statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
continuous_fst : continuous (fst : tsze R M → R) | continuous_fst | lemma | triv_sq_zero_ext.continuous_fst | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous",
"continuous_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_snd : continuous (snd : tsze R M → M) | continuous_snd | lemma | triv_sq_zero_ext.continuous_snd | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inl [has_zero M] : continuous (inl : R → tsze R M) | continuous_id.prod_mk continuous_const | lemma | triv_sq_zero_ext.continuous_inl | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous",
"continuous_const",
"continuous_inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inr [has_zero R] : continuous (inr : M → tsze R M) | continuous_const.prod_mk continuous_id | lemma | triv_sq_zero_ext.continuous_inr | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous",
"continuous_id",
"continuous_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_inl [has_zero M] : embedding (inl : R → tsze R M) | embedding_of_embedding_compose continuous_inl continuous_fst embedding_id | lemma | triv_sq_zero_ext.embedding_inl | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous_fst",
"continuous_inl",
"embedding",
"embedding_id",
"embedding_inl",
"embedding_of_embedding_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_inr [has_zero R] : embedding (inr : M → tsze R M) | embedding_of_embedding_compose continuous_inr continuous_snd embedding_id | lemma | triv_sq_zero_ext.embedding_inr | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous_inr",
"continuous_snd",
"embedding",
"embedding_id",
"embedding_inr",
"embedding_of_embedding_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_clm [comm_semiring R] [add_comm_monoid M] [module R M] : tsze R M →L[R] R | { to_fun := fst,
.. continuous_linear_map.fst R R M } | def | triv_sq_zero_ext.fst_clm | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"comm_semiring",
"continuous_linear_map.fst",
"module"
] | `triv_sq_zero_ext.fst` as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_clm [comm_semiring R] [add_comm_monoid M] [module R M] : tsze R M →L[R] M | { to_fun := snd,
cont := continuous_snd,
.. continuous_linear_map.snd R R M } | def | triv_sq_zero_ext.snd_clm | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"comm_semiring",
"cont",
"continuous_linear_map.snd",
"continuous_snd",
"module"
] | `triv_sq_zero_ext.snd` as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_clm [comm_semiring R] [add_comm_monoid M] [module R M] : R →L[R] tsze R M | { to_fun := inl,
.. continuous_linear_map.inl R R M } | def | triv_sq_zero_ext.inl_clm | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"comm_semiring",
"continuous_linear_map.inl",
"module"
] | `triv_sq_zero_ext.inl` as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_clm [comm_semiring R] [add_comm_monoid M] [module R M] : M →L[R] tsze R M | { to_fun := inr,
.. continuous_linear_map.inr R R M } | def | triv_sq_zero_ext.inr_clm | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"comm_semiring",
"continuous_linear_map.inr",
"module"
] | `triv_sq_zero_ext.inr` as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_semiring [semiring R] [add_comm_monoid M] [module R M] [module Rᵐᵒᵖ M]
[topological_semiring R] [has_continuous_add M]
[has_continuous_smul R M] [has_continuous_smul Rᵐᵒᵖ M] :
-- note: lean times out looking for the non_assoc_semiring instance without this hint
@topological_semiring (tsze R M) _ (no... | {} | lemma | triv_sq_zero_ext.topological_semiring | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"has_continuous_add",
"has_continuous_smul",
"module",
"semiring",
"topological_semiring"
] | This is not an instance due to complaints by the `fails_quickly` linter. At any rate, we only
really care about the `topological_ring` instance below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_inl [add_comm_monoid R] [add_comm_monoid M] {f : α → R} {a : R} (h : has_sum f a) :
has_sum (λ x, inl (f x)) (inl a : tsze R M) | h.map (⟨inl, inl_zero _, inl_add _⟩ : R →+ tsze R M) continuous_inl | lemma | triv_sq_zero_ext.has_sum_inl | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"continuous_inl",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_inr [add_comm_monoid R] [add_comm_monoid M] {f : α → M} {a : M} (h : has_sum f a) :
has_sum (λ x, inr (f x)) (inr a : tsze R M) | h.map (⟨inr, inr_zero _, inr_add _⟩ : M →+ tsze R M) continuous_inr | lemma | triv_sq_zero_ext.has_sum_inr | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"continuous_inr",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_fst [add_comm_monoid R] [add_comm_monoid M] {f : α → tsze R M} {a : tsze R M}
