statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
lipschitz_with_sup_right (z : α) : lipschitz_with 1 (λ x, x ⊔ z) | lipschitz_with.of_dist_le_mul $ λ x y, by
{ rw [nonneg.coe_one, one_mul, dist_eq_norm, dist_eq_norm], exact norm_sup_sub_sup_le_norm x y z, } | lemma | lipschitz_with_sup_right | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"lipschitz_with",
"nonneg.coe_one",
"norm_sup_sub_sup_le_norm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_pos : lipschitz_with 1 (has_pos_part.pos : α → α) | lipschitz_with_sup_right 0 | lemma | lipschitz_with_pos | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"lipschitz_with",
"lipschitz_with_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_pos : continuous (has_pos_part.pos : α → α) | lipschitz_with.continuous lipschitz_with_pos | lemma | continuous_pos | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"continuous",
"lipschitz_with.continuous",
"lipschitz_with_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_neg' : continuous (has_neg_part.neg : α → α) | continuous_pos.comp continuous_neg | lemma | continuous_neg' | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_nonneg {E} [normed_lattice_add_comm_group E] : is_closed {x : E | 0 ≤ x} | begin
suffices : {x : E | 0 ≤ x} = has_neg_part.neg ⁻¹' {(0 : E)},
by { rw this, exact is_closed.preimage continuous_neg' is_closed_singleton, },
ext1 x,
simp only [set.mem_preimage, set.mem_singleton_iff, set.mem_set_of_eq, neg_eq_zero_iff],
end | lemma | is_closed_nonneg | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"continuous_neg'",
"is_closed",
"is_closed.preimage",
"is_closed_singleton",
"normed_lattice_add_comm_group",
"set.mem_preimage",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_le_of_is_closed_nonneg {G} [ordered_add_comm_group G] [topological_space G]
[has_continuous_sub G] (h : is_closed {x : G | 0 ≤ x}) :
is_closed {p : G × G | p.fst ≤ p.snd} | begin
have : {p : G × G | p.fst ≤ p.snd} = (λ p : G × G, p.snd - p.fst) ⁻¹' {x : G | 0 ≤ x},
by { ext1 p, simp only [sub_nonneg, set.preimage_set_of_eq], },
rw this,
exact is_closed.preimage (continuous_snd.sub continuous_fst) h,
end | lemma | is_closed_le_of_is_closed_nonneg | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"continuous_fst",
"has_continuous_sub",
"is_closed",
"is_closed.preimage",
"ordered_add_comm_group",
"set.preimage_set_of_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_lattice_add_comm_group.order_closed_topology {E} [normed_lattice_add_comm_group E] :
order_closed_topology E | ⟨is_closed_le_of_is_closed_nonneg is_closed_nonneg⟩ | instance | normed_lattice_add_comm_group.order_closed_topology | analysis.normed.order | src/analysis/normed/order/lattice.lean | [
"topology.order.lattice",
"analysis.normed.group.basic",
"algebra.order.lattice_group"
] | [
"normed_lattice_add_comm_group",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.thickening' (hs : is_upper_set s) (ε : ℝ) :
is_upper_set (thickening ε s) | by { rw ←ball_mul_one, exact hs.mul_left } | lemma | is_upper_set.thickening' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.thickening' (hs : is_lower_set s) (ε : ℝ) :
is_lower_set (thickening ε s) | by { rw ←ball_mul_one, exact hs.mul_left } | lemma | is_lower_set.thickening' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.cthickening' (hs : is_upper_set s) (ε : ℝ) :
is_upper_set (cthickening ε s) | by { rw cthickening_eq_Inter_thickening'', exact is_upper_set_Inter₂ (λ δ hδ, hs.thickening' _) } | lemma | is_upper_set.cthickening' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"is_upper_set",
"is_upper_set_Inter₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.cthickening' (hs : is_lower_set s) (ε : ℝ) :
is_lower_set (cthickening ε s) | by { rw cthickening_eq_Inter_thickening'', exact is_lower_set_Inter₂ (λ δ hδ, hs.thickening' _) } | lemma | is_lower_set.cthickening' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"is_lower_set",
"is_lower_set_Inter₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_interior_subset' (s : set α) :
(upper_closure (interior s) : set α) ⊆ interior (upper_closure s) | upper_closure_min (interior_mono subset_upper_closure) (upper_closure s).upper.interior | lemma | upper_closure_interior_subset' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"interior",
"interior_mono",
"subset_upper_closure",
"upper_closure",
"upper_closure_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_interior_subset' (s : set α) :
(upper_closure (interior s) : set α) ⊆ interior (upper_closure s) | upper_closure_min (interior_mono subset_upper_closure) (upper_closure s).upper.interior | lemma | lower_closure_interior_subset' | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"interior",
"interior_mono",
"subset_upper_closure",
"upper_closure",
"upper_closure_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.mem_interior_of_forall_lt (hs : is_upper_set s) (hx : x ∈ closure s)
