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|---|---|---|---|---|---|---|---|---|---|---|
proj (i : ι) (p : path.homotopic.quotient as bs) : path.homotopic.quotient (as i) (bs i) | p.map_fn ⟨_, continuous_apply i⟩ | def | path.homotopic.proj | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"continuous_apply",
"path.homotopic.quotient"
] | Abbreviation for projection onto the ith coordinate | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_pi (i : ι) (paths : Π i, path.homotopic.quotient (as i) (bs i)) :
proj i (pi paths) = paths i | begin
apply quotient.induction_on_pi paths,
intro, unfold proj,
rw [pi_lift, ← path.homotopic.map_lift],
congr, ext, refl,
end | lemma | path.homotopic.proj_pi | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.map_lift",
"path.homotopic.quotient",
"quotient.induction_on_pi"
] | Lemmas showing projection is the inverse of pi | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_proj (p : path.homotopic.quotient as bs) : pi (λ i, proj i p) = p | begin
apply quotient.induction_on p,
intro, unfold proj,
simp_rw ← path.homotopic.map_lift,
rw pi_lift,
congr, ext, refl,
end | lemma | path.homotopic.pi_proj | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.map_lift",
"path.homotopic.quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_homotopy (h₁ : path.homotopy p₁ p₁') (h₂ : path.homotopy p₂ p₂') :
path.homotopy (p₁.prod p₂) (p₁'.prod p₂') | continuous_map.homotopy_rel.prod h₁ h₂ | def | path.homotopic.prod_homotopy | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"continuous_map.homotopy_rel.prod",
"path.homotopy"
] | The product of homotopies h₁ and h₂.
This is `homotopy_rel.prod` specialized for path homotopies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (q₁ : path.homotopic.quotient a₁ a₂) (q₂ : path.homotopic.quotient b₁ b₂) :
path.homotopic.quotient (a₁, b₁) (a₂, b₂) | quotient.map₂ path.prod (λ p₁ p₁' h₁ p₂ p₂' h₂, nonempty.map2 prod_homotopy h₁ h₂) q₁ q₂ | def | path.homotopic.prod | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"nonempty.map2",
"path.homotopic.quotient",
"path.prod",
"quotient.map₂"
] | The product of path classes q₁ and q₂. This is `path.prod` descended to the quotient | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_lift : prod ⟦p₁⟧ ⟦p₂⟧ = ⟦p₁.prod p₂⟧ | rfl | lemma | path.homotopic.prod_lift | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_eq_prod_comp : (prod q₁ q₂) ⬝ (prod r₁ r₂) = prod (q₁ ⬝ r₁) (q₂ ⬝ r₂) | begin
apply quotient.induction_on₂ q₁ q₂,
apply quotient.induction_on₂ r₁ r₂,
intros,
simp only [prod_lift, ← path.homotopic.comp_lift, path.trans_prod_eq_prod_trans],
end | lemma | path.homotopic.comp_prod_eq_prod_comp | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.comp_lift",
"path.trans_prod_eq_prod_trans"
] | Products commute with path composition.
This is `trans_prod_eq_prod_trans` descended to the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_left (p : path.homotopic.quotient c₁ c₂) : path.homotopic.quotient c₁.1 c₂.1 | p.map_fn ⟨_, continuous_fst⟩ | def | path.homotopic.proj_left | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.quotient"
] | Abbreviation for projection onto the left coordinate of a path class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_right (p : path.homotopic.quotient c₁ c₂) : path.homotopic.quotient c₁.2 c₂.2 | p.map_fn ⟨_, continuous_snd⟩ | def | path.homotopic.proj_right | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.quotient"
] | Abbreviation for projection onto the right coordinate of a path class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_left_prod : proj_left (prod q₁ q₂) = q₁ | begin
apply quotient.induction_on₂ q₁ q₂,
intros p₁ p₂,
unfold proj_left,
rw [prod_lift, ← path.homotopic.map_lift],
congr, ext, refl,
end | lemma | path.homotopic.proj_left_prod | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.map_lift"
] | Lemmas showing projection is the inverse of product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj_right_prod : proj_right (prod q₁ q₂) = q₂ | begin
apply quotient.induction_on₂ q₁ q₂,
intros p₁ p₂,
unfold proj_right,
rw [prod_lift, ← path.homotopic.map_lift],
congr, ext, refl,
end | lemma | path.homotopic.proj_right_prod | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.map_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_proj_left_proj_right (p : path.homotopic.quotient (a₁, b₁) (a₂, b₂))
: prod (proj_left p) (proj_right p) = p | begin
apply quotient.induction_on p,
intro p',
unfold proj_left, unfold proj_right,
simp only [← path.homotopic.map_lift, prod_lift],
congr, ext; refl,
end | lemma | path.homotopic.prod_proj_left_proj_right | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.map_lift",
"path.homotopic.quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_right_to_Ico_mod : continuous_within_at (to_Ico_mod hp a) (Ici x) x | begin
intros s h,
rw [filter.mem_map, mem_nhds_within_iff_exists_mem_nhds_inter],
haveI : nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩,
simp_rw mem_nhds_iff_exists_Ioo_subset at h ⊢,
obtain ⟨l, u, hxI, hIs⟩ := h,
let d := to_Ico_div hp a x • p,
have hd := to_Ico_mod_mem_Ico hp a x,
simp_rw [subset_def, mem_inter_if... | lemma | continuous_right_to_Ico_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_within_at",
"filter.mem_map",
"lt_min_iff",
"mem_nhds_iff_exists_Ioo_subset",
