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 |
|---|---|---|---|---|---|---|---|---|---|---|
to_comm_semiring {R} [comm_semiring R] [set_like S R] [subsemiring_class S R] :
comm_semiring s | subtype.coe_injective.comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring_class.to_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"comm_semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of a `comm_semiring` is a `comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_semiring {R} [ordered_semiring R] [set_like S R] [subsemiring_class S R] :
ordered_semiring s | subtype.coe_injective.ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring_class.to_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ordered_semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of an `ordered_semiring` is an `ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_strict_ordered_semiring {R} [strict_ordered_semiring R] [set_like S R]
[subsemiring_class S R] :
strict_ordered_semiring s | subtype.coe_injective.strict_ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring_class.to_strict_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"set_like",
"strict_ordered_semiring",
"subsemiring_class"
] | A subsemiring of an `strict_ordered_semiring` is an `strict_ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_comm_semiring {R} [ordered_comm_semiring R] [set_like S R]
[subsemiring_class S R] : ordered_comm_semiring s | subtype.coe_injective.ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring_class.to_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ordered_comm_semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of an `ordered_comm_semiring` is an `ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_strict_ordered_comm_semiring {R} [strict_ordered_comm_semiring R] [set_like S R]
[subsemiring_class S R] :
strict_ordered_comm_semiring s | subtype.coe_injective.strict_ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring_class.to_strict_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"set_like",
"strict_ordered_comm_semiring",
"subsemiring_class"
] | A subsemiring of an `strict_ordered_comm_semiring` is an `strict_ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_semiring {R} [linear_ordered_semiring R] [set_like S R]
[subsemiring_class S R] : linear_ordered_semiring s | subtype.coe_injective.linear_ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | subsemiring_class.to_linear_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"linear_ordered_semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of a `linear_ordered_semiring` is a `linear_ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_comm_semiring {R} [linear_ordered_comm_semiring R] [set_like S R]
[subsemiring_class S R] : linear_ordered_comm_semiring s | subtype.coe_injective.linear_ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | subsemiring_class.to_linear_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"linear_ordered_comm_semiring",
"set_like",
"subsemiring_class"
] | A subsemiring of a `linear_ordered_comm_semiring` is a `linear_ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsemiring (R : Type u) [non_assoc_semiring R] extends submonoid R, add_submonoid R | structure | subsemiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid",
"non_assoc_semiring",
"submonoid"
] | A subsemiring of a semiring `R` is a subset `s` that is both a multiplicative and an additive
submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier {s : subsemiring R} {x : R} : x ∈ s.carrier ↔ x ∈ s | iff.rfl | lemma | subsemiring.mem_carrier | 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 | |
ext {S T : subsemiring R} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | subsemiring.ext | 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",
"subsemiring"
] | Two subsemirings are equal if they have the same elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy (S : subsemiring R) (s : set R) (hs : s = ↑S) : subsemiring R | { carrier := s,
..S.to_add_submonoid.copy s hs,
..S.to_submonoid.copy s hs } | def | subsemiring.copy | 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"
] | Copy of a subsemiring with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : subsemiring R) (s : set R) (hs : s = ↑S) :
(S.copy s hs : set R) = s | rfl | lemma | subsemiring.coe_copy | 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 | |
copy_eq (S : subsemiring R) (s : set R) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | subsemiring.copy_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"
] | [
"set_like.coe_injective",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_injective : function.injective (to_submonoid : subsemiring R → submonoid R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | subsemiring.to_submonoid_injective | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"submonoid",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_strict_mono : strict_mono (to_submonoid : subsemiring R → submonoid R) | λ _ _, id | lemma | subsemiring.to_submonoid_strict_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"