(h : has_sum f a) : has_sum (λ x, fst (f x)) (fst a) | h.map (⟨fst, fst_zero, fst_add⟩ : tsze R M →+ R) continuous_fst | lemma | triv_sq_zero_ext.has_sum_fst | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"continuous_fst",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_snd [add_comm_monoid R] [add_comm_monoid M] {f : α → tsze R M} {a : tsze R M}
(h : has_sum f a) : has_sum (λ x, snd (f x)) (snd a) | h.map (⟨snd, snd_zero, snd_add⟩ : tsze R M →+ M) continuous_snd | lemma | triv_sq_zero_ext.has_sum_snd | topology.instances | src/topology/instances/triv_sq_zero_ext.lean | [
"algebra.triv_sq_zero_ext",
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"add_comm_monoid",
"continuous_snd",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [has_one Y] : ⇑(1 : locally_constant X Y) = (1 : X → Y) | rfl | lemma | locally_constant.coe_one | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply [has_one Y] (x : X) : (1 : locally_constant X Y) x = 1 | rfl | lemma | locally_constant.one_apply | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv [has_inv Y] (f : locally_constant X Y) : ⇑(f⁻¹) = f⁻¹ | rfl | lemma | locally_constant.coe_inv | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply [has_inv Y] (f : locally_constant X Y) (x : X) :
f⁻¹ x = (f x)⁻¹ | rfl | lemma | locally_constant.inv_apply | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [has_mul Y] (f g : locally_constant X Y) :
⇑(f * g) = f * g | rfl | lemma | locally_constant.coe_mul | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply [has_mul Y] (f g : locally_constant X Y) (x : X) :
(f * g) x = f x * g x | rfl | lemma | locally_constant.mul_apply | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_monoid_hom [mul_one_class Y] : locally_constant X Y →* (X → Y) | { to_fun := coe_fn,
map_one' := rfl,
map_mul' := λ _ _, rfl } | def | locally_constant.coe_fn_monoid_hom | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant",
"mul_one_class"
] | `coe_fn` is a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_monoid_hom [mul_one_class Y] : Y →* locally_constant X Y | { to_fun := const X,
map_one' := rfl,
map_mul' := λ _ _, rfl, } | def | locally_constant.const_monoid_hom | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant",
"mul_one_class"
] | The constant-function embedding, as a multiplicative monoid hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_fn (hU : is_clopen U) : locally_constant X Y | indicator 1 hU | def | locally_constant.char_fn | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"is_clopen",
"locally_constant"
] | Characteristic functions are locally constant functions taking `x : X` to `1` if `x ∈ U`,
where `U` is a clopen set, and `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_char_fn (hU : is_clopen U) : (char_fn Y hU : X → Y) = set.indicator U 1 | rfl | lemma | locally_constant.coe_char_fn | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"is_clopen",
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_fn_eq_one [nontrivial Y] (x : X) (hU : is_clopen U) :
char_fn Y hU x = (1 : Y) ↔ x ∈ U | set.indicator_eq_one_iff_mem _ | lemma | locally_constant.char_fn_eq_one | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"is_clopen",
"nontrivial",
"set.indicator_eq_one_iff_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_fn_eq_zero [nontrivial Y] (x : X) (hU : is_clopen U) :
char_fn Y hU x = (0 : Y) ↔ x ∉ U | set.indicator_eq_zero_iff_not_mem _ | lemma | locally_constant.char_fn_eq_zero | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"is_clopen",
"nontrivial",
"set.indicator_eq_zero_iff_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_fn_inj [nontrivial Y] (hU : is_clopen U) (hV : is_clopen V)
(h : char_fn Y hU = char_fn Y hV) : U = V | set.indicator_one_inj Y $ coe_inj.mpr h | lemma | locally_constant.char_fn_inj | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"is_clopen",
"nontrivial",
"set.indicator_one_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div [has_div Y] (f g : locally_constant X Y) :
⇑(f / g) = f / g | rfl | lemma | locally_constant.coe_div | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_apply [has_div Y] (f g : locally_constant X Y) (x : X) :
(f / g) x = f x / g x | rfl | lemma | locally_constant.div_apply | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_ring_hom [non_assoc_semiring Y] : Y →+* locally_constant X Y | { to_fun := const X,
.. const_monoid_hom,