"mem_nhds_within_iff_exists_mem_nhds_inter",
"nontrivial",
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_eq_self",
"to_Ico_mod_mem_Ico",
"to_Ico_mod_sub_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_left_to_Ioc_mod : continuous_within_at (to_Ioc_mod hp a) (Iic x) x | begin
rw (funext (λ y, eq.trans (by rw neg_neg) $ to_Ioc_mod_neg _ _ _) :
to_Ioc_mod hp a = (λ x, p - x) ∘ to_Ico_mod hp (-a) ∘ has_neg.neg),
exact ((continuous_sub_left _).continuous_at.comp_continuous_within_at $
(continuous_right_to_Ico_mod _ _ _).comp continuous_neg.continuous_within_at $ λ y, neg_le_ne... | lemma | continuous_left_to_Ioc_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_at.comp_continuous_within_at",
"continuous_right_to_Ico_mod",
"continuous_within_at",
"to_Ico_mod",
"to_Ioc_mod",
"to_Ioc_mod_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eventually_eq_to_Ioc_mod : to_Ico_mod hp a =ᶠ[𝓝 x] to_Ioc_mod hp a | is_open.mem_nhds (by {rw Ico_eq_locus_Ioc_eq_Union_Ioo, exact is_open_Union (λ i, is_open_Ioo)}) $
(not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp).1 $ not_modeq_iff_ne_mod_zmultiples.2 hx | lemma | to_Ico_mod_eventually_eq_to_Ioc_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"Ico_eq_locus_Ioc_eq_Union_Ioo",
"is_open.mem_nhds",
"is_open_Ioo",
"is_open_Union",
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_to_Ico_mod : continuous_at (to_Ico_mod hp a) x | let h := to_Ico_mod_eventually_eq_to_Ioc_mod hp a hx in continuous_at_iff_continuous_left_right.2 $
⟨(continuous_left_to_Ioc_mod hp a x).congr_of_eventually_eq
(h.filter_mono nhds_within_le_nhds) h.eq_of_nhds, continuous_right_to_Ico_mod hp a x⟩ | lemma | continuous_at_to_Ico_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_at",
"continuous_left_to_Ioc_mod",
"continuous_right_to_Ico_mod",
"nhds_within_le_nhds",
"to_Ico_mod",
"to_Ico_mod_eventually_eq_to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_to_Ioc_mod : continuous_at (to_Ioc_mod hp a) x | let h := to_Ico_mod_eventually_eq_to_Ioc_mod hp a hx in continuous_at_iff_continuous_left_right.2 $
⟨continuous_left_to_Ioc_mod hp a x, (continuous_right_to_Ico_mod hp a x).congr_of_eventually_eq
(h.symm.filter_mono nhds_within_le_nhds) h.symm.eq_of_nhds⟩ | lemma | continuous_at_to_Ioc_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_at",
"continuous_right_to_Ico_mod",
"nhds_within_le_nhds",
"to_Ico_mod_eventually_eq_to_Ioc_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_circle [linear_ordered_add_comm_group 𝕜] [topological_space 𝕜] [order_topology 𝕜] (p : 𝕜) | 𝕜 ⧸ zmultiples p | def | add_circle | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"linear_ordered_add_comm_group",
"order_topology",
"topological_space"
] | The "additive circle": `𝕜 ⧸ (ℤ ∙ p)`. See also `circle` and `real.angle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_nsmul {n : ℕ} {x : 𝕜} : (↑(n • x) : add_circle p) = n • (x : add_circle p) | rfl | lemma | add_circle.coe_nsmul | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul {n : ℤ} {x : 𝕜} : (↑(n • x) : add_circle p) = n • (x : add_circle p) | rfl | lemma | add_circle.coe_zsmul | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : 𝕜) : (↑(x + y) : add_circle p) = (x : add_circle p) + (y : add_circle p) | rfl | lemma | add_circle.coe_add | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (x y : 𝕜) : (↑(x - y) : add_circle p) = (x : add_circle p) - (y : add_circle p) | rfl | lemma | add_circle.coe_sub | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg {x : 𝕜} : (↑(-x) : add_circle p) = -(x : add_circle p) | rfl | lemma | add_circle.coe_neg | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero_iff {x : 𝕜} : (x : add_circle p) = 0 ↔ ∃ (n : ℤ), n • p = x | by simp [add_subgroup.mem_zmultiples_iff] | lemma | add_circle.coe_eq_zero_iff | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero_of_pos_iff (hp : 0 < p) {x : 𝕜} (hx : 0 < x) :
(x : add_circle p) = 0 ↔ ∃ (n : ℕ), n • p = x | begin
rw coe_eq_zero_iff,
split;
rintros ⟨n, rfl⟩,
{ replace hx : 0 < n,
{ contrapose! hx,
simpa only [←neg_nonneg, ←zsmul_neg, zsmul_neg'] using zsmul_nonneg hp.le (neg_nonneg.2 hx) },
exact ⟨n.to_nat, by rw [← coe_nat_zsmul, int.to_nat_of_nonneg hx.le]⟩, },
{ exact ⟨(n : ℤ), by simp⟩, },
end | lemma | add_circle.coe_eq_zero_of_pos_iff | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"int.to_nat_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_period : (p : add_circle p) = 0 | (quotient_add_group.eq_zero_iff p).2 $ mem_zmultiples p | lemma | add_circle.coe_period | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_period (x : 𝕜) : ((x + p : 𝕜) : add_circle p) = x | by rw [coe_add, ←eq_sub_iff_add_eq', sub_self, coe_period] | lemma | add_circle.coe_add_period | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mk' :
continuous (quotient_add_group.mk' (zmultiples p) : 𝕜 → add_circle p) | continuous_coinduced_rng | lemma | add_circle.continuous_mk' | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"continuous",
"continuous_coinduced_rng"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_Ico : add_circle p ≃ Ico a (a + p) | quotient_add_group.equiv_Ico_mod hp.out a | def | add_circle.equiv_Ico | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"quotient_add_group.equiv_Ico_mod"
] | The equivalence between `add_circle p` and the half-open interval `[a, a + p)`, whose inverse