] | [
"strict_mono",
"submonoid",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid_mono : monotone (to_submonoid : subsemiring R → submonoid R) | to_submonoid_strict_mono.monotone | lemma | subsemiring.to_submonoid_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"
] | [
"monotone",
"submonoid",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_injective :
function.injective (to_add_submonoid : subsemiring R → add_submonoid R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | subsemiring.to_add_submonoid_injective | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_strict_mono :
strict_mono (to_add_submonoid : subsemiring R → add_submonoid R) | λ _ _, id | lemma | subsemiring.to_add_submonoid_strict_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"
] | [
"add_submonoid",
"strict_mono",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_mono : monotone (to_add_submonoid : subsemiring R → add_submonoid R) | to_add_submonoid_strict_mono.monotone | lemma | subsemiring.to_add_submonoid_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"
] | [
"add_submonoid",
"monotone",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (s : set R) (sm : submonoid R) (hm : ↑sm = s)
(sa : add_submonoid R) (ha : ↑sa = s) :
subsemiring R | { carrier := s,
zero_mem' := ha ▸ sa.zero_mem,
one_mem' := hm ▸ sm.one_mem,
add_mem' := λ x y, by simpa only [← ha] using sa.add_mem,
mul_mem' := λ x y, by simpa only [← hm] using sm.mul_mem } | def | subsemiring.mk' | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid",
"mk'",
"submonoid",
"subsemiring"
] | Construct a `subsemiring R` from a set `s`, a submonoid `sm`, and an additive
submonoid `sa` such that `x ∈ s ↔ x ∈ sm ↔ x ∈ sa`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_submonoid R} (ha : ↑sa = s) :
(subsemiring.mk' s sm hm sa ha : set R) = s | rfl | lemma | subsemiring.coe_mk' | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid",
"submonoid",
"subsemiring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk' {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_submonoid R} (ha : ↑sa = s) {x : R} :
x ∈ subsemiring.mk' s sm hm sa ha ↔ x ∈ s | iff.rfl | lemma | subsemiring.mem_mk' | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"add_submonoid",
"submonoid",
"subsemiring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_submonoid {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_submonoid R} (ha : ↑sa = s) :
(subsemiring.mk' s sm hm sa ha).to_submonoid = sm | set_like.coe_injective hm.symm | lemma | subsemiring.mk'_to_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"
] | [
"add_submonoid",
"set_like.coe_injective",
"submonoid",
"subsemiring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_add_submonoid {s : set R} {sm : submonoid R} (hm : ↑sm = s)
{sa : add_submonoid R} (ha : ↑sa =s) :
(subsemiring.mk' s sm hm sa ha).to_add_submonoid = sa | set_like.coe_injective ha.symm | lemma | subsemiring.mk'_to_add_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"
] | [
"add_submonoid",
"set_like.coe_injective",
"submonoid",
"subsemiring.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem : (1 : R) ∈ s | one_mem s | theorem | subsemiring.one_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [] | A subsemiring contains the semiring's 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem : (0 : R) ∈ s | zero_mem s | theorem | subsemiring.zero_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [] | A subsemiring contains the semiring's 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list_prod_mem {R : Type*} [semiring R] (s : subsemiring R) {l : list R} :
(∀x ∈ l, x ∈ s) → l.prod ∈ s | list_prod_mem | lemma | subsemiring.list_prod_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"list_prod_mem",
"semiring",
"subsemiring"
] | Product of a list of elements in a `subsemiring` is in the `subsemiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiset_prod_mem {R} [comm_semiring R] (s : subsemiring R) (m : multiset R) :
(∀a ∈ m, a ∈ s) → m.prod ∈ s | multiset_prod_mem m | lemma | subsemiring.multiset_prod_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"comm_semiring",
"multiset",
"multiset_prod_mem",
"subsemiring"
] | Product of a multiset of elements in a `subsemiring` of a `comm_semiring`
is in the `subsemiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiset_sum_mem (m : multiset R) :
(∀a ∈ m, a ∈ s) → m.sum ∈ s | multiset_sum_mem m | lemma | subsemiring.multiset_sum_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"multiset"
] | Sum of a multiset of elements in a `subsemiring` of a `semiring` is
in the `add_subsemiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mem {R : Type*} [comm_semiring R] (s : subsemiring R)
{ι : Type*} {t : finset ι} {f : ι → R} (h : ∀c ∈ t, f c ∈ s) :
∏ i in t, f i ∈ s | prod_mem h | lemma | subsemiring.prod_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"comm_semiring",
"finset",
"prod_mem",
"subsemiring"
] | Product of elements of a subsemiring of a `comm_semiring` indexed by a `finset` is in the
subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_mem (s : subsemiring R)