.. const_add_monoid_hom, } | def | locally_constant.const_ring_hom | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"locally_constant",
"non_assoc_semiring"
] | The constant-function embedding, as a ring hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_smul [has_smul R Y] (r : R) (f : locally_constant X Y) : ⇑(r • f) = r • f | rfl | lemma | locally_constant.coe_smul | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"has_smul",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply [has_smul R Y] (r : R) (f : locally_constant X Y) (x : X) :
(r • f) x = r • (f x) | rfl | lemma | locally_constant.smul_apply | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"has_smul",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map (r : R) :
⇑(algebra_map R (locally_constant X Y) r) = algebra_map R (X → Y) r | rfl | lemma | locally_constant.coe_algebra_map | topology.locally_constant | src/topology/locally_constant/algebra.lean | [
"algebra.algebra.pi",
"topology.locally_constant.basic"
] | [
"algebra_map",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_locally_constant (f : X → Y) : Prop | ∀ s : set Y, is_open (f ⁻¹' s) | def | is_locally_constant | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_open"
] | A function between topological spaces is locally constant if the preimage of any set is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tfae (f : X → Y) :
tfae [is_locally_constant f,
∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,
∀ x, is_open {x' | f x' = f x},
∀ y, is_open (f ⁻¹' {y}),
∀ x, ∃ (U : set X) (hU : is_open U) (hx : x ∈ U), ∀ x' ∈ U, f x' = f x] | begin
tfae_have : 1 → 4, from λ h y, h {y},
tfae_have : 4 → 3, from λ h x, h (f x),
tfae_have : 3 → 2, from λ h x, is_open.mem_nhds (h x) rfl,
tfae_have : 2 → 5,
{ intros h x,
rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩,
exact ⟨U, hU, hx, eq⟩ },
tfae_have : 5 → 1,
{ intros h s,
refine is_... | lemma | is_locally_constant.tfae | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_discrete [discrete_topology X] (f : X → Y) :
is_locally_constant f | λ s, is_open_discrete _ | lemma | is_locally_constant.of_discrete | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"discrete_topology",
"is_locally_constant",
"is_open_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_fiber {f : X → Y} (hf : is_locally_constant f) (y : Y) :
is_open {x | f x = y} | hf {y} | lemma | is_locally_constant.is_open_fiber | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_fiber {f : X → Y} (hf : is_locally_constant f) (y : Y) :
is_closed {x | f x = y} | ⟨hf {y}ᶜ⟩ | lemma | is_locally_constant.is_closed_fiber | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_closed",
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_fiber {f : X → Y} (hf : is_locally_constant f) (y : Y) :
is_clopen {x | f x = y} | ⟨is_open_fiber hf _, is_closed_fiber hf _⟩ | lemma | is_locally_constant.is_clopen_fiber | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_clopen",
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_exists_open (f : X → Y) :
is_locally_constant f ↔ ∀ x, ∃ (U : set X) (hU : is_open U) (hx : x ∈ U), ∀ x' ∈ U, f x' = f x | (is_locally_constant.tfae f).out 0 4 | lemma | is_locally_constant.iff_exists_open | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_locally_constant.tfae",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_eventually_eq (f : X → Y) :
is_locally_constant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x | (is_locally_constant.tfae f).out 0 1 | lemma | is_locally_constant.iff_eventually_eq | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_locally_constant.tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_open {f : X → Y} (hf : is_locally_constant f) (x : X) :
∃ (U : set X) (hU : is_open U) (hx : x ∈ U), ∀ x' ∈ U, f x' = f x | (iff_exists_open f).1 hf x | lemma | is_locally_constant.exists_open | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq {f : X → Y} (hf : is_locally_constant f) (x : X) :
∀ᶠ y in 𝓝 x, f y = f x | (iff_eventually_eq f).1 hf x | lemma | is_locally_constant.eventually_eq | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous [topological_space Y] {f : X → Y} (hf : is_locally_constant f) :
continuous f | ⟨λ U hU, hf _⟩ | lemma | is_locally_constant.continuous | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous",
"is_locally_constant",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_continuous {_ : topological_space Y} [discrete_topology Y] (f : X → Y) :
