.subtype s | def | ring_hom.dom_restrict | 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"
] | [
"subsemiring_class.subtype"
] | Restriction of a ring homomorphism to a subsemiring of the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_apply (f : R →+* S) {s : σR} (x : s) : f.dom_restrict s x = f x | rfl | lemma | ring_hom.restrict_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (f : R →+* S) (s : σS) (h : ∀ x, f x ∈ s) : R →+* s | { to_fun := λ n, ⟨f n, h n⟩,
.. (f : R →* S).cod_restrict s h,
.. (f : R →+ S).cod_restrict s h } | def | ring_hom.cod_restrict | 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"
] | [] | Restriction of a ring homomorphism to a subsemiring of the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) :
s' →+* s | (f.dom_restrict s').cod_restrict s (λ x, h x x.2) | def | ring_hom.restrict | 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"
] | [] | The ring homomorphism from the preimage of `s` to `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_apply (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) (x : s') :
(f.restrict s' s h x : S) = f x | rfl | lemma | ring_hom.coe_restrict_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_restrict (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) :
(subsemiring_class.subtype s).comp (f.restrict s' s h) = f.comp (subsemiring_class.subtype s') | rfl | lemma | ring_hom.comp_restrict | 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"
] | [
"subsemiring_class.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_restrict (f : R →+* S) : R →+* f.srange | f.cod_restrict f.srange f.mem_srange_self | def | ring_hom.srange_restrict | 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"
] | [] | 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_srange_restrict (f : R →+* S) (x : R) :
(f.srange_restrict x : S) = f x | rfl | lemma | ring_hom.coe_srange_restrict | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_restrict_surjective (f : R →+* S) : function.surjective f.srange_restrict | λ ⟨y, hy⟩, let ⟨x, hx⟩ := mem_srange.mp hy in ⟨x, subtype.ext hx⟩ | lemma | ring_hom.srange_restrict_surjective | 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"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_top_iff_surjective {f : R →+* S} :
f.srange = (⊤ : subsemiring S) ↔ function.surjective f | set_like.ext'_iff.trans $ iff.trans (by rw [coe_srange, coe_top]) set.range_iff_surjective | lemma | ring_hom.srange_top_iff_surjective | 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"
] | [
"set.range_iff_surjective",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_top_of_surjective (f : R →+* S) (hf : function.surjective f) :
f.srange = (⊤ : subsemiring S) | srange_top_iff_surjective.2 hf | lemma | ring_hom.srange_top_of_surjective | 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"
] | [
"subsemiring"
] | The range of a surjective ring homomorphism is the whole of the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_slocus (f g : R →+* S) : subsemiring R | { carrier := {x | f x = g x}, .. (f : R →* S).eq_mlocus g, .. (f : R →+ S).eq_mlocus g } | def | ring_hom.eq_slocus | 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"
] | [
"subsemiring"
] | The subsemiring of elements `x : R` such that `f x = g x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_slocus_same (f : R →+* S) : f.eq_slocus f = ⊤ | set_like.ext $ λ _, eq_self_iff_true _ | lemma | ring_hom.eq_slocus_same | 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"
] | [
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_sclosure {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_slocus g, from closure_le.2 h | lemma | ring_hom.eq_on_sclosure | 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"
] | [
"closure",
"set.eq_on"
] | If two ring homomorphisms are equal on a set, then they are equal on its subsemiring closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_eq_on_stop {f g : R →+* S} (h : set.eq_on f g (⊤ : subsemiring R)) :
f = g | ext $ λ x, h trivial | lemma | ring_hom.eq_of_eq_on_stop | 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"
] | [
"set.eq_on",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq_on_sdense {s : set R} (hs : closure s = ⊤) {f g : R →+* S} (h : s.eq_on f g) :
f = g | eq_of_eq_on_stop $ hs ▸ eq_on_sclosure h | lemma | ring_hom.eq_of_eq_on_sdense | 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"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sclosure_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.sclosure_preimage_le | 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"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sclosure (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 _ _)