is the natural quotient map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_Ioc : add_circle p ≃ Ioc a (a + p) | quotient_add_group.equiv_Ioc_mod hp.out a | def | add_circle.equiv_Ioc | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"quotient_add_group.equiv_Ioc_mod"
] | The equivalence between `add_circle p` and the half-open interval `(a, a + p]`, whose inverse
is the natural quotient map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_Ico (f : 𝕜 → B) : add_circle p → B | restrict _ f ∘ add_circle.equiv_Ico p a | def | add_circle.lift_Ico | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"add_circle.equiv_Ico"
] | Given a function on `𝕜`, return the unique function on `add_circle p` agreeing with `f` on
`[a, a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_Ioc (f : 𝕜 → B) : add_circle p → B | restrict _ f ∘ add_circle.equiv_Ioc p a | def | add_circle.lift_Ioc | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"add_circle.equiv_Ioc"
] | Given a function on `𝕜`, return the unique function on `add_circle p` agreeing with `f` on
`(a, a + p]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_eq_coe_iff_of_mem_Ico {x y : 𝕜}
(hx : x ∈ Ico a (a + p)) (hy : y ∈ Ico a (a + p)) : (x : add_circle p) = y ↔ x = y | begin
refine ⟨λ h, _, by tauto⟩,
suffices : (⟨x, hx⟩ : Ico a (a + p)) = ⟨y, hy⟩, by exact subtype.mk.inj this,
apply_fun equiv_Ico p a at h,
rw [←(equiv_Ico p a).right_inv ⟨x, hx⟩, ←(equiv_Ico p a).right_inv ⟨y, hy⟩],
exact h
end | lemma | add_circle.coe_eq_coe_iff_of_mem_Ico | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Ico_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico a (a + p)) : lift_Ico p a f ↑x = f x | begin
have : (equiv_Ico p a) x = ⟨x, hx⟩,
{ rw equiv.apply_eq_iff_eq_symm_apply,
refl, },
rw [lift_Ico, comp_apply, this],
refl,
end | lemma | add_circle.lift_Ico_coe_apply | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"equiv.apply_eq_iff_eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Ioc_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ioc a (a + p)) : lift_Ioc p a f ↑x = f x | begin
have : (equiv_Ioc p a) x = ⟨x, hx⟩,
{ rw equiv.apply_eq_iff_eq_symm_apply,
refl, },
rw [lift_Ioc, comp_apply, this],
refl,
end | lemma | add_circle.lift_Ioc_coe_apply | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"equiv.apply_eq_iff_eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_equiv_Ico_symm : continuous (equiv_Ico p a).symm | continuous_quotient_mk.comp continuous_subtype_coe | lemma | add_circle.continuous_equiv_Ico_symm | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous",
"continuous_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_equiv_Ioc_symm : continuous (equiv_Ioc p a).symm | continuous_quotient_mk.comp continuous_subtype_coe | lemma | add_circle.continuous_equiv_Ioc_symm | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous",
"continuous_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_equiv_Ico : continuous_at (equiv_Ico p a) x | begin
induction x using quotient_add_group.induction_on',
rw [continuous_at, filter.tendsto, quotient_add_group.nhds_eq, filter.map_map],
exact (continuous_at_to_Ico_mod hp.out a hx).cod_restrict _,
end | lemma | add_circle.continuous_at_equiv_Ico | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_at",
"continuous_at_to_Ico_mod",
"filter.map_map",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_equiv_Ioc : continuous_at (equiv_Ioc p a) x | begin
induction x using quotient_add_group.induction_on',
rw [continuous_at, filter.tendsto, quotient_add_group.nhds_eq, filter.map_map],
exact (continuous_at_to_Ioc_mod hp.out a hx).cod_restrict _,
end | lemma | add_circle.continuous_at_equiv_Ioc | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous_at",
"continuous_at_to_Ioc_mod",
"filter.map_map",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_image_Ico_eq : (coe : 𝕜 → add_circle p) '' Ico a (a + p) = univ | by { rw image_eq_range, exact (equiv_Ico p a).symm.range_eq_univ } | lemma | add_circle.coe_image_Ico_eq | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | The image of the closed-open interval `[a, a + p)` under the quotient map `𝕜 → add_circle p` is
the entire space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_image_Ioc_eq : (coe : 𝕜 → add_circle p) '' Ioc a (a + p) = univ | by { rw image_eq_range, exact (equiv_Ioc p a).symm.range_eq_univ } | lemma | add_circle.coe_image_Ioc_eq | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | The image of the closed-open interval `[a, a + p)` under the quotient map `𝕜 → add_circle p` is
the entire space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_image_Icc_eq : (coe : 𝕜 → add_circle p) '' Icc a (a + p) = univ | eq_top_mono (image_subset _ Ico_subset_Icc_self) $ coe_image_Ico_eq _ _ | lemma | add_circle.coe_image_Icc_eq | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"eq_top_mono"
] | The image of the closed interval `[0, p]` under the quotient map `𝕜 → add_circle p` is the
entire space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_add_circle (hp : p ≠ 0) (hq : q ≠ 0) : add_circle p ≃+ add_circle q | quotient_add_group.congr _ _ (add_aut.mul_right $ (units.mk0 p hp)⁻¹ * units.mk0 q hq) $
by rw [add_monoid_hom.map_zmultiples, add_monoid_hom.coe_coe, add_aut.mul_right_apply,