(h : ∀ i, x i < y i) :
y ∈ interior s | begin
casesI nonempty_fintype ι,
obtain ⟨ε, hε, hxy⟩ := pi.exists_forall_pos_add_lt h,
obtain ⟨z, hz, hxz⟩ := metric.mem_closure_iff.1 hx _ hε,
rw dist_pi_lt_iff hε at hxz,
have hyz : ∀ i, z i < y i,
{ refine λ i, (hxy _).trans_le' (sub_le_iff_le_add'.1 $ (le_abs_self _).trans _),
rw [←real.norm_eq_abs,... | lemma | is_upper_set.mem_interior_of_forall_lt | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"ball_pi",
"closure",
"dist_pi_lt_iff",
"interior",
"is_upper_set",
"le_abs_self",
"nonempty_fintype",
"pi.exists_forall_pos_add_lt",
"real.ball_eq_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.mem_interior_of_forall_lt (hs : is_lower_set s) (hx : x ∈ closure s)
(h : ∀ i, y i < x i) :
y ∈ interior s | begin
casesI nonempty_fintype ι,
obtain ⟨ε, hε, hxy⟩ := pi.exists_forall_pos_add_lt h,
obtain ⟨z, hz, hxz⟩ := metric.mem_closure_iff.1 hx _ hε,
rw dist_pi_lt_iff hε at hxz,
have hyz : ∀ i, y i < z i,
{ refine λ i, (lt_sub_iff_add_lt.2 $ hxy _).trans_le (sub_le_comm.1 $ (le_abs_self _).trans _),
rw [←rea... | lemma | is_lower_set.mem_interior_of_forall_lt | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"ball_pi",
"closure",
"dist_pi_lt_iff",
"interior",
"is_lower_set",
"le_abs_self",
"nonempty_fintype",
"pi.exists_forall_pos_add_lt",
"real.ball_eq_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_inf_sup (x y : ι → ℝ) : dist (x ⊓ y) (x ⊔ y) = dist x y | begin
refine congr_arg coe (finset.sup_congr rfl $ λ i _, _),
simp only [real.nndist_eq', sup_eq_max, inf_eq_min, max_sub_min_eq_abs, pi.inf_apply,
pi.sup_apply, real.nnabs_of_nonneg, abs_nonneg, real.to_nnreal_abs],
end | lemma | dist_inf_sup | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"abs_nonneg",
"finset.sup_congr",
"inf_eq_min",
"max_sub_min_eq_abs",
"pi.inf_apply",
"pi.sup_apply",
"real.nnabs_of_nonneg",
"real.nndist_eq'",
"real.to_nnreal_abs",
"sup_eq_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mono_left : monotone_on (λ x, dist x y) (Ici y) | begin
refine λ y₁ hy₁ y₂ hy₂ hy, nnreal.coe_le_coe.2 (finset.sup_mono_fun $ λ i _, _),
rw [real.nndist_eq, real.nnabs_of_nonneg (sub_nonneg_of_le (‹y ≤ _› i : y i ≤ y₁ i)),
real.nndist_eq, real.nnabs_of_nonneg (sub_nonneg_of_le (‹y ≤ _› i : y i ≤ y₂ i))],
exact real.to_nnreal_mono (sub_le_sub_right (hy _) _),... | lemma | dist_mono_left | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"finset.sup_mono_fun",
"monotone_on",
"real.nnabs_of_nonneg",
"real.nndist_eq",
"real.to_nnreal_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mono_right : monotone_on (dist x) (Ici x) | by simpa only [dist_comm] using dist_mono_left | lemma | dist_mono_right | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"dist_comm",
"dist_mono_left",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_anti_left : antitone_on (λ x, dist x y) (Iic y) | begin
refine λ y₁ hy₁ y₂ hy₂ hy, nnreal.coe_le_coe.2 (finset.sup_mono_fun $ λ i _, _),
rw [real.nndist_eq', real.nnabs_of_nonneg (sub_nonneg_of_le (‹_ ≤ y› i : y₂ i ≤ y i)),
real.nndist_eq', real.nnabs_of_nonneg (sub_nonneg_of_le (‹_ ≤ y› i : y₁ i ≤ y i))],
exact real.to_nnreal_mono (sub_le_sub_left (hy _) _)... | lemma | dist_anti_left | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"antitone_on",
"finset.sup_mono_fun",
"real.nnabs_of_nonneg",
"real.nndist_eq'",
"real.to_nnreal_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_anti_right : antitone_on (dist x) (Iic x) | by simpa only [dist_comm] using dist_anti_left | lemma | dist_anti_right | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"antitone_on",
"dist_anti_left",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_dist_of_le (ha : a₂ ≤ a₁) (h₁ : a₁ ≤ b₁) (hb : b₁ ≤ b₂) : dist a₁ b₁ ≤ dist a₂ b₂ | (dist_mono_right h₁ (h₁.trans hb) hb).trans $
dist_anti_left (ha.trans $ h₁.trans hb) (h₁.trans hb) ha | lemma | dist_le_dist_of_le | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"dist_anti_left",
"dist_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.bdd_below : bounded s → bdd_below s | begin
rintro ⟨r, hr⟩,
obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty,
{ exact bdd_below_empty },
{ exact ⟨x - const _ r, λ y hy i, sub_le_comm.1
(abs_sub_le_iff.1 $ (dist_le_pi_dist _ _ _).trans $ hr _ hx _ hy).1⟩ }
end | lemma | metric.bounded.bdd_below | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_below",
"bdd_below_empty",
"dist_le_pi_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.bdd_above : bounded s → bdd_above s | begin
rintro ⟨r, hr⟩,
obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty,
{ exact bdd_above_empty },
{ exact ⟨x + const _ r, λ y hy i, sub_le_iff_le_add'.1 $
(abs_sub_le_iff.1 $ (dist_le_pi_dist _ _ _).trans $ hr _ hx _ hy).2⟩ }
end | lemma | metric.bounded.bdd_above | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_above_empty",
"dist_le_pi_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.bounded : bdd_below s → bdd_above s → bounded s | begin