is_locally_constant f ↔ continuous f | ⟨is_locally_constant.continuous, λ h s, h.is_open_preimage s (is_open_discrete _)⟩ | lemma | is_locally_constant.iff_continuous | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous",
"discrete_topology",
"is_locally_constant",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_constant (f : X → Y) (h : ∀ x y, f x = f y) :
is_locally_constant f | (iff_eventually_eq f).2 $ λ x, eventually_of_forall $ λ x', h _ _ | lemma | is_locally_constant.of_constant | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (y : Y) : is_locally_constant (function.const X y) | of_constant _ $ λ _ _, rfl | lemma | is_locally_constant.const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {f : X → Y} (hf : is_locally_constant f) (g : Y → Z) :
is_locally_constant (g ∘ f) | λ s, by { rw set.preimage_comp, exact hf _ } | lemma | is_locally_constant.comp | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk {Y'} {f : X → Y} {f' : X → Y'} (hf : is_locally_constant f)
(hf' : is_locally_constant f') :
is_locally_constant (λ x, (f x, f' x)) | (iff_eventually_eq _).2 $ λ x, (hf.eventually_eq x).mp $ (hf'.eventually_eq x).mono $
λ x' hf' hf, prod.ext hf hf' | lemma | is_locally_constant.prod_mk | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂}
(hf : is_locally_constant f) (hg : is_locally_constant g) (h : Y₁ → Y₂ → Z) :
is_locally_constant (λ x, h (f x) (g x)) | (hf.prod_mk hg).comp (λ x : Y₁ × Y₂, h x.1 x.2) | lemma | is_locally_constant.comp₂ | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous [topological_space Y] {g : Y → Z} {f : X → Y}
(hg : is_locally_constant g) (hf : continuous f) :
is_locally_constant (g ∘ f) | λ s, by { rw set.preimage_comp, exact hf.is_open_preimage _ (hg _) } | lemma | is_locally_constant.comp_continuous | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous",
"is_locally_constant",
"set.preimage_comp",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_of_is_preconnected {f : X → Y} (hf : is_locally_constant f)
{s : set X} (hs : is_preconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) :
f x = f y | begin
let U := f ⁻¹' {f y},
suffices : x ∉ Uᶜ, from not_not.1 this,
intro hxV,
specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩,
{ simp only [union_compl_self, subset_univ] },
{ simpa only [inter_empty, not_nonempty_empty, inter_compl_self] using hs }
end | lemma | is_locally_constant.apply_eq_of_is_preconnected | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_preconnected"
] | A locally constant function is constant on any preconnected set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_eq_of_preconnected_space [preconnected_space X]
{f : X → Y} (hf : is_locally_constant f) (x y : X) :
f x = f y | hf.apply_eq_of_is_preconnected is_preconnected_univ trivial trivial | lemma | is_locally_constant.apply_eq_of_preconnected_space | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_const [preconnected_space X] {f : X → Y} (hf : is_locally_constant f) (x : X) :
f = function.const X (f x) | funext $ λ y, hf.apply_eq_of_preconnected_space y x | lemma | is_locally_constant.eq_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_const [preconnected_space X] [nonempty Y] {f : X → Y} (hf : is_locally_constant f) :
∃ y, f = function.const X y | begin
casesI is_empty_or_nonempty X,
{ exact ⟨classical.arbitrary Y, funext $ h.elim⟩ },
{ exact ⟨f (classical.arbitrary X), hf.eq_const _⟩ },
end | lemma | is_locally_constant.exists_eq_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"classical.arbitrary",
"is_empty_or_nonempty",
"is_locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_is_const [preconnected_space X] {f : X → Y} :
is_locally_constant f ↔ ∀ x y, f x = f y | ⟨λ h x y, h.apply_eq_of_is_preconnected is_preconnected_univ trivial trivial, of_constant _⟩ | lemma | is_locally_constant.iff_is_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_finite [compact_space X] {f : X → Y} (hf : is_locally_constant f) :
(set.range f).finite | begin
letI : topological_space Y := ⊥, haveI := discrete_topology_bot Y,
rw @iff_continuous X Y ‹_› ‹_› at hf,
exact (is_compact_range hf).finite_of_discrete
end | lemma | is_locally_constant.range_finite | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"compact_space",
"discrete_topology_bot",
"finite",
"is_compact_range",
"is_locally_constant",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one [has_one Y] : is_locally_constant (1 : X → Y) | const 1 | lemma | is_locally_constant.one | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv [has_inv Y] ⦃f : X → Y⦄ (hf : is_locally_constant f) :