units.coe_mul, units.coe_mk0, units.coe_inv, units.coe_mk0, mul_inv_cancel_left₀ hp] | def | add_circle.equiv_add_circle | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_aut.mul_right",
"add_aut.mul_right_apply",
"add_circle",
"mul_inv_cancel_left₀",
"units.coe_inv",
"units.coe_mk0",
"units.coe_mul",
"units.mk0"
] | The rescaling equivalence between additive circles with different periods. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_add_circle_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
equiv_add_circle p q hp hq (x : 𝕜) = (x * (p⁻¹ * q) : 𝕜) | rfl | lemma | add_circle.equiv_add_circle_apply_mk | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_add_circle_symm_apply_mk (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜) :
(equiv_add_circle p q hp hq).symm (x : 𝕜) = (x * (q⁻¹ * p) : 𝕜) | rfl | lemma | add_circle.equiv_add_circle_symm_apply_mk | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_equiv_Ico_mk_apply (x : 𝕜) :
(equiv_Ico p 0 $ quotient_add_group.mk x : 𝕜) = int.fract (x / p) * p | to_Ico_mod_eq_fract_mul _ x | lemma | add_circle.coe_equiv_Ico_mk_apply | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"int.fract",
"to_Ico_mod_eq_fract_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_order_of_period_div {n : ℕ} (h : 0 < n) : add_order_of ((p / n : 𝕜) : add_circle p) = n | begin
rw [add_order_of_eq_iff h],
replace h : 0 < (n : 𝕜) := nat.cast_pos.2 h,
refine ⟨_, λ m hn h0, _⟩; simp only [ne, ← coe_nsmul, nsmul_eq_mul],
{ rw [mul_div_cancel' _ h.ne', coe_period] },
rw coe_eq_zero_of_pos_iff p hp.out (mul_pos (nat.cast_pos.2 h0) $ div_pos hp.out h),
rintro ⟨k, hk⟩,
rw [mul_di... | lemma | add_circle.add_order_of_period_div | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"div_pos",
"eq_div_iff",
"mul_comm",
"mul_div",
"mul_div_cancel'",
"mul_left_injective₀",
"mul_right_comm",
"nat.cast_inj",
"nat.cast_mul",
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_mul_add_order_of_div_eq {n : ℕ} (m : ℕ) (hn : 0 < n) :
m.gcd n * add_order_of (↑(↑m / ↑n * p) : add_circle p) = n | begin
rw [mul_comm_div, ← nsmul_eq_mul, coe_nsmul, add_order_of_nsmul''],
{ rw [add_order_of_period_div hn, nat.gcd_comm, nat.mul_div_cancel'],
exacts [n.gcd_dvd_left m, hp] },
{ rw [← add_order_of_pos_iff, add_order_of_period_div hn], exacts [hn, hp] },
end | lemma | add_circle.gcd_mul_add_order_of_div_eq | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"mul_comm_div",
"nat.gcd_comm",
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_order_of_div_of_gcd_eq_one {m n : ℕ} (hn : 0 < n) (h : m.gcd n = 1) :
add_order_of (↑(↑m / ↑n * p) : add_circle p) = n | by { convert gcd_mul_add_order_of_div_eq p m hn, rw [h, one_mul] } | lemma | add_circle.add_order_of_div_of_gcd_eq_one | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_order_of_div_of_gcd_eq_one' {m : ℤ} {n : ℕ} (hn : 0 < n) (h : m.nat_abs.gcd n = 1) :
add_order_of (↑(↑m / ↑n * p) : add_circle p) = n | begin
induction m,
{ simp only [int.of_nat_eq_coe, int.cast_coe_nat, int.nat_abs_of_nat] at h ⊢,
exact add_order_of_div_of_gcd_eq_one hn h, },
{ simp only [int.cast_neg_succ_of_nat, neg_div, neg_mul, coe_neg, order_of_neg],
exact add_order_of_div_of_gcd_eq_one hn h, },
end | lemma | add_circle.add_order_of_div_of_gcd_eq_one' | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"int.cast_coe_nat",
"int.cast_neg_succ_of_nat",
"neg_div",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_order_of_coe_rat {q : ℚ} : add_order_of (↑(↑q * p) : add_circle p) = q.denom | begin
have : (↑(q.denom : ℤ) : 𝕜) ≠ 0, { norm_cast, exact q.pos.ne.symm, },
rw [← @rat.num_denom q, rat.cast_mk_of_ne_zero _ _ this, int.cast_coe_nat, rat.num_denom,
add_order_of_div_of_gcd_eq_one' q.pos q.cop],
apply_instance,
end | lemma | add_circle.add_order_of_coe_rat | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"int.cast_coe_nat",
"rat.cast_mk_of_ne_zero",
"rat.num_denom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_order_of_eq_pos_iff {u : add_circle p} {n : ℕ} (h : 0 < n) :
add_order_of u = n ↔ ∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n * p) = u | begin
refine ⟨quotient_add_group.induction_on' u (λ k hk, _), _⟩, swap,
{ rintros ⟨m, h₀, h₁, rfl⟩, exact add_order_of_div_of_gcd_eq_one h h₁ },
have h0 := add_order_of_nsmul_eq_zero (k : add_circle p),
rw [hk, ← coe_nsmul, coe_eq_zero_iff] at h0,
obtain ⟨a, ha⟩ := h0,
have h0 : (_ : 𝕜) ≠ 0 := nat.cast_ne_... | lemma | add_circle.add_order_of_eq_pos_iff | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"add_div",
"add_smul",
"div_eq_iff",
"int.cast_coe_nat",
"int.cast_mul",
"int.coe_nat_lt",
"int.mod_lt_of_pos",
"int.mod_nonneg",
"int.to_nat_of_nonneg",
"mul_assoc",
"mul_comm",
"mul_div",
"mul_div_cancel",
"mul_div_right_comm",
"nat.mul_left_eq_self_iff",
"nsmul_eq_mu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_gcd_eq_one_of_is_of_fin_add_order {u : add_circle p} (h : is_of_fin_add_order u) :
∃ m : ℕ, m.gcd (add_order_of u) = 1 ∧
m < (add_order_of u) ∧
↑(((m : 𝕜) / add_order_of u) * p) = u | let ⟨m, hl, hg, he⟩ := (add_order_of_eq_pos_iff $ add_order_of_pos' h).1 rfl in ⟨m, hg, hl, he⟩ | lemma | add_circle.exists_gcd_eq_one_of_is_of_fin_add_order | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"is_of_fin_add_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_add_order_of_equiv {n : ℕ} (hn : 0 < n) :