rintro ⟨a, ha⟩ ⟨b, hb⟩,
refine ⟨dist a b, λ x hx y hy, _⟩,
rw ←dist_inf_sup,
exact dist_le_dist_of_le (le_inf (ha hx) $ ha hy) inf_le_sup (sup_le (hb hx) $ hb hy),
end | lemma | bdd_below.bounded | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below",
"dist_le_dist_of_le",
"inf_le_sup",
"le_inf",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.bounded : bdd_above s → bdd_below s → bounded s | flip bdd_below.bounded | lemma | bdd_above.bounded | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below",
"bdd_below.bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_iff_bdd_below_bdd_above : bounded s ↔ bdd_below s ∧ bdd_above s | ⟨λ h, ⟨h.bdd_below, h.bdd_above⟩, λ h, h.1.bounded h.2⟩ | lemma | bounded_iff_bdd_below_bdd_above | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.bounded_inter (hs : bdd_below s) (ht : bdd_above t) : bounded (s ∩ t) | (hs.mono $ inter_subset_left _ _).bounded $ ht.mono $ inter_subset_right _ _ | lemma | bdd_below.bounded_inter | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.bounded_inter (hs : bdd_above s) (ht : bdd_below t) : bounded (s ∩ t) | (hs.mono $ inter_subset_left _ _).bounded $ ht.mono $ inter_subset_right _ _ | lemma | bdd_above.bounded_inter | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.exists_subset_ball (hs : is_upper_set s) (hx : x ∈ closure s) (hδ : 0 < δ) :
∃ y, closed_ball y (δ/4) ⊆ closed_ball x δ ∧ closed_ball y (δ/4) ⊆ interior s | begin
refine ⟨x + const _ (3/4*δ), closed_ball_subset_closed_ball' _, _⟩,
{ rw dist_self_add_left,
refine (add_le_add_left (pi_norm_const_le $ 3 / 4 * δ) _).trans_eq _,
simp [real.norm_of_nonneg, hδ.le, zero_le_three],
ring_nf },
obtain ⟨y, hy, hxy⟩ := metric.mem_closure_iff.1 hx _ (div_pos hδ zero_lt... | lemma | is_upper_set.exists_subset_ball | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"abs_sub_le_iff",
"closure",
"div_pos",
"interior",
"is_upper_set",
"real.norm_of_nonneg",
"subset_closure",
"zero_le_three",
"zero_lt_four"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.exists_subset_ball (hs : is_lower_set s) (hx : x ∈ closure s) (hδ : 0 < δ) :
∃ y, closed_ball y (δ/4) ⊆ closed_ball x δ ∧ closed_ball y (δ/4) ⊆ interior s | begin
refine ⟨x - const _ (3/4*δ), closed_ball_subset_closed_ball' _, _⟩,
{ rw dist_self_sub_left,
refine (add_le_add_left (pi_norm_const_le $ 3 / 4 * δ) _).trans_eq _,
simp [real.norm_of_nonneg, hδ.le, zero_le_three],
ring_nf },
obtain ⟨y, hy, hxy⟩ := metric.mem_closure_iff.1 hx _ (div_pos hδ zero_lt... | lemma | is_lower_set.exists_subset_ball | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"abs_sub_le_iff",
"closure",
"div_pos",
"interior",
"is_lower_set",
"real.norm_of_nonneg",
"subset_closure",
"zero_le_three",
"zero_lt_four"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_antichain.interior_eq_empty [nonempty ι] (hs : is_antichain (≤) s) : interior s = ∅ | begin
casesI nonempty_fintype ι,
refine eq_empty_of_forall_not_mem (λ x hx, _),
have hx' := interior_subset hx,
rw [mem_interior_iff_mem_nhds, metric.mem_nhds_iff] at hx,
obtain ⟨ε, hε, hx⟩ := hx,
refine hs.not_lt hx' (hx _) (lt_add_of_pos_right _ (by positivity : 0 < const ι (ε / 2))),
simpa [const, @pi_... | lemma | is_antichain.interior_eq_empty | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"abs_of_nonneg",
"interior",
"interior_subset",
"is_antichain",
"mem_interior_iff_mem_nhds",
"metric.mem_nhds_iff",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.upper_closure (hs : is_closed s) (hs' : bdd_below s) :
is_closed (upper_closure s : set (ι → ℝ)) | begin
casesI nonempty_fintype ι,
refine is_seq_closed.is_closed (λ f x hf hx, _),
choose g hg hgf using hf,
obtain ⟨a, ha⟩ := hx.bdd_above_range,
obtain ⟨b, hb, φ, hφ, hbf⟩ := tendsto_subseq_of_bounded (hs'.bounded_inter
bdd_above_Iic) (λ n, ⟨hg n, (hgf _).trans $ ha $ mem_range_self _⟩),
exact ⟨b, clo... | lemma | is_closed.upper_closure | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above_Iic",
"bdd_below",
"closure_minimal",
"is_closed",
"is_seq_closed.is_closed",
"le_of_tendsto_of_tendsto'",
"nonempty_fintype",
"tendsto_subseq_of_bounded",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.lower_closure (hs : is_closed s) (hs' : bdd_above s) :
is_closed (lower_closure s : set (ι → ℝ)) | begin
casesI nonempty_fintype ι,
refine is_seq_closed.is_closed (λ f x hf hx, _),
choose g hg hfg using hf,
haveI : bounded_ge_nhds_class ℝ := by apply_instance,
obtain ⟨a, ha⟩ := hx.bdd_below_range,
obtain ⟨b, hb, φ, hφ, hbf⟩ := tendsto_subseq_of_bounded (hs'.bounded_inter
bdd_below_Ici) (λ n, ⟨hg n, (... | lemma | is_closed.lower_closure | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"bdd_below_Ici",
"bounded_ge_nhds_class",
"closure_minimal",
"is_closed",
"is_seq_closed.is_closed",
"le_of_tendsto_of_tendsto'",
"lower_closure",
"nonempty_fintype",