(sclosure_preimage_le _ _))
(closure_le.2 $ set.image_subset _ subset_closure) | lemma | ring_hom.map_sclosure | 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"
] | [
"closure",
"closure_mono",
"set.image_subset",
"set.subset_preimage_image",
"subset_closure"
] | The image under a ring homomorphism of the subsemiring generated by a set equals
the subsemiring generated by the image of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion {S T : subsemiring R} (h : S ≤ T) : S →+* T | S.subtype.cod_restrict _ (λ x, h x.2) | def | subsemiring.inclusion | 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"
] | [
"subsemiring"
] | The ring homomorphism associated to an inclusion of subsemirings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
srange_subtype (s : subsemiring R) : s.subtype.srange = s | set_like.coe_injective $ (coe_srange _).trans subtype.range_coe | lemma | subsemiring.srange_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"
] | [
"set_like.coe_injective",
"subsemiring",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_bot_sup_bot_prod (s : subsemiring R) (t : subsemiring 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 | subsemiring.prod_bot_sup_bot_prod | 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"
] | [
"bot_le",
"le_sup_left",
"le_sup_right",
"prod.fst_mul_snd",
"subsemiring",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring_congr (h : s = t) : s ≃+* t | { map_mul' := λ _ _, rfl, map_add' := λ _ _, rfl, ..equiv.set_congr $ congr_arg _ h } | def | ring_equiv.subsemiring_congr | 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"
] | [
"equiv.set_congr"
] | Makes the identity isomorphism from a proof two subsemirings of a multiplicative
monoid are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sof_left_inverse {g : S → R} {f : R →+* S} (h : function.left_inverse g f) :
R ≃+* f.srange | { to_fun := λ x, f.srange_restrict x,
inv_fun := λ x, (g ∘ f.srange.subtype) x,
left_inv := h,
right_inv := λ x, subtype.ext $
let ⟨x', hx'⟩ := ring_hom.mem_srange.mp x.prop in
show f (g x) = x, by rw [←hx', h x'],
..f.srange_restrict } | def | ring_equiv.sof_left_inverse | 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"
] | [
"inv_fun",
"subtype.ext"
] | Restrict a ring homomorphism with a left inverse to a ring isomorphism to its
`ring_hom.srange`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sof_left_inverse_apply
{g : S → R} {f : R →+* S} (h : function.left_inverse g f) (x : R) :
↑(sof_left_inverse h x) = f x | rfl | lemma | ring_equiv.sof_left_inverse_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sof_left_inverse_symm_apply
{g : S → R} {f : R →+* S} (h : function.left_inverse g f) (x : f.srange) :
(sof_left_inverse h).symm x = g x | rfl | lemma | ring_equiv.sof_left_inverse_symm_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring_map (e : R ≃+* S) (s : subsemiring R) :
s ≃+* s.map e.to_ring_hom | { ..e.to_add_equiv.add_submonoid_map s.to_add_submonoid,
..e.to_mul_equiv.submonoid_map s.to_submonoid } | def | ring_equiv.subsemiring_map | 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"
] | [
"subsemiring"
] | Given an equivalence `e : R ≃+* S` of semirings and a subsemiring `s` of `R`,
`subsemiring_map e s` is the induced equivalence between `s` and `s.map e` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def [has_smul R' α] {S : subsemiring R'} (g : S) (m : α) : g • m = (g : R') • m | rfl | lemma | subsemiring.smul_def | 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"
] | [
"has_smul",
"subsemiring"
] | 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 : subsemiring R') :
smul_comm_class S α β | S.to_submonoid.smul_comm_class_left | instance | subsemiring.smul_comm_class_left | 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"
] | [
"has_smul",
"smul_comm_class",
"subsemiring"
] | 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 : subsemiring R') :
smul_comm_class α S β | S.to_submonoid.smul_comm_class_right | instance | subsemiring.smul_comm_class_right | 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"
] | [
"has_smul",
"smul_comm_class",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center.smul_comm_class_left : smul_comm_class (center R') R' R' | submonoid.center.smul_comm_class_left | instance | subsemiring.center.smul_comm_class_left | 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"
] | [
"smul_comm_class",