is_locally_constant f⁻¹ | hf.comp (λ x, x⁻¹) | lemma | is_locally_constant.inv | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul [has_mul Y] ⦃f g : X → Y⦄ (hf : is_locally_constant f) (hg : is_locally_constant g) :
is_locally_constant (f * g) | hf.comp₂ hg (*) | lemma | is_locally_constant.mul | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div [has_div Y] ⦃f g : X → Y⦄ (hf : is_locally_constant f) (hg : is_locally_constant g) :
is_locally_constant (f / g) | hf.comp₂ hg (/) | lemma | is_locally_constant.div | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc {α β : Type*} (f : X → α) (g : α → β)
(h : is_locally_constant (g ∘ f)) (inj : function.injective g) : is_locally_constant f | begin
rw (is_locally_constant.tfae f).out 0 3,
intros a,
have : f ⁻¹' {a} = (g ∘ f) ⁻¹' { g a },
{ ext x,
simp only [mem_singleton_iff, function.comp_app, mem_preimage],
exact ⟨λ h, by rw h, λ h, inj h⟩ },
rw this,
apply h,
end | lemma | is_locally_constant.desc | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"is_locally_constant.tfae"
] | If a composition of a function `f` followed by an injection `g` is locally
constant, then the locally constant property descends to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_constant_on_connected_components [locally_connected_space X] {f : X → Y}
(h : ∀ x, ∀ y ∈ connected_component x, f y = f x) :
is_locally_constant f | begin
rw iff_exists_open,
exact λ x, ⟨connected_component x, is_open_connected_component, mem_connected_component, h x⟩,
end | lemma | is_locally_constant.of_constant_on_connected_components | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"connected_component",
"is_locally_constant",
"is_open_connected_component",
"locally_connected_space",
"mem_connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_constant_on_preconnected_clopens [locally_connected_space X] {f : X → Y}
(h : ∀ U : set X, is_preconnected U → is_clopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) :
is_locally_constant f | of_constant_on_connected_components (λ x, h (connected_component x)
is_preconnected_connected_component is_clopen_connected_component x mem_connected_component) | lemma | is_locally_constant.of_constant_on_preconnected_clopens | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"connected_component",
"is_clopen",
"is_clopen_connected_component",
"is_locally_constant",
"is_preconnected",
"is_preconnected_connected_component",
"locally_connected_space",
"mem_connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_constant (X Y : Type*) [topological_space X] | (to_fun : X → Y)
(is_locally_constant : is_locally_constant to_fun) | structure | locally_constant | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant",
"topological_space"
] | A (bundled) locally constant function from a topological space `X` to a type `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (f : locally_constant X Y) : f.to_fun = f | rfl | lemma | locally_constant.to_fun_eq_coe | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : X → Y) (h) : ⇑(⟨f, h⟩ : locally_constant X Y) = f | rfl | lemma | locally_constant.coe_mk | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : locally_constant X Y} (h : f = g) (x : X) : f x = g x | congr_arg (λ h : locally_constant X Y, h x) h | theorem | locally_constant.congr_fun | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg (f : locally_constant X Y) {x y : X} (h : x = y) : f x = f y | congr_arg (λ x : X, f x) h | theorem | locally_constant.congr_arg | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (locally_constant X Y) (X → Y) coe_fn | | ⟨f, hf⟩ ⟨g, hg⟩ h := have f = g, from h, by subst f | theorem | locally_constant.coe_injective | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {f g : locally_constant X Y} : (f : X → Y) = g ↔ f = g | coe_injective.eq_iff | theorem | locally_constant.coe_inj | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : locally_constant X Y⦄ (h : ∀ x, f x = g x) : f = g | coe_injective (funext h) | theorem | locally_constant.ext | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : locally_constant X Y} : f = g ↔ ∀ x, f x = g x | ⟨λ h x, h ▸ rfl, λ h, ext h⟩ | theorem | locally_constant.ext_iff | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous f | f.is_locally_constant.continuous | lemma | locally_constant.continuous | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map : C(X, Y) | ⟨f, f.continuous⟩ | def | locally_constant.to_continuous_map | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [] | We can turn a locally-constant function into a bundled `continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_eq_coe : f.to_continuous_map = f | rfl | lemma | locally_constant.to_continuous_map_eq_coe | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_continuous_map : ((f : C(X, Y)) : X → Y) = (f : X → Y) | rfl | lemma | locally_constant.coe_continuous_map | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map_injective :
