statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_local_min_on.is_local_min (hf : is_local_min_on f s a) (hs : s ∈ 𝓝 a) : is_local_min f a | have 𝓝 a ≤ 𝓟 s, from le_principal_iff.2 hs,
hf.filter_mono $ le_inf le_rfl this | lemma | is_local_min_on.is_local_min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min",
"is_local_min_on",
"le_inf",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.is_local_max (hf : is_local_max_on f s a) (hs : s ∈ 𝓝 a) : is_local_max f a | have 𝓝 a ≤ 𝓟 s, from le_principal_iff.2 hs,
hf.filter_mono $ le_inf le_rfl this | lemma | is_local_max_on.is_local_max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_max_on",
"le_inf",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.is_local_extr (hf : is_local_extr_on f s a) (hs : s ∈ 𝓝 a) :
is_local_extr f a | hf.elim (λ hf, (hf.is_local_min hs).is_extr) (λ hf, (hf.is_local_max hs).is_extr) | lemma | is_local_extr_on.is_local_extr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.is_local_min (hf : is_min_on f s a) (hs : s ∈ 𝓝 a) : is_local_min f a | hf.localize.is_local_min hs | lemma | is_min_on.is_local_min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.is_local_max (hf : is_max_on f s a) (hs : s ∈ 𝓝 a) : is_local_max f a | hf.localize.is_local_max hs | lemma | is_max_on.is_local_max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.is_local_extr (hf : is_extr_on f s a) (hs : s ∈ 𝓝 a) : is_local_extr f a | hf.localize.is_local_extr hs | lemma | is_extr_on.is_local_extr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_extr_on",
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.not_nhds_le_map [topological_space β]
(hf : is_local_min_on f s a) [ne_bot (𝓝[<] (f a))] :
¬𝓝 (f a) ≤ map f (𝓝[s] a) | λ hle,
have ∀ᶠ y in 𝓝[<] (f a), f a ≤ y,
from (eventually_map.2 hf).filter_mono (inf_le_left.trans hle),
let ⟨y, hy⟩ := (this.and self_mem_nhds_within).exists in hy.1.not_lt hy.2 | lemma | is_local_min_on.not_nhds_le_map | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on",
"self_mem_nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.not_nhds_le_map [topological_space β]
(hf : is_local_max_on f s a) [ne_bot (𝓝[>] (f a))] :
¬𝓝 (f a) ≤ map f (𝓝[s] a) | @is_local_min_on.not_nhds_le_map α βᵒᵈ _ _ _ _ _ ‹_› hf ‹_› | lemma | is_local_max_on.not_nhds_le_map | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_local_min_on.not_nhds_le_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.not_nhds_le_map [topological_space β]
(hf : is_local_extr_on f s a) [ne_bot (𝓝[<] (f a))] [ne_bot (𝓝[>] (f a))] :
¬𝓝 (f a) ≤ map f (𝓝[s] a) | hf.elim (λ h, h.not_nhds_le_map) (λ h, h.not_nhds_le_map) | lemma | is_local_extr_on.not_nhds_le_map | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on_const {b : β} : is_local_min_on (λ _, b) s a | is_min_filter_const | lemma | is_local_min_on_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on",
"is_min_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on_const {b : β} : is_local_max_on (λ _, b) s a | is_max_filter_const | lemma | is_local_max_on_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_max_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on_const {b : β} : is_local_extr_on (λ _, b) s a | is_extr_filter_const | lemma | is_local_extr_on_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_extr_filter_const",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_const {b : β} : is_local_min (λ _, b) a | is_min_filter_const | lemma | is_local_min_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min",
"is_min_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_const {b : β} : is_local_max (λ _, b) a | is_max_filter_const | lemma | is_local_max_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_max_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_const {b : β} : is_local_extr (λ _, b) a | is_extr_filter_const | lemma | is_local_extr_const | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_extr_filter_const",
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.comp_mono (hf : is_local_min f a) {g : β → γ} (hg : monotone g) :
is_local_min (g ∘ f) a | hf.comp_mono hg | lemma | is_local_min.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.comp_mono (hf : is_local_max f a) {g : β → γ} (hg : monotone g) :
is_local_max (g ∘ f) a | hf.comp_mono hg | lemma | is_local_max.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.comp_mono (hf : is_local_extr f a) {g : β → γ} (hg : monotone g) :
is_local_extr (g ∘ f) a | hf.comp_mono hg | lemma | is_local_extr.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.comp_antitone (hf : is_local_min f a) {g : β → γ}
(hg : antitone g) :
is_local_max (g ∘ f) a | hf.comp_antitone hg | lemma | is_local_min.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.comp_antitone (hf : is_local_max f a) {g : β → γ}