{ι : Type*} {t : finset ι} {f : ι → R} (h : ∀c ∈ t, f c ∈ s) :
∑ i in t, f i ∈ s | sum_mem h | lemma | subsemiring.sum_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"finset",
"subsemiring"
] | Sum of elements in an `subsemiring` of an `semiring` indexed by a `finset`
is in the `add_subsemiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_non_assoc_semiring : non_assoc_semiring s | { mul_zero := λ x, subtype.eq $ mul_zero x,
zero_mul := λ x, subtype.eq $ zero_mul x,
right_distrib := λ x y z, subtype.eq $ right_distrib x y z,
left_distrib := λ x y z, subtype.eq $ left_distrib x y z,
nat_cast := λ n, ⟨n, coe_nat_mem s n⟩,
nat_cast_zero := by simp [nat.cast]; refl,
nat_cast_succ := λ _, ... | instance | subsemiring.to_non_assoc_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"coe_nat_mem",
"left_distrib",
"mul_zero",
"nat.cast",
"non_assoc_semiring",
"right_distrib",
"zero_mul"
] | A subsemiring of a `non_assoc_semiring` inherits a `non_assoc_semiring` structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one : ((1 : s) : R) = (1 : R) | rfl | lemma | subsemiring.coe_one | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem {R : Type*} [semiring R] (s : subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :
x^n ∈ s | pow_mem hx n | lemma | subsemiring.pow_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"pow_mem",
"semiring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_semiring {R} [semiring R] (s : subsemiring R) : semiring s | { ..s.to_non_assoc_semiring, ..s.to_submonoid.to_monoid } | instance | subsemiring.to_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"subsemiring"
] | A subsemiring of a `semiring` is a `semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pow {R} [semiring R] (s : subsemiring R) (x : s) (n : ℕ) :
((x^n : s) : R) = (x^n : R) | begin
induction n with n ih,
{ simp, },
{ simp [pow_succ, ih], },
end | lemma | subsemiring.coe_pow | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ih",
"pow_succ",
"semiring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_comm_semiring {R} [comm_semiring R] (s : subsemiring R) : comm_semiring s | { mul_comm := λ _ _, subtype.eq $ mul_comm _ _, ..s.to_semiring} | instance | subsemiring.to_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"comm_semiring",
"mul_comm",
"subsemiring"
] | A subsemiring of a `comm_semiring` is a `comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype : s →+* R | { to_fun := coe, .. s.to_submonoid.subtype, .. s.to_add_submonoid.subtype } | def | subsemiring.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"
] | [] | The natural ring hom from a subsemiring of semiring `R` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_semiring {R} [ordered_semiring R] (s : subsemiring R) : ordered_semiring s | subtype.coe_injective.ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring.to_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ordered_semiring",
"subsemiring"
] | A subsemiring of an `ordered_semiring` is an `ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_strict_ordered_semiring {R} [strict_ordered_semiring R] (s : subsemiring R) :
strict_ordered_semiring s | subtype.coe_injective.strict_ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring.to_strict_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"strict_ordered_semiring",
"subsemiring"
] | A subsemiring of a `strict_ordered_semiring` is a `strict_ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_comm_semiring {R} [ordered_comm_semiring R] (s : subsemiring R) :
ordered_comm_semiring s | subtype.coe_injective.ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring.to_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"ordered_comm_semiring",
"subsemiring"
] | A subsemiring of an `ordered_comm_semiring` is an `ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_strict_ordered_comm_semiring {R} [strict_ordered_comm_semiring R] (s : subsemiring R) :
strict_ordered_comm_semiring s | subtype.coe_injective.strict_ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | subsemiring.to_strict_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"strict_ordered_comm_semiring",
"subsemiring"
] | A subsemiring of a `strict_ordered_comm_semiring` is a `strict_ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_semiring {R} [linear_ordered_semiring R] (s : subsemiring R) :
linear_ordered_semiring s | subtype.coe_injective.linear_ordered_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | subsemiring.to_linear_ordered_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"linear_ordered_semiring",
"subsemiring"
] | A subsemiring of a `linear_ordered_semiring` is a `linear_ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_comm_semiring {R} [linear_ordered_comm_semiring R] (s : subsemiring R) :
linear_ordered_comm_semiring s | subtype.coe_injective.linear_ordered_comm_semiring coe
rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | subsemiring.to_linear_ordered_comm_semiring | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"linear_ordered_comm_semiring",
"subsemiring"