"submonoid.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' | submonoid.center.smul_comm_class_right | instance | subsemiring.center.smul_comm_class_right | 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"
] | [
"smul_comm_class",
"submonoid.center.smul_comm_class_right"
] | The center of a semiring acts commutatively on that semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_comm_semiring_of_comm {s : set R'} (hcomm : ∀ (a ∈ s) (b ∈ s), a * b = b * a) :
comm_semiring (closure s) | { mul_comm := λ x y,
begin
ext,
simp only [subsemiring.coe_mul],
refine closure_induction₂ x.prop y.prop hcomm
(λ x, by simp only [zero_mul, mul_zero])
(λ x, by simp only [zero_mul, mul_zero])
(λ x, by simp only [one_mul, mul_one])
(λ x, by simp only [one_mul, mul_one])
(λ x y z h₁ h₂,... | def | subsemiring.closure_comm_semiring_of_comm | 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"
] | [
"closure",
"comm_semiring",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_zero",
"one_mul",
"subsemiring.coe_mul",
"zero_mul"
] | If all the elements of a set `s` commute, then `closure s` is a commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_submonoid (R : Type*) [strict_ordered_semiring R] : submonoid R | { carrier := {x | 0 < x},
one_mem' := show (0 : R) < 1, from zero_lt_one,
mul_mem' := λ x y (hx : 0 < x) (hy : 0 < y), mul_pos hx hy } | def | pos_submonoid | 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"
] | [
"strict_ordered_semiring",
"submonoid",
"zero_lt_one"
] | Submonoid of positive elements of an ordered semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_pos_monoid {R : Type*} [strict_ordered_semiring R] (u : Rˣ) :
↑u ∈ pos_submonoid R ↔ (0 : R) < u | iff.rfl | lemma | mem_pos_monoid | 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"
] | [
"pos_submonoid",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_mul_action : mul_action M (subsemiring 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 | subsemiring.pointwise_mul_action | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"mul_action",
"mul_semiring_action.to_ring_hom",
"one_smul",
"ring_hom.ext",
"subsemiring"
] | The action on a subsemiring 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 : subsemiring R) :
a • S = S.map (mul_semiring_action.to_ring_hom _ _ a) | rfl | lemma | subsemiring.pointwise_smul_def | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"mul_semiring_action.to_ring_hom",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pointwise_smul (m : M) (S : subsemiring R) : ↑(m • S) = m • (S : set R) | rfl | lemma | subsemiring.coe_pointwise_smul | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_add_submonoid (m : M) (S : subsemiring R) :
(m • S).to_add_submonoid = m • S.to_add_submonoid | rfl | lemma | subsemiring.pointwise_smul_to_add_submonoid | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul (m : M) (r : R) (S : subsemiring R) : r ∈ S → m • r ∈ m • S | (set.smul_mem_smul_set : _ → _ ∈ m • (S : set R)) | lemma | subsemiring.smul_mem_pointwise_smul | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"set.smul_mem_smul_set",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_pointwise_iff_exists (m : M) (r : R) (S : subsemiring R) :
r ∈ m • S ↔ ∃ (s : R), s ∈ S ∧ m • s = r | (set.mem_smul_set : r ∈ m • (S : set R) ↔ _) | lemma | subsemiring.mem_smul_pointwise_iff_exists | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"set.mem_smul_set",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_bot (a : M) : a • (⊥ : subsemiring R) = ⊥ | map_bot _ | lemma | subsemiring.smul_bot | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup (a : M) (S T : subsemiring R) : a • (S ⊔ T) = a • S ⊔ a • T | map_sup _ _ _ | lemma | subsemiring.smul_sup | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closure (a : M) (s : set R) : a • closure s = closure (a • s) | ring_hom.map_sclosure _ _ | lemma | subsemiring.smul_closure | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"closure",
"ring_hom.map_sclosure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_central_scalar [mul_semiring_action Mᵐᵒᵖ R] [is_central_scalar M R] :
is_central_scalar M (subsemiring R) | ⟨λ a S, congr_arg (λ f, S.map f) $ ring_hom.ext $ by exact op_smul_eq_smul _⟩ | instance | subsemiring.pointwise_central_scalar | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"is_central_scalar",
"mul_semiring_action",
"ring_hom.ext",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul_iff {a : M} {S : subsemiring R} {x : R} :
a • x ∈ a • S ↔ x ∈ S | smul_mem_smul_set_iff | lemma | subsemiring.smul_mem_pointwise_smul_iff | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : subsemiring R} {x : R} :