{u : add_circle p | add_order_of u = n} ≃ {m | m < n ∧ m.gcd n = 1} | equiv.symm $ equiv.of_bijective
(λ m, ⟨↑((m : 𝕜) / n * p), add_order_of_div_of_gcd_eq_one hn (m.prop.2)⟩)
begin
refine ⟨λ m₁ m₂ h, subtype.ext _, λ u, _⟩,
{ simp_rw [subtype.ext_iff, subtype.coe_mk] at h,
rw [← sub_eq_zero, ← coe_sub, ← sub_mul, ← sub_div, coe_coe, coe_coe, ← int.cast_coe_nat m₁,
← i... | def | add_circle.set_add_order_of_equiv | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"coe_coe",
"eq_div_iff",
"equiv.of_bijective",
"equiv.symm",
"int.cast_coe_nat",
"int.cast_sub",
"int.eq_zero_of_abs_lt_dvd",
"mul_comm",
"mul_div_right_comm",
"mul_smul_comm",
"nat.cast_inj",
"nat.cast_nonneg",
"nsmul_eq_mul",
"smul_smul",
"sub_div",
"subtype.coe_mk",
... | The natural bijection between points of order `n` and natural numbers less than and coprime to
`n`. The inverse of the map sends `m ↦ (m/n * p : add_circle p)` where `m` is coprime to `n` and
satisfies `0 ≤ m < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_add_order_of_eq_totient {n : ℕ} :
nat.card {u : add_circle p // add_order_of u = n} = n.totient | begin
rcases n.eq_zero_or_pos with rfl | hn,
{ simp only [nat.totient_zero, add_order_of_eq_zero_iff],
rcases em (∃ (u : add_circle p), ¬ is_of_fin_add_order u) with ⟨u, hu⟩ | h,
{ haveI : infinite {u : add_circle p // ¬is_of_fin_add_order u},
{ erw infinite_coe_iff,
exact infinite_not_is_of_f... | lemma | add_circle.card_add_order_of_eq_totient | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"em",
"infinite",
"is_empty",
"is_of_fin_add_order",
"nat.card",
"nat.card_congr",
"nat.card_eq_zero_of_infinite",
"nat.card_of_is_empty",
"nat.gcd_comm",
"nat.totient_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_set_of_add_order_eq {n : ℕ} (hn : 0 < n) :
{u : add_circle p | add_order_of u = n}.finite | finite_coe_iff.mp $ nat.finite_of_card_ne_zero $ by simpa only [coe_set_of,
card_add_order_of_eq_totient p] using (nat.totient_pos hn).ne' | lemma | add_circle.finite_set_of_add_order_eq | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"finite",
"nat.finite_of_card_ne_zero",
"nat.totient_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space [fact (0 < p)] : compact_space $ add_circle p | begin
rw [← is_compact_univ_iff, ← coe_image_Icc_eq p 0],
exact is_compact_Icc.image (add_circle.continuous_mk' p),
end | instance | add_circle.compact_space | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle",
"add_circle.continuous_mk'",
"compact_space",
"fact",
"is_compact_univ_iff"
] | The "additive circle" `ℝ ⧸ (ℤ ∙ p)` is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_add_circle | add_circle (1 : ℝ) | abbreviation | unit_add_circle | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle"
] | The unit circle `ℝ ⧸ ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
endpoint_ident : Icc a (a + p) → Icc a (a + p) → Prop
| mk : endpoint_ident
⟨a, left_mem_Icc.mpr $ le_add_of_nonneg_right hp.out.le⟩
⟨a + p, right_mem_Icc.mpr $ le_add_of_nonneg_right hp.out.le⟩ | inductive | add_circle.endpoint_ident | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [] | The relation identifying the endpoints of `Icc a (a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_Icc_quot : 𝕋 ≃ quot (endpoint_ident p a) | { to_fun := λ x, quot.mk _ $ inclusion Ico_subset_Icc_self (equiv_Ico _ _ x),
inv_fun := λ x, quot.lift_on x coe $ by { rintro _ _ ⟨_⟩, exact (coe_add_period p a).symm },
left_inv := (equiv_Ico p a).symm_apply_apply,
right_inv := quot.ind $ by
{ rintro ⟨x, hx⟩,
have := _,
rcases ne_or_eq x (a + p) with ... | def | add_circle.equiv_Icc_quot | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"inv_fun",
"ne_or_eq"
] | The equivalence between `add_circle p` and the quotient of `[a, a + p]` by the relation
identifying the endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_Icc_quot_comp_mk_eq_to_Ico_mod : equiv_Icc_quot p a ∘ quotient.mk' =
λ x, quot.mk _ ⟨to_Ico_mod hp.out a x, Ico_subset_Icc_self $ to_Ico_mod_mem_Ico _ _ x⟩ | rfl | lemma | add_circle.equiv_Icc_quot_comp_mk_eq_to_Ico_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"quotient.mk'",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_Icc_quot_comp_mk_eq_to_Ioc_mod : equiv_Icc_quot p a ∘ quotient.mk' =
λ x, quot.mk _ ⟨to_Ioc_mod hp.out a x, Ioc_subset_Icc_self $ to_Ioc_mod_mem_Ioc _ _ x⟩ | begin
rw equiv_Icc_quot_comp_mk_eq_to_Ico_mod, funext,
by_cases a ≡ x [PMOD p],
{ simp_rw [(modeq_iff_to_Ico_mod_eq_left hp.out).1 h, (modeq_iff_to_Ioc_mod_eq_right hp.out).1 h],
exact quot.sound endpoint_ident.mk },
{ simp_rw (not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp.out).1 h }
end | lemma | add_circle.equiv_Icc_quot_comp_mk_eq_to_Ioc_mod | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"quotient.mk'",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeo_Icc_quot : 𝕋 ≃ₜ quot (endpoint_ident p a) | { to_equiv := equiv_Icc_quot p a,
continuous_to_fun := begin
simp_rw [quotient_map_quotient_mk.continuous_iff,
continuous_iff_continuous_at, continuous_at_iff_continuous_left_right],
intro x, split,
work_on_goal 1 { erw equiv_Icc_quot_comp_mk_eq_to_Ioc_mod },
work_on_goal 2 { erw equiv_Icc_quot_... | def | add_circle.homeo_Icc_quot | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"add_circle.continuous_mk'",
"continuous_at_iff_continuous_left_right",
"continuous_iff_continuous_at",