(hg : antitone g) :
is_local_min (g ∘ f) a | hf.comp_antitone hg | lemma | is_local_max.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.comp_antitone (hf : is_local_extr f a) {g : β → γ}
(hg : antitone g) :
is_local_extr (g ∘ f) a | hf.comp_antitone hg | lemma | is_local_extr.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.comp_mono (hf : is_local_min_on f s a) {g : β → γ} (hg : monotone g) :
is_local_min_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_local_min_on.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.comp_mono (hf : is_local_max_on f s a) {g : β → γ} (hg : monotone g) :
is_local_max_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_local_max_on.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.comp_mono (hf : is_local_extr_on f s a) {g : β → γ} (hg : monotone g) :
is_local_extr_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_local_extr_on.comp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.comp_antitone (hf : is_local_min_on f s a) {g : β → γ}
(hg : antitone g) :
is_local_max_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_local_min_on.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.comp_antitone (hf : is_local_max_on f s a) {g : β → γ}
(hg : antitone g) :
is_local_min_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_local_max_on.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.comp_antitone (hf : is_local_extr_on f s a) {g : β → γ}
(hg : antitone g) :
is_local_extr_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_local_extr_on.comp_antitone | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"antitone",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_local_min f a) {g : α → γ} (hg : is_local_min g a) :
is_local_min (λ x, op (f x) (g x)) a | hf.bicomp_mono hop hg | lemma | is_local_min.bicomp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_local_max f a) {g : α → γ} (hg : is_local_max g a) :
is_local_max (λ x, op (f x) (g x)) a | hf.bicomp_mono hop hg | lemma | is_local_max.bicomp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_local_min_on f s a) {g : α → γ} (hg : is_local_min_on g s a) :
is_local_min_on (λ x, op (f x) (g x)) s a | hf.bicomp_mono hop hg | lemma | is_local_min_on.bicomp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_local_max_on f s a) {g : α → γ} (hg : is_local_max_on g s a) :
is_local_max_on (λ x, op (f x) (g x)) s a | hf.bicomp_mono hop hg | lemma | is_local_max_on.bicomp_mono | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.comp_continuous [topological_space δ] {g : δ → α} {b : δ}
(hf : is_local_min f (g b)) (hg : continuous_at g b) :
is_local_min (f ∘ g) b | hg hf | lemma | is_local_min.comp_continuous | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_at",
"is_local_min",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.comp_continuous [topological_space δ] {g : δ → α} {b : δ}
(hf : is_local_max f (g b)) (hg : continuous_at g b) :
is_local_max (f ∘ g) b | hg hf | lemma | is_local_max.comp_continuous | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_at",
"is_local_max",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.comp_continuous [topological_space δ] {g : δ → α} {b : δ}
(hf : is_local_extr f (g b)) (hg : continuous_at g b) :
is_local_extr (f ∘ g) b | hf.comp_tendsto hg | lemma | is_local_extr.comp_continuous | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_at",
"is_local_extr",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.comp_continuous_on [topological_space δ] {s : set δ} {g : δ → α} {b : δ}
(hf : is_local_min f (g b)) (hg : continuous_on g s) (hb : b ∈ s) :
is_local_min_on (f ∘ g) s b | hf.comp_tendsto (hg b hb) | lemma | is_local_min.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"is_local_min",
"is_local_min_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.comp_continuous_on [topological_space δ] {s : set δ} {g : δ → α} {b : δ}
(hf : is_local_max f (g b)) (hg : continuous_on g s) (hb : b ∈ s) :
is_local_max_on (f ∘ g) s b | hf.comp_tendsto (hg b hb) | lemma | is_local_max.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"is_local_max",
"is_local_max_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.comp_continuous_on [topological_space δ] {s : set δ} (g : δ → α) {b : δ}
(hf : is_local_extr f (g b)) (hg : continuous_on g s) (hb : b ∈ s) :
is_local_extr_on (f ∘ g) s b | hf.elim (λ hf, (hf.comp_continuous_on hg hb).is_extr)
(λ hf, (is_local_max.comp_continuous_on hf hg hb).is_extr) | lemma | is_local_extr.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"is_local_extr",
"is_local_extr_on",
"is_local_max.comp_continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.comp_continuous_on [topological_space δ] {t : set α} {s : set δ} {g : δ → α}