] | A subsemiring of a `linear_ordered_comm_semiring` is a `linear_ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nsmul_mem {x : R} (hx : x ∈ s) (n : ℕ) :
n • x ∈ s | nsmul_mem hx n | lemma | subsemiring.nsmul_mem | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_submonoid {s : subsemiring R} {x : R} : x ∈ s.to_submonoid ↔ x ∈ s | iff.rfl | lemma | subsemiring.mem_to_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submonoid (s : subsemiring R) : (s.to_submonoid : set R) = s | rfl | lemma | subsemiring.coe_to_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_add_submonoid {s : subsemiring R} {x : R} :
x ∈ s.to_add_submonoid ↔ x ∈ s | iff.rfl | lemma | subsemiring.mem_to_add_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_submonoid (s : subsemiring R) : (s.to_add_submonoid : set R) = s | rfl | lemma | subsemiring.coe_to_add_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : R) : x ∈ (⊤ : subsemiring R) | set.mem_univ x | lemma | subsemiring.mem_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"
] | [
"set.mem_univ",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : subsemiring R) : set R) = set.univ | rfl | lemma | subsemiring.coe_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 | |
top_equiv : (⊤ : subsemiring R) ≃+* R | { to_fun := λ r, r,
inv_fun := λ r, ⟨r, subsemiring.mem_top r⟩,
left_inv := λ r, set_like.eta r _,
right_inv := λ r, set_like.coe_mk r _,
map_mul' := (⊤ : subsemiring R).coe_mul,
map_add' := (⊤ : subsemiring R).coe_add, } | def | subsemiring.top_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"
] | [
"inv_fun",
"set_like.coe_mk",
"set_like.eta",
"subsemiring",
"subsemiring.mem_top"
] | The ring equiv between the top element of `subsemiring R` and `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap (f : R →+* S) (s : subsemiring S) : subsemiring R | { carrier := f ⁻¹' s,
.. s.to_submonoid.comap (f : R →* S), .. s.to_add_submonoid.comap (f : R →+ S) } | def | subsemiring.comap | 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 preimage of a subsemiring along a ring homomorphism is a subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (s : subsemiring S) (f : R →+* S) : (s.comap f : set R) = f ⁻¹' s | rfl | lemma | subsemiring.coe_comap | 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_comap {s : subsemiring S} {f : R →+* S} {x : R} : x ∈ s.comap f ↔ f x ∈ s | iff.rfl | lemma | subsemiring.mem_comap | 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_comap (s : subsemiring T) (g : S →+* T) (f : R →+* S) :
(s.comap g).comap f = s.comap (g.comp f) | rfl | lemma | subsemiring.comap_comap | 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 (f : R →+* S) (s : subsemiring R) : subsemiring S | { carrier := f '' s,
.. s.to_submonoid.map (f : R →* S), .. s.to_add_submonoid.map (f : R →+ S) } | def | 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"
] | The image of a subsemiring along a ring homomorphism is a subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (f : R →+* S) (s : subsemiring R) : (s.map f : set S) = f '' s | rfl | lemma | subsemiring.coe_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {f : R →+* S} {s : subsemiring R} {y : S} :
y ∈ s.map f ↔ ∃ x ∈ s, f x = y | set.mem_image_iff_bex | lemma | subsemiring.mem_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"
] | [
"mem_map",
"set.mem_image_iff_bex",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap {f : R →+* S} {s : subsemiring R} {t : subsemiring S} :
s.map f ≤ t ↔ s ≤ t.comap f | set.image_subset_iff | lemma | subsemiring.map_le_iff_le_comap | 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.image_subset_iff",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange : subsemiring S | ((⊤ : subsemiring R).map f).copy (set.range f) set.image_univ.symm | def | ring_hom.srange | 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",
"subsemiring"
] | The range of a ring homomorphism is a subsemiring. See Note [range copy pattern]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_srange : (f.srange : set S) = set.range f | rfl | lemma | ring_hom.coe_srange | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_srange {f : R →+* S} {y : S} : y ∈ f.srange ↔ ∃ x, f x = y | iff.rfl | lemma | ring_hom.mem_srange | 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_eq_map (f : R →+* S) : f.srange = (⊤ : subsemiring R).map f | by { ext, simp } | lemma | ring_hom.srange_eq_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_srange_self (f : R →+* S) (x : R) : f x ∈ f.srange | mem_srange.mpr ⟨x, rfl⟩ | lemma | ring_hom.mem_srange_self | 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 | |
map_srange : f.srange.map g = (g.comp f).srange | by simpa only [srange_eq_map] using (⊤ : subsemiring R).map_map g f | lemma | ring_hom.map_srange | 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 | |
fintype_srange [fintype R] [decidable_eq S] (f : R →+* S) : fintype (srange f) | set.fintype_range f | instance | ring_hom.fintype_srange | 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"
] | [
"fintype",
"set.fintype_range"
] | The range of a morphism of semirings is a fintype, if the domain is a fintype.