"continuous_left_to_Ioc_mod",
"continuous_quot_lift",
"continuous_right_to_Ico_mod",
"continuous_subtype_coe"
] | The natural map from `[a, a + p] ⊂ 𝕜` with endpoints identified to `𝕜 / ℤ • p`, as a
homeomorphism of topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_Ico_eq_lift_Icc {f : 𝕜 → B} (h : f a = f (a + p)) : lift_Ico p a f =
quot.lift (restrict (Icc a $ a + p) f) (by { rintro _ _ ⟨_⟩, exact h }) ∘ equiv_Icc_quot p a | rfl | lemma | add_circle.lift_Ico_eq_lift_Icc | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Ico_continuous [topological_space B] {f : 𝕜 → B} (hf : f a = f (a + p))
(hc : continuous_on f $ Icc a (a + p)) : continuous (lift_Ico p a f) | begin
rw lift_Ico_eq_lift_Icc hf,
refine continuous.comp _ (homeo_Icc_quot p a).continuous_to_fun,
exact continuous_coinduced_dom.mpr (continuous_on_iff_continuous_restrict.mp hc),
end | lemma | add_circle.lift_Ico_continuous | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous",
"continuous.comp",
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Ico_zero_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico 0 p) :
lift_Ico p 0 f ↑x = f x | lift_Ico_coe_apply (by rwa zero_add) | lemma | add_circle.lift_Ico_zero_coe_apply | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_Ico_zero_continuous [topological_space B] {f : 𝕜 → B}
(hf : f 0 = f p) (hc : continuous_on f $ Icc 0 p) : continuous (lift_Ico p 0 f) | lift_Ico_continuous (by rwa zero_add : f 0 = f (0 + p)) (by rwa zero_add) | lemma | add_circle.lift_Ico_zero_continuous | topology.instances | src/topology/instances/add_circle.lean | [
"data.nat.totient",
"algebra.ring.add_aut",
"group_theory.divisible",
"group_theory.order_of_element",
"algebra.order.floor",
"algebra.order.to_interval_mod",
"topology.instances.real"
] | [
"continuous",
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.subfield_eq_of_closed {K : subfield ℂ} (hc : is_closed (K : set ℂ)) :
K = of_real.field_range ∨ K = ⊤ | begin
suffices : range (coe : ℝ → ℂ) ⊆ K,
{ rw [range_subset_iff, ← coe_algebra_map] at this,
have := (subalgebra.is_simple_order_of_finrank finrank_real_complex).eq_bot_or_eq_top
(subfield.to_intermediate_field K this).to_subalgebra,
simp_rw ← set_like.coe_set_eq at this ⊢,
convert this using 2,
... | lemma | complex.subfield_eq_of_closed | topology.instances | src/topology/instances/complex.lean | [
"analysis.complex.basic",
"field_theory.intermediate_field",
"topology.algebra.uniform_ring"
] | [
"algebra.coe_bot",
"closure",
"closure_mono",
"dense_range.closure_range",
"image_closure_subset_closure_image",
"is_closed",
"is_closed.closure_eq",
"ring_hom.coe_field_range",
"set.range",
"set_like.coe_set_eq",
"set_like.mem_coe",
"subalgebra.is_simple_order_of_finrank",
"subfield",
"su... | The only closed subfields of `ℂ` are `ℝ` and `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.uniform_continuous_ring_hom_eq_id_or_conj (K : subfield ℂ) {ψ : K →+* ℂ}
(hc : uniform_continuous ψ) : ψ.to_fun = K.subtype ∨ ψ.to_fun = conj ∘ K.subtype | begin
letI : topological_division_ring ℂ := topological_division_ring.mk,
letI : topological_ring K.topological_closure :=
subring.topological_ring K.topological_closure.to_subring,
set ι : K → K.topological_closure := subfield.inclusion K.le_topological_closure,
have ui : uniform_inducing ι :=
⟨ by {... | lemma | complex.uniform_continuous_ring_hom_eq_id_or_conj | topology.instances | src/topology/instances/complex.lean | [
"analysis.complex.basic",
"field_theory.intermediate_field",
"topology.algebra.uniform_ring"
] | [
"closure",
"complex.subfield_eq_of_closed",
"continuity",
"continuous",
"dense",
"dense_embedding.subtype",
"dense_embedding_id",
"dense_inducing.extend_eq",
"dense_inducing.extend_ring_hom",
"dense_range.comp",
"filter.comap_comap",
"function.surjective.dense_range",
"ring_equiv.subfield_co... | Let `K` a subfield of `ℂ` and let `ψ : K →+* ℂ` a ring homomorphism. Assume that `ψ` is uniform
continuous, then `ψ` is either the inclusion map or the composition of the inclusion map with the
complex conjugation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_topology.first_countable_topology [discrete_topology α] :
first_countable_topology α | { nhds_generated_countable := by { rw nhds_discrete, exact is_countably_generated_pure } } | instance | discrete_topology.first_countable_topology | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"nhds_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.second_countable_topology_of_encodable
[hd : discrete_topology α] [encodable α] :
second_countable_topology α | begin
haveI : ∀ (i : α), second_countable_topology ↥({i} : set α),
from λ i, { is_open_generated_countable :=
⟨{univ}, countable_singleton _, by simp only [eq_iff_true_of_subsingleton]⟩, },
exact second_countable_topology_of_countable_cover (singletons_open_iff_discrete.mpr hd)
(Union_of_singleton α),... | instance | discrete_topology.second_countable_topology_of_encodable | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"encodable",
"eq_iff_true_of_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_topological_space_eq_generate_from_of_pred_succ_order {α} [partial_order α]
[pred_order α] [succ_order α] [no_min_order α] [no_max_order α] :
(⊥ : topological_space α) = generate_from {s | ∃ a, s = Ioi a ∨ s = Iio a} | begin
refine (eq_bot_of_singletons_open (λ a, _)).symm,