{b : δ} (hf : is_local_min_on f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : continuous_on g s)
(hb : b ∈ s) :
is_local_min_on (f ∘ g) s b | hf.comp_tendsto (tendsto_nhds_within_mono_right (image_subset_iff.mpr hst)
(continuous_within_at.tendsto_nhds_within_image (hg b hb))) | lemma | is_local_min_on.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"continuous_within_at.tendsto_nhds_within_image",
"is_local_min_on",
"tendsto_nhds_within_mono_right",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.comp_continuous_on [topological_space δ] {t : set α} {s : set δ} {g : δ → α}
{b : δ} (hf : is_local_max_on f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : continuous_on g s)
(hb : b ∈ s) :
is_local_max_on (f ∘ g) s b | hf.comp_tendsto (tendsto_nhds_within_mono_right (image_subset_iff.mpr hst)
(continuous_within_at.tendsto_nhds_within_image (hg b hb))) | lemma | is_local_max_on.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"continuous_within_at.tendsto_nhds_within_image",
"is_local_max_on",
"tendsto_nhds_within_mono_right",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.comp_continuous_on [topological_space δ] {t : set α} {s : set δ} (g : δ → α)
{b : δ} (hf : is_local_extr_on f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : continuous_on g s)
(hb : b ∈ s) :
is_local_extr_on (f ∘ g) s b | hf.elim (λ hf, (hf.comp_continuous_on hst hg hb).is_extr)
(λ hf, (is_local_max_on.comp_continuous_on hf hst hg hb).is_extr) | lemma | is_local_extr_on.comp_continuous_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"continuous_on",
"is_local_extr_on",
"is_local_max_on.comp_continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.add (hf : is_local_min f a) (hg : is_local_min g a) :
is_local_min (λ x, f x + g x) a | hf.add hg | lemma | is_local_min.add | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.add (hf : is_local_max f a) (hg : is_local_max g a) :
is_local_max (λ x, f x + g x) a | hf.add hg | lemma | is_local_max.add | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.add (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :
is_local_min_on (λ x, f x + g x) s a | hf.add hg | lemma | is_local_min_on.add | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.add (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :
is_local_max_on (λ x, f x + g x) s a | hf.add hg | lemma | is_local_max_on.add | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.neg (hf : is_local_min f a) : is_local_max (λ x, -f x) a | hf.neg | lemma | is_local_min.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.neg (hf : is_local_max f a) : is_local_min (λ x, -f x) a | hf.neg | lemma | is_local_max.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.neg (hf : is_local_extr f a) : is_local_extr (λ x, -f x) a | hf.neg | lemma | is_local_extr.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.neg (hf : is_local_min_on f s a) : is_local_max_on (λ x, -f x) s a | hf.neg | lemma | is_local_min_on.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.neg (hf : is_local_max_on f s a) : is_local_min_on (λ x, -f x) s a | hf.neg | lemma | is_local_max_on.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.neg (hf : is_local_extr_on f s a) : is_local_extr_on (λ x, -f x) s a | hf.neg | lemma | is_local_extr_on.neg | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.sub (hf : is_local_min f a) (hg : is_local_max g a) :
is_local_min (λ x, f x - g x) a | hf.sub hg | lemma | is_local_min.sub | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.sub (hf : is_local_max f a) (hg : is_local_min g a) :
is_local_max (λ x, f x - g x) a | hf.sub hg | lemma | is_local_max.sub | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.sub (hf : is_local_min_on f s a) (hg : is_local_max_on g s a) :
is_local_min_on (λ x, f x - g x) s a | hf.sub hg | lemma | is_local_min_on.sub | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.sub (hf : is_local_max_on f s a) (hg : is_local_min_on g s a) :
is_local_max_on (λ x, f x - g x) s a | hf.sub hg | lemma | is_local_max_on.sub | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.sup (hf : is_local_min f a) (hg : is_local_min g a) :
is_local_min (λ x, f x ⊔ g x) a | hf.sup hg | lemma | is_local_min.sup | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.sup (hf : is_local_max f a) (hg : is_local_max g a) :
is_local_max (λ x, f x ⊔ g x) a | hf.sup hg | lemma | is_local_max.sup | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.sup (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :
is_local_min_on (λ x, f x ⊔ g x) s a | hf.sup hg | lemma | is_local_min_on.sup | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.sup (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :
is_local_max_on (λ x, f x ⊔ g x) s a | hf.sup hg | lemma | is_local_max_on.sup | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.inf (hf : is_local_min f a) (hg : is_local_min g a) :
is_local_min (λ x, f x ⊓ g x) a | hf.inf hg | lemma | is_local_min.inf | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.inf (hf : is_local_max f a) (hg : is_local_max g a) :
is_local_max (λ x, f x ⊓ g x) a | hf.inf hg | lemma | is_local_max.inf | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.inf (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :
is_local_min_on (λ x, f x ⊓ g x) s a | hf.inf hg | lemma | is_local_min_on.inf | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.inf (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :
is_local_max_on (λ x, f x ⊓ g x) s a | hf.inf hg | lemma | is_local_max_on.inf | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.min (hf : is_local_min f a) (hg : is_local_min g a) :
is_local_min (λ x, min (f x) (g x)) a | hf.min hg | lemma | is_local_min.min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.min (hf : is_local_max f a) (hg : is_local_max g a) :
is_local_max (λ x, min (f x) (g x)) a | hf.min hg | lemma | is_local_max.min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.min (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :
is_local_min_on (λ x, min (f x) (g x)) s a | hf.min hg | lemma | is_local_min_on.min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.min (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :
is_local_max_on (λ x, min (f x) (g x)) s a | hf.min hg | lemma | is_local_max_on.min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.max (hf : is_local_min f a) (hg : is_local_min g a) :
is_local_min (λ x, max (f x) (g x)) a | hf.max hg | lemma | is_local_min.max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.max (hf : is_local_max f a) (hg : is_local_max g a) :
is_local_max (λ x, max (f x) (g x)) a | hf.max hg | lemma | is_local_max.max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.max (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :
is_local_min_on (λ x, max (f x) (g x)) s a | hf.max hg | lemma | is_local_min_on.max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.max (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :
is_local_max_on (λ x, max (f x) (g x)) s a | hf.max hg | lemma | is_local_max_on.max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_local_max_on {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝[s] a] f)
(hfga : f a = g a) (h : is_local_max_on f s a) : is_local_max_on g s a | hle.is_max_filter hfga h | lemma | filter.eventually_le.is_local_max_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.congr {f g : α → β} {a : α} (h : is_local_max_on f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : is_local_max_on g s a | h.congr heq $ heq.eq_of_nhds_within hmem | lemma | is_local_max_on.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_max_on_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : is_local_max_on f s a ↔ is_local_max_on g s a | heq.is_max_filter_iff $ heq.eq_of_nhds_within hmem | lemma | filter.eventually_eq.is_local_max_on_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_local_min_on {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝[s] a] g)
(hfga : f a = g a) (h : is_local_min_on f s a) : is_local_min_on g s a | hle.is_min_filter hfga h | lemma | filter.eventually_le.is_local_min_on | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.congr {f g : α → β} {a : α} (h : is_local_min_on f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : is_local_min_on g s a | h.congr heq $ heq.eq_of_nhds_within hmem | lemma | is_local_min_on.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_min_on_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : is_local_min_on f s a ↔ is_local_min_on g s a | heq.is_min_filter_iff $ heq.eq_of_nhds_within hmem | lemma | filter.eventually_eq.is_local_min_on_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.congr {f g : α → β} {a : α} (h : is_local_extr_on f s a)
(heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : is_local_extr_on g s a | h.congr heq $ heq.eq_of_nhds_within hmem | lemma | is_local_extr_on.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_extr_on_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : is_local_extr_on f s a ↔ is_local_extr_on g s a | heq.is_extr_filter_iff $ heq.eq_of_nhds_within hmem | lemma | filter.eventually_eq.is_local_extr_on_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_local_max {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝 a] f) (hfga : f a = g a)
(h : is_local_max f a) : is_local_max g a | hle.is_max_filter hfga h | lemma | filter.eventually_le.is_local_max | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max.congr {f g : α → β} {a : α} (h : is_local_max f a) (heq : f =ᶠ[𝓝 a] g) :