function.injective (to_continuous_map : locally_constant X Y → C(X, Y)) | λ _ _ h, ext (continuous_map.congr_fun h) | lemma | locally_constant.to_continuous_map_injective | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous_map.congr_fun",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (X : Type*) {Y : Type*} [topological_space X] (y : Y) :
locally_constant X Y | ⟨function.const X y, is_locally_constant.const _⟩ | def | locally_constant.const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_locally_constant.const",
"locally_constant",
"topological_space"
] | The constant locally constant function on `X` with value `y : Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const (y : Y) : (const X y : X → Y) = function.const X y | rfl | lemma | locally_constant.coe_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_clopen {X : Type*} [topological_space X] {U : set X} [∀ x, decidable (x ∈ U)]
(hU : is_clopen U) : locally_constant X (fin 2) | { to_fun := λ x, if x ∈ U then 0 else 1,
is_locally_constant := begin
rw (is_locally_constant.tfae (λ x, if x ∈ U then (0 : fin 2) else 1)).out 0 3,
intros e,
fin_cases e,
{ convert hU.1 using 1,
ext,
simp only [mem_singleton_iff, fin.one_eq_zero_iff, mem_preimage, ite_eq_left_iff,
... | def | locally_constant.of_clopen | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fin.one_eq_zero_iff",
"is_clopen",
"is_closed_compl_iff",
"is_locally_constant",
"is_locally_constant.tfae",
"ite_eq_left_iff",
"locally_constant",
"nat.succ_succ_ne_one",
"topological_space"
] | The locally constant function to `fin 2` associated to a clopen set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_clopen_fiber_zero {X : Type*} [topological_space X] {U : set X}
[∀ x, decidable (x ∈ U)] (hU : is_clopen U) : of_clopen hU ⁻¹' ({0} : set (fin 2)) = U | begin
ext,
simp only [of_clopen, mem_singleton_iff, fin.one_eq_zero_iff, coe_mk, mem_preimage,
ite_eq_left_iff, nat.succ_succ_ne_one],
tauto,
end | lemma | locally_constant.of_clopen_fiber_zero | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fin.one_eq_zero_iff",
"is_clopen",
"ite_eq_left_iff",
"nat.succ_succ_ne_one",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_clopen_fiber_one {X : Type*} [topological_space X] {U : set X}
[∀ x, decidable (x ∈ U)] (hU : is_clopen U) : of_clopen hU ⁻¹' ({1} : set (fin 2)) = Uᶜ | begin
ext,
simp only [of_clopen, mem_singleton_iff, coe_mk, fin.zero_eq_one_iff, mem_preimage,
ite_eq_right_iff, mem_compl_iff, nat.succ_succ_ne_one],
tauto,
end | lemma | locally_constant.of_clopen_fiber_one | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fin.zero_eq_one_iff",
"is_clopen",
"ite_eq_right_iff",
"nat.succ_succ_ne_one",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_constant_eq_of_fiber_zero_eq {X : Type*} [topological_space X]
(f g : locally_constant X (fin 2)) (h : f ⁻¹' ({0} : set (fin 2)) = g ⁻¹' {0}) : f = g | begin
simp only [set.ext_iff, mem_singleton_iff, mem_preimage] at h,
ext1 x,
exact fin.fin_two_eq_of_eq_zero_iff (h x)
end | lemma | locally_constant.locally_constant_eq_of_fiber_zero_eq | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fin.fin_two_eq_of_eq_zero_iff",
"locally_constant",
"set.ext_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_finite [compact_space X] (f : locally_constant X Y) :
(set.range f).finite | f.is_locally_constant.range_finite | lemma | locally_constant.range_finite | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"compact_space",
"finite",
"locally_constant",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_of_is_preconnected (f : locally_constant X Y) {s : set X} (hs : is_preconnected s)
{x y : X} (hx : x ∈ s) (hy : y ∈ s) :
f x = f y | f.is_locally_constant.apply_eq_of_is_preconnected hs hx hy | lemma | locally_constant.apply_eq_of_is_preconnected | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"is_preconnected",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_of_preconnected_space [preconnected_space X] (f : locally_constant X Y) (x y : X) :
f x = f y | f.is_locally_constant.apply_eq_of_is_preconnected is_preconnected_univ trivial trivial | lemma | locally_constant.apply_eq_of_preconnected_space | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_const [preconnected_space X] (f : locally_constant X Y) (x : X) :