Note: this instance can form a diamond with `subtype.fintype` in the
presence of `fintype S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_bot : ((⊥ : subsemiring R) : set R) = set.range (coe : ℕ → R) | (nat.cast_ring_hom R).coe_srange | lemma | subsemiring.coe_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"
] | [
"nat.cast_ring_hom",
"set.range",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : R} : x ∈ (⊥ : subsemiring R) ↔ ∃ n : ℕ, ↑n=x | ring_hom.mem_srange | lemma | subsemiring.mem_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"
] | [
"ring_hom.mem_srange",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (p p' : subsemiring R) : ((p ⊓ p' : subsemiring R) : set R) = p ∩ p' | rfl | lemma | subsemiring.coe_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 | |
mem_inf {p p' : subsemiring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' | iff.rfl | lemma | subsemiring.mem_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 | |
coe_Inf (S : set (subsemiring R)) :
((Inf S : subsemiring R) : set R) = ⋂ s ∈ S, ↑s | rfl | lemma | subsemiring.coe_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 | |
mem_Inf {S : set (subsemiring R)} {x : R} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | set.mem_Inter₂ | lemma | subsemiring.mem_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"
] | [
"set.mem_Inter₂",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_submonoid (s : set (subsemiring R)) :
(Inf s).to_submonoid = ⨅ t ∈ s, subsemiring.to_submonoid t | mk'_to_submonoid _ _ | lemma | subsemiring.Inf_to_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_add_submonoid (s : set (subsemiring R)) :
(Inf s).to_add_submonoid = ⨅ t ∈ s, subsemiring.to_add_submonoid t | mk'_to_add_submonoid _ _ | lemma | subsemiring.Inf_to_add_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff' (A : subsemiring R) : A = ⊤ ↔ ∀ x : R, x ∈ A | eq_top_iff.trans ⟨λ h m, h $ mem_top m, λ h m _, h m⟩ | lemma | subsemiring.eq_top_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"
] | [
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center (R) [semiring R] : subsemiring R | { carrier := set.center R,
zero_mem' := set.zero_mem_center R,
add_mem' := λ a b, set.add_mem_center,
.. submonoid.center R } | def | subsemiring.center | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set.add_mem_center",
"set.center",
"set.zero_mem_center",
"submonoid.center",
"subsemiring"
] | 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) [semiring R] : ↑(center R) = set.center R | rfl | lemma | subsemiring.coe_center | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_to_submonoid (R) [semiring R] : (center R).to_submonoid = submonoid.center R | rfl | lemma | subsemiring.center_to_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"
] | [
"semiring",
"submonoid.center"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_center_iff {R} [semiring R] {z : R} : z ∈ center R ↔ ∀ g, g * z = z * g | iff.rfl | lemma | subsemiring.mem_center_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"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_mem_center {R} [semiring R] [decidable_eq R] [fintype R] :
decidable_pred (∈ center R) | λ _, decidable_of_iff' _ mem_center_iff | instance | subsemiring.decidable_mem_center | 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"
] | [
"decidable_of_iff'",
"fintype",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_eq_top (R) [comm_semiring R] : center R = ⊤ | set_like.coe_injective (set.center_eq_univ R) | lemma | subsemiring.center_eq_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"
] | [
"comm_semiring",
"set.center_eq_univ",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer {R} [semiring R] (s : set R) : subsemiring R | { carrier := s.centralizer,
zero_mem' := set.zero_mem_centralizer _,
add_mem' := λ x y hx hy, set.add_mem_centralizer hx hy,
..submonoid.centralizer s } | def | subsemiring.centralizer | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set.add_mem_centralizer",
"set.zero_mem_centralizer",
"submonoid.centralizer",
"subsemiring"
] | The centralizer of a set as subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_centralizer {R} [semiring R] (s : set R) : (centralizer s : set R) = s.centralizer | rfl | lemma | subsemiring.coe_centralizer | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_to_submonoid {R} [semiring R] (s : set R) :
(centralizer s).to_submonoid = submonoid.centralizer s | rfl | lemma | subsemiring.centralizer_to_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"