x ∈ a • S ↔ a⁻¹ • x ∈ S | mem_smul_set_iff_inv_smul_mem | lemma | subsemiring.mem_pointwise_smul_iff_inv_smul_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_pointwise_smul_iff {a : M} {S : subsemiring R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S | mem_inv_smul_set_iff | lemma | subsemiring.mem_inv_pointwise_smul_iff | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_pointwise_smul_iff {a : M} {S T : subsemiring R} :
a • S ≤ a • T ↔ S ≤ T | set_smul_subset_set_smul_iff | lemma | subsemiring.pointwise_smul_le_pointwise_smul_iff | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_subset_iff {a : M} {S T : subsemiring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T | set_smul_subset_iff | lemma | subsemiring.pointwise_smul_subset_iff | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_pointwise_smul_iff {a : M} {S T : subsemiring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T | subset_set_smul_iff | lemma | subsemiring.subset_pointwise_smul_iff | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subsemiring R)
(x : R) : a • x ∈ a • S ↔ x ∈ S | smul_mem_smul_set_iff₀ ha (S : set R) x | lemma | subsemiring.smul_mem_pointwise_smul_iff₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : subsemiring R) (x : R) :
x ∈ a • S ↔ a⁻¹ • x ∈ S | mem_smul_set_iff_inv_smul_mem₀ ha (S : set R) x | lemma | subsemiring.mem_pointwise_smul_iff_inv_smul_mem₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subsemiring R) (x : R) :
x ∈ a⁻¹ • S ↔ a • x ∈ S | mem_inv_smul_set_iff₀ ha (S : set R) x | lemma | subsemiring.mem_inv_pointwise_smul_iff₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subsemiring R} :
a • S ≤ a • T ↔ S ≤ T | set_smul_subset_set_smul_iff₀ ha | lemma | subsemiring.pointwise_smul_le_pointwise_smul_iff₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : subsemiring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T | set_smul_subset_iff₀ ha | lemma | subsemiring.pointwise_smul_le_iff₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subsemiring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T | subset_set_smul_iff₀ ha | lemma | subsemiring.le_pointwise_smul_iff₀ | ring_theory.subsemiring | src/ring_theory/subsemiring/pointwise.lean | [
"algebra.group_ring_action.basic",
"ring_theory.subsemiring.basic",
"group_theory.submonoid.pointwise",
"data.set.pointwise.basic"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation extends R →*₀ Γ₀ | (map_add_le_max' : ∀ x y, to_fun (x + y) ≤ max (to_fun x) (to_fun y)) | structure | valuation | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | The type of `Γ₀`-valued valuations on `R`.
When you extend this structure, make sure to extend `valuation_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_class extends monoid_with_zero_hom_class F R Γ₀ | (map_add_le_max (f : F) (x y : R) : f (x + y) ≤ max (f x) (f y)) | class | valuation_class | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"monoid_with_zero_hom_class"
] | `valuation_class F α β` states that `F` is a type of valuations.
You should also extend this typeclass when you extend `valuation`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (v : valuation R Γ₀) : v.to_fun = v | rfl | lemma | valuation.to_fun_eq_coe | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {v₁ v₂ : valuation R Γ₀} (h : ∀ r, v₁ r = v₂ r) : v₁ = v₂ | fun_like.ext _ _ h | lemma | valuation.ext | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"fun_like.ext",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : ⇑(v : R →*₀ Γ₀) = v | rfl | lemma | valuation.coe_coe | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : v 0 = 0 | v.map_zero' | lemma | valuation.map_zero | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : v 1 = 1 | v.map_one' | lemma | valuation.map_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul : ∀ x y, v (x * y) = v x * v y | v.map_mul' | lemma | valuation.map_mul | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add : ∀ x y, v (x + y) ≤ max (v x) (v y) | v.map_add_le_max' | lemma | valuation.map_add | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_le {x y g} (hx : v x ≤ g) (hy : v y ≤ g) : v (x + y) ≤ g | le_trans (v.map_add x y) $ max_le hx hy | lemma | valuation.map_add_le | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_lt {x y g} (hx : v x < g) (hy : v y < g) : v (x + y) < g | lt_of_le_of_lt (v.map_add x y) $ max_lt hx hy | lemma | valuation.map_add_lt | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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