have h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a),
{ suffices h_singleton_eq_inter' : {a} = Iic a ∩ Ici a,
by rw [h_singleton_eq_inter', ←Ioi_pred, ←Iio_succ],
rw [inter_comm, Ici_inter_Iic, Icc_self a], },
rw h_singleton_eq_inter,
... | lemma | bot_topological_space_eq_generate_from_of_pred_succ_order | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"eq_bot_of_singletons_open",
"is_open.inter",
"no_max_order",
"no_min_order",
"pred_order",
"succ_order",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_iff_order_topology_of_pred_succ' [partial_order α]
[pred_order α] [succ_order α] [no_min_order α] [no_max_order α] :
discrete_topology α ↔ order_topology α | begin
refine ⟨λ h, ⟨_⟩, λ h, ⟨_⟩⟩,
{ rw h.eq_bot,
exact bot_topological_space_eq_generate_from_of_pred_succ_order, },
{ rw h.topology_eq_generate_intervals,
exact bot_topological_space_eq_generate_from_of_pred_succ_order.symm, },
end | lemma | discrete_topology_iff_order_topology_of_pred_succ' | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"bot_topological_space_eq_generate_from_of_pred_succ_order",
"discrete_topology",
"no_max_order",
"no_min_order",
"order_topology",
"pred_order",
"succ_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.order_topology_of_pred_succ' [h : discrete_topology α] [partial_order α]
[pred_order α] [succ_order α] [no_min_order α] [no_max_order α] :
order_topology α | discrete_topology_iff_order_topology_of_pred_succ'.1 h | instance | discrete_topology.order_topology_of_pred_succ' | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"no_max_order",
"no_min_order",
"order_topology",
"pred_order",
"succ_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.bot_topological_space_eq_generate_from
{α} [linear_order α] [pred_order α] [succ_order α] :
(⊥ : topological_space α) = generate_from {s | ∃ a, s = Ioi a ∨ s = Iio a} | begin
refine (eq_bot_of_singletons_open (λ a, _)).symm,
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a,
by rw [inter_comm, Ici_inter_Iic, Icc_self a],
by_cases ha_top : is_top a,
{ rw [ha_top.Iic_eq, inter_comm, inter_univ] at h_singleton_eq_inter,
by_cases ha_bot : is_bot a,
{ rw ha_bot.Ici_eq at ... | lemma | linear_order.bot_topological_space_eq_generate_from | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"eq_bot_of_singletons_open",
"is_bot",
"is_bot_iff_is_min",
"is_open.inter",
"is_open_univ",
"is_top",
"is_top_iff_is_max",
"pred_order",
"succ_order",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_iff_order_topology_of_pred_succ
[linear_order α] [pred_order α] [succ_order α] :
discrete_topology α ↔ order_topology α | begin
refine ⟨λ h, ⟨_⟩, λ h, ⟨_⟩⟩,
{ rw h.eq_bot,
exact linear_order.bot_topological_space_eq_generate_from, },
{ rw h.topology_eq_generate_intervals,
exact linear_order.bot_topological_space_eq_generate_from.symm, },
end | lemma | discrete_topology_iff_order_topology_of_pred_succ | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"linear_order.bot_topological_space_eq_generate_from",
"order_topology",
"pred_order",
"succ_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.order_topology_of_pred_succ [h : discrete_topology α] [linear_order α]
[pred_order α] [succ_order α] :
order_topology α | discrete_topology_iff_order_topology_of_pred_succ.mp h | instance | discrete_topology.order_topology_of_pred_succ | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"order_topology",
"pred_order",
"succ_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.metrizable_space [discrete_topology α] : metrizable_space α | begin
unfreezingI { obtain rfl := discrete_topology.eq_bot α },
exact @uniform_space.metrizable_space α ⊥ (is_countably_generated_principal _) _,
end | instance | discrete_topology.metrizable_space | topology.instances | src/topology/instances/discrete.lean | [
"order.succ_pred.basic",
"topology.order.basic",
"topology.metric_space.metrizable_uniformity"
] | [
"discrete_topology",
"uniform_space.metrizable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_coe : embedding (coe : ℝ≥0 → ℝ≥0∞) | ⟨⟨begin
refine le_antisymm _ _,
{ rw [@order_topology.topology_eq_generate_intervals ℝ≥0∞ _,
← coinduced_le_iff_le_induced],
refine le_generate_from (assume s ha, _),
rcases ha with ⟨a, rfl | rfl⟩,
show is_open {b : ℝ≥0 | a < ↑b},
{ cases a; simp [none_eq_top, some_eq_coe, is_open_lt'] },
... | lemma | ennreal.embedding_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"coinduced_le_iff_le_induced",
"embedding",
"is_open",
"is_open_Iio",
"is_open_Ioi",
"is_open_const",
"is_open_gt'",
"is_open_lt'",
"le_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_ne_top : is_open {a : ℝ≥0∞ | a ≠ ⊤} | is_open_ne | lemma | ennreal.is_open_ne_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"is_open",
"is_open_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Ico_zero : is_open (Ico 0 b) | by { rw ennreal.Ico_eq_Iio, exact is_open_Iio} | lemma | ennreal.is_open_Ico_zero | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.Ico_eq_Iio",
"is_open",
"is_open_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_coe : open_embedding (coe : ℝ≥0 → ℝ≥0∞) | ⟨embedding_coe, by { convert is_open_ne_top, ext (x|_); simp [none_eq_top, some_eq_coe] }⟩ | lemma | ennreal.open_embedding_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_range_mem_nhds : range (coe : ℝ≥0 → ℝ≥0∞) ∈ 𝓝 (r : ℝ≥0∞) | is_open.mem_nhds open_embedding_coe.open_range $ mem_range_self _ | lemma | ennreal.coe_range_mem_nhds | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe {f : filter α} {m : α → ℝ≥0} {a : ℝ≥0} :