"tendsto_subseq_of_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.upper_closure (hs : is_clopen s) (hs' : bdd_below s) :
is_clopen (upper_closure s : set (ι → ℝ)) | ⟨hs.1.upper_closure, hs.2.upper_closure hs'⟩ | lemma | is_clopen.upper_closure | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_below",
"is_clopen",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.lower_closure (hs : is_clopen s) (hs' : bdd_above s) :
is_clopen (lower_closure s : set (ι → ℝ)) | ⟨hs.1.lower_closure, hs.2.lower_closure hs'⟩ | lemma | is_clopen.lower_closure | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"is_clopen",
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_upper_closure_comm (hs : bdd_below s) :
closure (upper_closure s : set (ι → ℝ)) = upper_closure (closure s) | (closure_minimal (upper_closure_anti subset_closure) $
is_closed_closure.upper_closure hs.closure).antisymm $
upper_closure_min (closure_mono subset_upper_closure) (upper_closure s).upper.closure | lemma | closure_upper_closure_comm | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_below",
"closure",
"closure_minimal",
"closure_mono",
"subset_closure",
"subset_upper_closure",
"upper_closure",
"upper_closure_anti",
"upper_closure_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_lower_closure_comm (hs : bdd_above s) :
closure (lower_closure s : set (ι → ℝ)) = lower_closure (closure s) | (closure_minimal (lower_closure_mono subset_closure) $
is_closed_closure.lower_closure hs.closure).antisymm $
lower_closure_min (closure_mono subset_lower_closure) (lower_closure s).lower.closure | lemma | closure_lower_closure_comm | analysis.normed.order | src/analysis/normed/order/upper_lower.lean | [
"algebra.order.field.pi",
"analysis.normed.group.pointwise",
"analysis.normed.order.basic",
"topology.algebra.order.upper_lower"
] | [
"bdd_above",
"closure",
"closure_minimal",
"closure_mono",
"lower_closure",
"lower_closure_min",
"lower_closure_mono",
"subset_closure",
"subset_lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_seminorm (R : Type*) [non_unital_non_assoc_ring R]
extends add_group_seminorm R | (mul_le' : ∀ x y : R, to_fun (x * y) ≤ to_fun x * to_fun y) | structure | ring_seminorm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"add_group_seminorm",
"non_unital_non_assoc_ring"
] | A seminorm on a ring `R` is a function `f : R → ℝ` that preserves zero, takes nonnegative
values, is subadditive and submultiplicative and such that `f (-x) = f x` for all `x ∈ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_norm (R : Type*) [non_unital_non_assoc_ring R]
extends ring_seminorm R, add_group_norm R | structure | ring_norm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"add_group_norm",
"non_unital_non_assoc_ring",
"ring_seminorm"
] | A function `f : R → ℝ` is a norm on a (nonunital) ring if it is a seminorm and `f x = 0`
implies `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_seminorm (R : Type*) [non_assoc_ring R]
extends add_group_seminorm R, monoid_with_zero_hom R ℝ | structure | mul_ring_seminorm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"add_group_seminorm",
"monoid_with_zero_hom",
"non_assoc_ring"
] | A multiplicative seminorm on a ring `R` is a function `f : R → ℝ` that preserves zero and
multiplication, takes nonnegative values, is subadditive and such that `f (-x) = f x` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_norm (R : Type*) [non_assoc_ring R] extends mul_ring_seminorm R, add_group_norm R | structure | mul_ring_norm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"add_group_norm",
"mul_ring_seminorm",
"non_assoc_ring"
] | A multiplicative norm on a ring `R` is a multiplicative ring seminorm such that `f x = 0`
implies `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_seminorm_class : ring_seminorm_class (ring_seminorm R) R ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero',
map_add_le_add := λ f, f.add_le',
map_mul_le_mul := λ f, f.mul_le',
map_neg_eq_map := λ f, f.neg' } | instance | ring_seminorm.ring_seminorm_class | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_seminorm",
"ring_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (p : ring_seminorm R) : p.to_fun = p | rfl | lemma | ring_seminorm.to_fun_eq_coe | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : ring_seminorm R} : (∀ x, p x = q x) → p = q | fun_like.ext p q | lemma | ring_seminorm.ext | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"fun_like.ext",
"ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff {p : ring_seminorm R} : p = 0 ↔ ∀ x, p x = 0 | fun_like.ext_iff | lemma | ring_seminorm.eq_zero_iff | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"eq_zero_iff",
"fun_like.ext_iff",
"ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_iff {p : ring_seminorm R} : p ≠ 0 ↔ ∃ x, p x ≠ 0 | by simp [eq_zero_iff] | lemma | ring_seminorm.ne_zero_iff | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"eq_zero_iff",
"ne_zero_iff",
"ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq R] (x : R) :
(1 : ring_seminorm R) x = if x = 0 then 0 else 1 | rfl | lemma | ring_seminorm.apply_one | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminorm_one_eq_one_iff_ne_zero (hp : p 1 ≤ 1) :
p 1 = 1 ↔ p ≠ 0 | begin
refine ⟨λ h, ne_zero_iff.mpr ⟨1, by {rw h, exact one_ne_zero}⟩, λ h, _⟩,
obtain hp0 | hp0 := (map_nonneg p (1 : R)).eq_or_gt,
{ cases h (ext $ λ x, (map_nonneg _ _).antisymm' _),
simpa only [hp0, mul_one, mul_zero] using map_mul_le_mul p x 1},
{ refine hp.antisymm ((le_mul_iff_one_le_left hp0).1 _),
... | lemma | ring_seminorm.seminorm_one_eq_one_iff_ne_zero | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"antisymm'",
"le_mul_iff_one_le_left",
"map_nonneg",
"mul_one",
"mul_zero",
"one_mul",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_ring_seminorm (R : Type*) [non_unital_semi_normed_ring R] :
ring_seminorm R | { to_fun := norm,
mul_le' := norm_mul_le,
..(norm_add_group_seminorm R) } | def | norm_ring_seminorm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"non_unital_semi_normed_ring",
"norm_mul_le",
"ring_seminorm"
] | The norm of a `non_unital_semi_normed_ring` as a `ring_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_norm_class : ring_norm_class (ring_norm R) R ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero',
map_add_le_add := λ f, f.add_le',
map_mul_le_mul := λ f, f.mul_le',
map_neg_eq_map := λ f, f.neg',
eq_zero_of_map_eq_zero := λ f, f.eq_zero_of_map_eq_zero' } | instance | ring_norm.ring_norm_class | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_norm",
"ring_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (p : ring_norm R) : p.to_fun = p | rfl | lemma | ring_norm.to_fun_eq_coe | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : ring_norm R} : (∀ x, p x = q x) → p = q | fun_like.ext p q | lemma | ring_norm.ext | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"fun_like.ext",
"ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq R] (x : R) : (1 : ring_norm R) x = if x = 0 then 0 else 1 | rfl | lemma | ring_norm.apply_one | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_seminorm_class : mul_ring_seminorm_class (mul_ring_seminorm R) R ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero',
map_one := λ f, f.map_one',
map_add_le_add := λ f, f.add_le',
map_mul := λ f, f.map_mul',
map_neg_eq_map := λ f, f.neg' } | instance | mul_ring_seminorm.mul_ring_seminorm_class | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"map_mul",
"map_one",
"mul_ring_seminorm",
"mul_ring_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (p : mul_ring_seminorm R) : p.to_fun = p | rfl | lemma | mul_ring_seminorm.to_fun_eq_coe | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"mul_ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : mul_ring_seminorm R} : (∀ x, p x = q x) → p = q | fun_like.ext p q | lemma | mul_ring_seminorm.ext | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"fun_like.ext",
"mul_ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (x : R) : (1 : mul_ring_seminorm R) x = if x = 0 then 0 else 1 | rfl | lemma | mul_ring_seminorm.apply_one | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"mul_ring_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_norm_class : mul_ring_norm_class (mul_ring_norm R) R ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_zero := λ f, f.map_zero',
map_one := λ f, f.map_one',
map_add_le_add := λ f, f.add_le',
map_mul := λ f, f.map_mul',
map_neg_eq_map := λ f, f.neg',
eq_zero_of_map_eq_zero := λ f, f.eq_zero_of_map_eq_zero' } | instance | mul_ring_norm.mul_ring_norm_class | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"map_mul",
"map_one",
"mul_ring_norm",
"mul_ring_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (p : mul_ring_norm R) : p.to_fun = p | rfl | lemma | mul_ring_norm.to_fun_eq_coe | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"mul_ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : mul_ring_norm R} : (∀ x, p x = q x) → p = q | fun_like.ext p q | lemma | mul_ring_norm.ext | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"fun_like.ext",
"mul_ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (x : R) : (1 : mul_ring_norm R) x = if x = 0 then 0 else 1 | rfl | lemma | mul_ring_norm.apply_one | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"mul_ring_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_seminorm.to_ring_norm {K : Type*} [field K] (f : ring_seminorm K) (hnt : f ≠ 0) :
ring_norm K | { eq_zero_of_map_eq_zero' := λ x hx,
begin
obtain ⟨c, hc⟩ := ring_seminorm.ne_zero_iff.mp hnt,
by_contradiction hn0,
have hc0 : f c = 0,
{ rw [← mul_one c, ← mul_inv_cancel hn0, ← mul_assoc, mul_comm c, mul_assoc],
exact le_antisymm (le_trans (map_mul_le_mul f _ _)
(by rw [← ring_seminor... | def | ring_seminorm.to_ring_norm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"by_contradiction",
"field",