f = const X (f x) | ext $ λ y, apply_eq_of_preconnected_space f _ _ | lemma | locally_constant.eq_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_const [preconnected_space X] [nonempty Y] (f : locally_constant X Y) :
∃ y, f = const X y | begin
rcases classical.em (nonempty X) with ⟨⟨x⟩⟩|hX,
{ exact ⟨f x, f.eq_const x⟩ },
{ exact ⟨classical.arbitrary Y, ext $ λ x, (hX ⟨x⟩).elim⟩ }
end | lemma | locally_constant.exists_eq_const | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : Y → Z) : locally_constant X Y → locally_constant X Z | λ g, ⟨f ∘ g, λ s, by { rw set.preimage_comp, apply g.is_locally_constant }⟩ | def | locally_constant.map | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"set.preimage_comp"
] | Push forward of locally constant maps under any map, by post-composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_apply (f : Y → Z) (g : locally_constant X Y) : ⇑(map f g) = f ∘ g | rfl | lemma | locally_constant.map_apply | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : @map X Y Y _ id = id | by { ext, refl } | lemma | locally_constant.map_id | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {Y₁ Y₂ Y₃ : Type*} (g : Y₂ → Y₃) (f : Y₁ → Y₂) :
@map X _ _ _ g ∘ map f = map (g ∘ f) | by { ext, refl } | lemma | locally_constant.map_comp | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip {X α β : Type*} [topological_space X] (f : locally_constant X (α → β)) (a : α) :
locally_constant X β | f.map (λ f, f a) | def | locally_constant.flip | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"locally_constant",
"topological_space"
] | Given a locally constant function to `α → β`, construct a family of locally constant
functions with values in β indexed by α. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unflip {X α β : Type*} [fintype α] [topological_space X] (f : α → locally_constant X β) :
locally_constant X (α → β) | { to_fun := λ x a, f a x,
is_locally_constant := begin
rw (is_locally_constant.tfae (λ x a, f a x)).out 0 3,
intros g,
have : (λ (x : X) (a : α), f a x) ⁻¹' {g} = ⋂ (a : α), (f a) ⁻¹' {g a}, by tidy,
rw this,
apply is_open_Inter,
intros a,
apply (f a).is_locally_constant,
end } | def | locally_constant.unflip | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fintype",
"is_locally_constant",
"is_locally_constant.tfae",
"is_open_Inter",
"locally_constant",
"topological_space"
] | If α is finite, this constructs a locally constant function to `α → β` given a
family of locally constant functions with values in β indexed by α. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unflip_flip {X α β : Type*} [fintype α] [topological_space X]
(f : locally_constant X (α → β)) : unflip f.flip = f | by { ext, refl } | lemma | locally_constant.unflip_flip | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fintype",
"locally_constant",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_unflip {X α β : Type*} [fintype α] [topological_space X]
(f : α → locally_constant X β) : (unflip f).flip = f | by { ext, refl } | lemma | locally_constant.flip_unflip | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"fintype",
"locally_constant",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : X → Y) :
locally_constant Y Z → locally_constant X Z | if hf : continuous f
then λ g, ⟨g ∘ f, g.is_locally_constant.comp_continuous hf⟩
else
begin
by_cases H : nonempty X,
{ introsI g, exact const X (g $ f $ classical.arbitrary X) },
{ intro g, refine ⟨λ x, (H ⟨x⟩).elim, _⟩,
intro s, rw is_open_iff_nhds, intro x, exact (H ⟨x⟩).elim }
end | def | locally_constant.comap | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"classical.arbitrary",
"continuous",
"is_open_iff_nhds",
"locally_constant"
] | Pull back of locally constant maps under any map, by pre-composition.
This definition only makes sense if `f` is continuous,
in which case it sends locally constant functions to their precomposition with `f`.
See also `locally_constant.coe_comap`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (f : X → Y) (g : locally_constant Y Z) (hf : continuous f) :
⇑(comap f g) = g ∘ f | by { rw [comap, dif_pos hf], refl } | lemma | locally_constant.coe_comap | topology.locally_constant | src/topology/locally_constant/basic.lean | [
"topology.subset_properties",
"topology.connected",
"topology.continuous_function.basic",
"algebra.indicator_function",
"tactic.tfae",
"tactic.fin_cases"
] | [
"continuous",
"locally_constant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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