] | [
"semiring",
"submonoid.centralizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_centralizer_iff {R} [semiring R] {s : set R} {z : R} :
z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g | iff.rfl | lemma | subsemiring.mem_centralizer_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"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_le_centralizer {R} [semiring R] (s) : center R ≤ centralizer s | s.center_subset_centralizer | lemma | subsemiring.center_le_centralizer | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_le {R} [semiring R] (s t : set R) (h : s ⊆ t) :
centralizer t ≤ centralizer s | set.centralizer_subset h | lemma | subsemiring.centralizer_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"
] | [
"semiring",
"set.centralizer_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_eq_top_iff_subset {R} [semiring R] {s : set R} :
centralizer s = ⊤ ↔ s ⊆ center R | set_like.ext'_iff.trans set.centralizer_eq_top_iff_subset | lemma | subsemiring.centralizer_eq_top_iff_subset | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set.centralizer_eq_top_iff_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_univ {R} [semiring R] : centralizer set.univ = center R | set_like.ext' (set.centralizer_univ R) | lemma | subsemiring.centralizer_univ | ring_theory.subsemiring | src/ring_theory/subsemiring/basic.lean | [
"algebra.module.basic",
"algebra.ring.equiv",
"algebra.ring.prod",
"algebra.order.ring.inj_surj",
"algebra.group_ring_action.subobjects",
"data.set.finite",
"group_theory.submonoid.centralizer",
"group_theory.submonoid.membership"
] | [
"semiring",
"set.centralizer_univ",
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (s : set R) : subsemiring R | Inf {S | s ⊆ S} | def | subsemiring.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",
"subsemiring"
] | The `subsemiring` generated by a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure {x : R} {s : set R} : x ∈ closure s ↔ ∀ S : subsemiring R, s ⊆ S → x ∈ S | mem_Inf | lemma | subsemiring.mem_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",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_le {s : set R} {t : subsemiring R} : closure s ≤ t ↔ s ⊆ t | ⟨set.subset.trans subset_closure, λ h, Inf_le h⟩ | lemma | subsemiring.closure_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"
] | [
"Inf_le",
"closure",
"subsemiring",
"subset_closure"
] | A subsemiring `S` includes `closure s` if and only if it includes `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_of_le {s : set R} {t : subsemiring R} (h₁ : s ⊆ t) (h₂ : t ≤ closure s) :
closure s = t | le_antisymm (closure_le.2 h₁) h₂ | lemma | subsemiring.closure_eq_of_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",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_equiv {f : R ≃+* S} {K : subsemiring R} {x : S} :
x ∈ K.map (f : R →+* S) ↔ f.symm x ∈ K | @set.mem_image_equiv _ _ ↑K f.to_equiv x | lemma | subsemiring.mem_map_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"
] | [
"set.mem_image_equiv",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_eq_comap_symm (f : R ≃+* S) (K : subsemiring R) :
K.map (f : R →+* S) = K.comap f.symm | set_like.coe_injective (f.to_equiv.image_eq_preimage K) | lemma | subsemiring.map_equiv_eq_comap_symm | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_equiv_eq_map_symm (f : R ≃+* S) (K : subsemiring S) :
K.comap (f : R →+* S) = K.map f.symm | (map_equiv_eq_comap_symm f.symm K).symm | lemma | subsemiring.comap_equiv_eq_map_symm | 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 | |
subsemiring_closure (M : submonoid R) : subsemiring R | { one_mem' := add_submonoid.mem_closure.mpr (λ y hy, hy M.one_mem),
mul_mem' := λ x y, mul_mem_class.mul_mem_add_closure,
..add_submonoid.closure (M : set R)} | def | submonoid.subsemiring_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"
] | [
"mul_mem_class.mul_mem_add_closure",
"submonoid",
"subsemiring"
] | The additive closure of a submonoid is a subsemiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsemiring_closure_coe :
(M.subsemiring_closure : set R) = add_submonoid.closure (M : set R) | rfl | lemma | submonoid.subsemiring_closure_coe | 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_closure_to_add_submonoid :
M.subsemiring_closure.to_add_submonoid = add_submonoid.closure (M : set R) | rfl | lemma | submonoid.subsemiring_closure_to_add_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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