"map_nonneg",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_one",
"ring_norm",
"ring_seminorm",
"ring_seminorm.to_fun_eq_coe",
"zero_mul"
] | A nonzero ring seminorm on a field `K` is a ring norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_ring_norm (R : Type*) [non_unital_normed_ring R] : ring_norm R | { ..norm_add_group_norm R, ..norm_ring_seminorm R } | def | norm_ring_norm | analysis.normed.ring | src/analysis/normed/ring/seminorm.lean | [
"analysis.normed.field.basic"
] | [
"non_unital_normed_ring",
"norm_ring_seminorm",
"ring_norm"
] | The norm of a normed_ring as a ring_norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_subspace.is_closed_direction_iff (s : affine_subspace 𝕜 Q) :
is_closed (s.direction : set W) ↔ is_closed (s : set Q) | begin
rcases s.eq_bot_or_nonempty with rfl|⟨x, hx⟩, { simp [is_closed_singleton] },
rw [← (isometry_equiv.vadd_const x).to_homeomorph.symm.is_closed_image,
affine_subspace.coe_direction_eq_vsub_set_right hx],
refl
end | lemma | affine_subspace.is_closed_direction_iff | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"affine_subspace",
"affine_subspace.coe_direction_eq_vsub_set_right",
"is_closed",
"is_closed_singleton",
"isometry_equiv.vadd_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
dist p₁ (homothety p₁ c p₂) = ‖c‖ * dist p₁ p₂ | by simp [homothety_def, norm_smul, ← dist_eq_norm_vsub, dist_comm] | lemma | dist_center_homothety | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_eq_norm_vsub",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_center_homothety (p₁ p₂ : P) (c : 𝕜) :
nndist p₁ (homothety p₁ c p₂) = ‖c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_center_homothety _ _ _ | lemma | nndist_center_homothety | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_center_homothety",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
dist (homothety p₁ c p₂) p₁ = ‖c‖ * dist p₁ p₂ | by rw [dist_comm, dist_center_homothety] | lemma | dist_homothety_center | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_center_homothety",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
nndist (homothety p₁ c p₂) p₁ = ‖c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_homothety_center _ _ _ | lemma | nndist_homothety_center | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_homothety_center",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_line_map_line_map (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
dist (line_map p₁ p₂ c₁) (line_map p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ | begin
rw dist_comm p₁ p₂,
simp only [line_map_apply, dist_eq_norm_vsub, vadd_vsub_vadd_cancel_right, ← sub_smul, norm_smul,
vsub_eq_sub],
end | lemma | dist_line_map_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_eq_norm_vsub",
"norm_smul",
"sub_smul",
"vadd_vsub_vadd_cancel_right",
"vsub_eq_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_line_map_line_map (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
nndist (line_map p₁ p₂ c₁) (line_map p₁ p₂ c₂) = nndist c₁ c₂ * nndist p₁ p₂ | nnreal.eq $ dist_line_map_line_map _ _ _ _ | lemma | nndist_line_map_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_line_map",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_with_line_map (p₁ p₂ : P) :
lipschitz_with (nndist p₁ p₂) (line_map p₁ p₂ : 𝕜 → P) | lipschitz_with.of_dist_le_mul $ λ c₁ c₂,
((dist_line_map_line_map p₁ p₂ c₁ c₂).trans (mul_comm _ _)).le | lemma | lipschitz_with_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_line_map",
"lipschitz_with",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_line_map_left (p₁ p₂ : P) (c : 𝕜) :
dist (line_map p₁ p₂ c) p₁ = ‖c‖ * dist p₁ p₂ | by simpa only [line_map_apply_zero, dist_zero_right] using dist_line_map_line_map p₁ p₂ c 0 | lemma | dist_line_map_left | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_line_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_line_map_left (p₁ p₂ : P) (c : 𝕜) :
nndist (line_map p₁ p₂ c) p₁ = ‖c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_line_map_left _ _ _ | lemma | nndist_line_map_left | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_left",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_left_line_map (p₁ p₂ : P) (c : 𝕜) :
dist p₁ (line_map p₁ p₂ c) = ‖c‖ * dist p₁ p₂ | (dist_comm _ _).trans (dist_line_map_left _ _ _) | lemma | dist_left_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_line_map_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_left_line_map (p₁ p₂ : P) (c : 𝕜) :
nndist p₁ (line_map p₁ p₂ c) = ‖c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_left_line_map _ _ _ | lemma | nndist_left_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_left_line_map",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_line_map_right (p₁ p₂ : P) (c : 𝕜) :