tendsto (λa, (m a : ℝ≥0∞)) f (𝓝 ↑a) ↔ tendsto m f (𝓝 a) | embedding_coe.tendsto_nhds_iff.symm | lemma | ennreal.tendsto_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : ℝ≥0 → ℝ≥0∞) | embedding_coe.continuous | lemma | ennreal.continuous_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe_iff {α} [topological_space α] {f : α → ℝ≥0} :
continuous (λa, (f a : ℝ≥0∞)) ↔ continuous f | embedding_coe.continuous_iff.symm | lemma | ennreal.continuous_coe_iff | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_coe {r : ℝ≥0} : 𝓝 (r : ℝ≥0∞) = (𝓝 r).map coe | (open_embedding_coe.map_nhds_eq r).symm | lemma | ennreal.nhds_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_coe_iff {α : Type*} {l : filter α} {x : ℝ≥0} {f : ℝ≥0∞ → α} :
tendsto f (𝓝 ↑x) l ↔ tendsto (f ∘ coe : ℝ≥0 → α) (𝓝 x) l | show _ ≤ _ ↔ _ ≤ _, by rw [nhds_coe, filter.map_map] | lemma | ennreal.tendsto_nhds_coe_iff | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"filter",
"filter.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_coe_iff {α : Type*} [topological_space α] {x : ℝ≥0} {f : ℝ≥0∞ → α} :
continuous_at f (↑x) ↔ continuous_at (f ∘ coe : ℝ≥0 → α) x | tendsto_nhds_coe_iff | lemma | ennreal.continuous_at_coe_iff | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_coe_coe {r p : ℝ≥0} :
𝓝 ((r : ℝ≥0∞), (p : ℝ≥0∞)) = (𝓝 (r, p)).map (λp:ℝ≥0×ℝ≥0, (p.1, p.2)) | ((open_embedding_coe.prod open_embedding_coe).map_nhds_eq (r, p)).symm | lemma | ennreal.nhds_coe_coe | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_real : continuous ennreal.of_real | (continuous_coe_iff.2 continuous_id).comp continuous_real_to_nnreal | lemma | ennreal.continuous_of_real | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous",
"continuous_id",
"continuous_real_to_nnreal",
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_real {f : filter α} {m : α → ℝ} {a : ℝ} (h : tendsto m f (𝓝 a)) :
tendsto (λa, ennreal.of_real (m a)) f (𝓝 (ennreal.of_real a)) | tendsto.comp (continuous.tendsto continuous_of_real _) h | lemma | ennreal.tendsto_of_real | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous.tendsto",
"ennreal.of_real",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_to_nnreal {a : ℝ≥0∞} (ha : a ≠ ⊤) :
tendsto ennreal.to_nnreal (𝓝 a) (𝓝 a.to_nnreal) | begin
lift a to ℝ≥0 using ha,
rw [nhds_coe, tendsto_map'_iff],
exact tendsto_id
end | lemma | ennreal.tendsto_to_nnreal | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_nnreal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_of_to_real_eventually_eq {l : filter α} {f g : α → ℝ≥0∞}
(hfi : ∀ᶠ x in l, f x ≠ ∞) (hgi : ∀ᶠ x in l, g x ≠ ∞)
(hfg : (λ x, (f x).to_real) =ᶠ[l] (λ x, (g x).to_real)) :
f =ᶠ[l] g | begin
filter_upwards [hfi, hgi, hfg] with _ hfx hgx _,
rwa ← ennreal.to_real_eq_to_real hfx hgx,
end | lemma | ennreal.eventually_eq_of_to_real_eventually_eq | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real_eq_to_real",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_to_nnreal : continuous_on ennreal.to_nnreal {a | a ≠ ∞} | λ a ha, continuous_at.continuous_within_at (tendsto_to_nnreal ha) | lemma | ennreal.continuous_on_to_nnreal | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"continuous_at.continuous_within_at",
"continuous_on",
"ennreal.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_to_real {a : ℝ≥0∞} (ha : a ≠ ⊤) : tendsto ennreal.to_real (𝓝 a) (𝓝 a.to_real) | nnreal.tendsto_coe.2 $ tendsto_to_nnreal ha | lemma | ennreal.tendsto_to_real | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"ennreal.to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_top_homeomorph_nnreal : {a | a ≠ ∞} ≃ₜ ℝ≥0 | { continuous_to_fun := continuous_on_iff_continuous_restrict.1 continuous_on_to_nnreal,
continuous_inv_fun := continuous_coe.subtype_mk _,
.. ne_top_equiv_nnreal } | def | ennreal.ne_top_homeomorph_nnreal | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [] | The set of finite `ℝ≥0∞` numbers is homeomorphic to `ℝ≥0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_top_homeomorph_nnreal : {a | a < ∞} ≃ₜ ℝ≥0 | by refine (homeomorph.set_congr $ set.ext $ λ x, _).trans ne_top_homeomorph_nnreal;
simp only [mem_set_of_eq, lt_top_iff_ne_top] | def | ennreal.lt_top_homeomorph_nnreal | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"homeomorph.set_congr",
"lt_top_iff_ne_top",
"set.ext"
] | The set of finite `ℝ≥0∞` numbers is homeomorphic to `ℝ≥0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_top : 𝓝 ∞ = ⨅ a ≠ ∞, 𝓟 (Ioi a) | nhds_top_order.trans $ by simp [lt_top_iff_ne_top, Ioi] | lemma | ennreal.nhds_top | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [
"lt_top_iff_ne_top",
"nhds_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top' : 𝓝 ∞ = ⨅ r : ℝ≥0, 𝓟 (Ioi r) | nhds_top.trans $ infi_ne_top _ | lemma | ennreal.nhds_top' | topology.instances | src/topology/instances/ennreal.lean | [
"topology.instances.nnreal",
"topology.algebra.order.monotone_continuity",
"topology.algebra.infinite_sum.real",
"topology.algebra.order.liminf_limsup",
"topology.metric_space.lipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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