is_local_max g a | h.congr heq heq.eq_of_nhds | lemma | is_local_max.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_max_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
is_local_max f a ↔ is_local_max g a | heq.is_max_filter_iff heq.eq_of_nhds | lemma | filter.eventually_eq.is_local_max_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_local_min {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝 a] g) (hfga : f a = g a)
(h : is_local_min f a) : is_local_min g a | hle.is_min_filter hfga h | lemma | filter.eventually_le.is_local_min | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min.congr {f g : α → β} {a : α} (h : is_local_min f a) (heq : f =ᶠ[𝓝 a] g) :
is_local_min g a | h.congr heq heq.eq_of_nhds | lemma | is_local_min.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_min_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
is_local_min f a ↔ is_local_min g a | heq.is_min_filter_iff heq.eq_of_nhds | lemma | filter.eventually_eq.is_local_min_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr.congr {f g : α → β} {a : α} (h : is_local_extr f a) (heq : f =ᶠ[𝓝 a] g) :
is_local_extr g a | h.congr heq heq.eq_of_nhds | lemma | is_local_extr.congr | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_local_extr_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
is_local_extr f a ↔ is_local_extr g a | heq.is_extr_filter_iff heq.eq_of_nhds | lemma | filter.eventually_eq.is_local_extr_iff | topology | src/topology/local_extr.lean | [
"order.filter.extr",
"topology.continuous_on"
] | [
"is_local_extr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_homeomorph (α : Type*) (β : Type*) [topological_space α] [topological_space β]
extends local_equiv α β | (open_source : is_open source)
(open_target : is_open target)
(continuous_to_fun : continuous_on to_fun source)
(continuous_inv_fun : continuous_on inv_fun target) | structure | local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on",
"inv_fun",
"is_open",
"local_equiv",
"topological_space"
] | local homeomorphisms, defined on open subsets of the space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm : local_homeomorph β α | { open_source := e.open_target,
open_target := e.open_source,
continuous_to_fun := e.continuous_inv_fun,
continuous_inv_fun := e.continuous_to_fun,
..e.to_local_equiv.symm } | def | local_homeomorph.symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | The inverse of a local homeomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (e : local_homeomorph α β) : α → β | e | def | local_homeomorph.simps.apply | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (e : local_homeomorph α β) : β → α | e.symm
initialize_simps_projections local_homeomorph
(to_local_equiv_to_fun → apply, to_local_equiv_inv_fun → symm_apply,
to_local_equiv_source → source, to_local_equiv_target → target, -to_local_equiv) | def | local_homeomorph.simps.symm_apply | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on : continuous_on e e.source | e.continuous_to_fun | lemma | local_homeomorph.continuous_on | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_symm : continuous_on e.symm e.target | e.continuous_inv_fun | lemma | local_homeomorph.continuous_on_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (e : local_equiv α β) (a b c d) :
(local_homeomorph.mk e a b c d : α → β) = e | rfl | lemma | local_homeomorph.mk_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe_symm (e : local_equiv α β) (a b c d) :
((local_homeomorph.mk e a b c d).symm : β → α) = e.symm | rfl | lemma | local_homeomorph.mk_coe_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_equiv_injective : injective (to_local_equiv : local_homeomorph α β → local_equiv α β) | | ⟨e, h₁, h₂, h₃, h₄⟩ ⟨e', h₁', h₂', h₃', h₄'⟩ rfl := rfl | lemma | local_homeomorph.to_local_equiv_injective | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (e : local_homeomorph α β) : e.to_fun = e | rfl | lemma | local_homeomorph.to_fun_eq_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_eq_coe (e : local_homeomorph α β) : e.inv_fun = e.symm | rfl | lemma | local_homeomorph.inv_fun_eq_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : (e.to_local_equiv : α → β) = e | rfl | lemma | local_homeomorph.coe_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_symm : (e.to_local_equiv.symm : β → α) = e.symm | rfl | lemma | local_homeomorph.coe_coe_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_apply {x : α} {y : β} (hx : x ∈ e.source) (hy : y ∈ e.target) :
x = e.symm y ↔ e x = y | e.to_local_equiv.eq_symm_apply hx hy | lemma | local_homeomorph.eq_symm_apply | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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