dist (line_map p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ | by simpa only [line_map_apply_one, dist_eq_norm'] using dist_line_map_line_map p₁ p₂ c 1 | lemma | dist_line_map_right | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_line_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_line_map_right (p₁ p₂ : P) (c : 𝕜) :
nndist (line_map p₁ p₂ c) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_line_map_right _ _ _ | lemma | nndist_line_map_right | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_right",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_right_line_map (p₁ p₂ : P) (c : 𝕜) :
dist p₂ (line_map p₁ p₂ c) = ‖1 - c‖ * dist p₁ p₂ | (dist_comm _ _).trans (dist_line_map_right _ _ _) | lemma | dist_right_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_line_map_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_right_line_map (p₁ p₂ : P) (c : 𝕜) :
nndist p₂ (line_map p₁ p₂ c) = ‖1 - c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_right_line_map _ _ _ | lemma | nndist_right_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_right_line_map",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
dist (homothety p₁ c p₂) p₂ = ‖1 - c‖ * dist p₁ p₂ | by rw [homothety_eq_line_map, dist_line_map_right] | lemma | dist_homothety_self | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_line_map_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_homothety_self (p₁ p₂ : P) (c : 𝕜) :
nndist (homothety p₁ c p₂) p₂ = ‖1 - c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_homothety_self _ _ _ | lemma | nndist_homothety_self | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_homothety_self",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
dist p₂ (homothety p₁ c p₂) = ‖1 - c‖ * dist p₁ p₂ | by rw [dist_comm, dist_homothety_self] | lemma | dist_self_homothety | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_homothety_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_self_homothety (p₁ p₂ : P) (c : 𝕜) :
nndist p₂ (homothety p₁ c p₂) = ‖1 - c‖₊ * nndist p₁ p₂ | nnreal.eq $ dist_self_homothety _ _ _ | lemma | nndist_self_homothety | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_self_homothety",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_left_midpoint (p₁ p₂ : P) :
dist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2:𝕜)‖⁻¹ * dist p₁ p₂ | by rw [midpoint, dist_comm, dist_line_map_left, inv_of_eq_inv, ← norm_inv] | lemma | dist_left_midpoint | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_line_map_left",
"inv_of_eq_inv",
"midpoint",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_left_midpoint (p₁ p₂ : P) :
nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖(2:𝕜)‖₊⁻¹ * nndist p₁ p₂ | nnreal.eq $ dist_left_midpoint _ _ | lemma | nndist_left_midpoint | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_left_midpoint",
"midpoint",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_midpoint_left (p₁ p₂ : P) :
dist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2:𝕜)‖⁻¹ * dist p₁ p₂ | by rw [dist_comm, dist_left_midpoint] | lemma | dist_midpoint_left | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_left_midpoint",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_midpoint_left (p₁ p₂ : P) :
nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖(2:𝕜)‖₊⁻¹ * nndist p₁ p₂ | nnreal.eq $ dist_midpoint_left _ _ | lemma | nndist_midpoint_left | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_midpoint_left",
"midpoint",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_midpoint_right (p₁ p₂ : P) :
dist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2:𝕜)‖⁻¹ * dist p₁ p₂ | by rw [midpoint_comm, dist_midpoint_left, dist_comm] | lemma | dist_midpoint_right | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_midpoint_left",
"midpoint",
"midpoint_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_midpoint_right (p₁ p₂ : P) :
nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2:𝕜)‖₊⁻¹ * nndist p₁ p₂ | nnreal.eq $ dist_midpoint_right _ _ | lemma | nndist_midpoint_right | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_midpoint_right",
"midpoint",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_right_midpoint (p₁ p₂ : P) :
dist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2:𝕜)‖⁻¹ * dist p₁ p₂ | by rw [dist_comm, dist_midpoint_right] | lemma | dist_right_midpoint | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_comm",
"dist_midpoint_right",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_right_midpoint (p₁ p₂ : P) :
nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖(2:𝕜)‖₊⁻¹ * nndist p₁ p₂ | nnreal.eq $ dist_right_midpoint _ _ | lemma | nndist_right_midpoint | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_right_midpoint",
"midpoint",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖(2 : 𝕜)‖ | begin
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, midpoint_vsub_midpoint];
try { apply_instance },
rw [midpoint_eq_smul_add, norm_smul, inv_of_eq_inv, norm_inv, ← div_eq_inv_mul],
exact div_le_div_of_le_of_nonneg (norm_add_le _ _) (norm_nonneg _),
end | lemma | dist_midpoint_midpoint_le' | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_eq_norm_vsub",
"div_eq_inv_mul",
"div_le_div_of_le_of_nonneg",
"inv_of_eq_inv",
"midpoint",
"midpoint_eq_smul_add",
"midpoint_vsub_midpoint",
"norm_inv",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_midpoint_midpoint_le' (p₁ p₂ p₃ p₄ : P) :
nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖(2 : 𝕜)‖₊ | dist_midpoint_midpoint_le' _ _ _ _ | lemma | nndist_midpoint_midpoint_le' | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_midpoint_midpoint_le'",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_with_line_map {p₁ p₂ : Q} (h : p₁ ≠ p₂) :
antilipschitz_with (nndist p₁ p₂)⁻¹ (line_map p₁ p₂ : 𝕜 → Q) | antilipschitz_with.of_le_mul_dist $ λ c₁ c₂, by rw [dist_line_map_line_map, nnreal.coe_inv,
← dist_nndist, mul_left_comm, inv_mul_cancel (dist_ne_zero.2 h), mul_one] | lemma | antilipschitz_with_line_map | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"antilipschitz_with",
"dist_line_map_line_map",
"dist_nndist",
"inv_mul_cancel",
"mul_left_comm",
"mul_one",
"nnreal.coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_homothety_mem_of_mem_interior (x : Q) {s : set Q} {y : Q} (hy : y ∈ interior s) :
∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s | begin
rw (normed_add_comm_group.nhds_basis_norm_lt (1 : 𝕜)).eventually_iff,
cases eq_or_ne y x with h h, { use 1, simp [h.symm, interior_subset hy], },
have hxy : 0 < ‖y -ᵥ x‖, { rwa [norm_pos_iff, vsub_ne_zero], },
obtain ⟨u, hu₁, hu₂, hu₃⟩ := mem_interior.mp hy,
obtain ⟨ε, hε, hyε⟩ := metric.is_open_iff.mp... | lemma | eventually_homothety_mem_of_mem_interior | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_eq_norm_vsub",
"div_pos",
"eq_or_ne",
"interior",
"interior_subset",
"lt_div_iff",
"metric.mem_ball",
"norm_smul",
"one_smul",
"sub_smul",
"vadd_vsub_eq_sub_vsub",
"vsub_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_homothety_image_subset_of_finite_subset_interior
(x : Q) {s : set Q} {t : set Q} (ht : t.finite) (h : t ⊆ interior s) :
∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ '' t ⊆ s | begin
suffices : ∀ y ∈ t, ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s,
{ simp_rw set.image_subset_iff,
exact (filter.eventually_all_finite ht).mpr this, },
intros y hy,
exact eventually_homothety_mem_of_mem_interior 𝕜 x (h hy),
end | lemma | eventually_homothety_image_subset_of_finite_subset_interior | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"eventually_homothety_mem_of_mem_interior",
"filter.eventually_all_finite",
"interior",
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2 | by simpa using dist_midpoint_midpoint_le' p₁ p₂ p₃ p₄ | lemma | dist_midpoint_midpoint_le | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_midpoint_midpoint_le'",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_midpoint_midpoint_le (p₁ p₂ p₃ p₄ : V) :
nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2 | dist_midpoint_midpoint_le _ _ _ _ | lemma | nndist_midpoint_midpoint_le | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"dist_midpoint_midpoint_le",
"midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_map.of_map_midpoint (f : P → Q)
(h : ∀ x y, f (midpoint ℝ x y) = midpoint ℝ (f x) (f y))
(hfc : continuous f) :
P →ᵃ[ℝ] Q | affine_map.mk' f
↑((add_monoid_hom.of_map_midpoint ℝ ℝ
((affine_equiv.vadd_const ℝ (f $ classical.arbitrary P)).symm ∘ f ∘
(affine_equiv.vadd_const ℝ (classical.arbitrary P))) (by simp)
(λ x y, by simp [h])).to_real_linear_map $ by apply_rules [continuous.vadd, continuous.vsub,
continuous_cons... | def | affine_map.of_map_midpoint | analysis.normed_space | src/analysis/normed_space/add_torsor.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.add_torsor",
"linear_algebra.affine_space.midpoint_zero",
"linear_algebra.affine_space.affine_subspace",
"topology.instances.real_vector_space"
] | [
"add_monoid_hom.of_map_midpoint",
"affine_equiv.vadd_const",
"affine_map.mk'",
"classical.arbitrary",
"continuous",
"continuous.vsub",
"continuous_const",
"continuous_id",
"midpoint"
] | A continuous map between two normed affine spaces is an affine map provided that
it sends midpoints to midpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_barycentric_coord [nontrivial ι] (b : affine_basis ι 𝕜 P) (i : ι) :
is_open_map (b.coord i) | affine_map.is_open_map_linear_iff.mp $ (b.coord i).linear.is_open_map_of_finite_dimensional $
(b.coord i).linear_surjective_iff.mpr (b.surjective_coord i) | lemma | is_open_map_barycentric_coord | analysis.normed_space | src/analysis/normed_space/add_torsor_bases.lean | [
"analysis.normed_space.finite_dimension",
"analysis.calculus.affine_map",
"analysis.convex.combination",
"linear_algebra.affine_space.finite_dimensional"
] | [
"affine_basis",
"is_open_map",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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