statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
norm_eq_abs (n : ℤ) : ‖n‖ = |n| | rfl | lemma | int.norm_eq_abs | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_nat (n : ℕ) : ‖(n : ℤ)‖ = n | by simp [int.norm_eq_abs] | lemma | int.norm_coe_nat | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"int.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.nnreal.coe_nat_abs (n : ℤ) : (n.nat_abs : ℝ≥0) = ‖n‖₊ | nnreal.eq $ calc ((n.nat_abs : ℝ≥0) : ℝ)
= (n.nat_abs : ℤ) : by simp only [int.cast_coe_nat, nnreal.coe_nat_cast]
... = |n| : by simp only [int.coe_nat_abs, int.cast_abs]
... = ‖n‖ : rfl | lemma | nnreal.coe_nat_abs | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"int.cast_abs",
"int.cast_coe_nat",
"int.coe_nat_abs",
"nnreal.coe_nat_cast",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_floor_nnreal_iff (z : ℤ) (c : ℝ≥0) : |z| ≤ ⌊c⌋₊ ↔ ‖z‖₊ ≤ c | begin
rw [int.abs_eq_nat_abs, int.coe_nat_le, nat.le_floor_iff (zero_le c)],
congr',
exact nnreal.coe_nat_abs z,
end | lemma | int.abs_le_floor_nnreal_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"int.abs_eq_nat_abs",
"int.coe_nat_le",
"nat.le_floor_iff",
"nnreal.coe_nat_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cast_real (r : ℚ) : ‖(r : ℝ)‖ = ‖r‖ | rfl | lemma | rat.norm_cast_real | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.int.norm_cast_rat (m : ℤ) : ‖(m : ℚ)‖ = ‖m‖ | by rw [← rat.norm_cast_real, ← int.norm_cast_real]; congr' 1; norm_cast | lemma | int.norm_cast_rat | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"int.norm_cast_real",
"rat.norm_cast_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a^n‖ ≤ ‖n‖ * ‖a‖ | by rcases n.eq_coe_or_neg with ⟨n, rfl | rfl⟩; simpa using norm_pow_le_mul_norm n a | lemma | norm_zpow_le_mul_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_pow_le_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_zpow_le_mul_norm (n : ℤ) (a : α) : ‖a^n‖₊ ≤ ‖n‖₊ * ‖a‖₊ | by simpa only [← nnreal.coe_le_coe, nnreal.coe_mul] using norm_zpow_le_mul_norm n a | lemma | nnnorm_zpow_le_mul_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"nnreal.coe_le_coe",
"nnreal.coe_mul",
"norm_zpow_le_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv (hf : lipschitz_with K f) : lipschitz_with K (λ x, (f x)⁻¹) | λ x y, (edist_inv_inv _ _).trans_le $ hf x y | lemma | lipschitz_with.inv | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"edist_inv_inv",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul' (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (λ x, f x * g x) | λ x y, calc
edist (f x * g x) (f y * g y) ≤ edist (f x) (f y) + edist (g x) (g y) : edist_mul_mul_le _ _ _ _
... ≤ Kf * edist x y + Kg * edist x y : add_le_add (hf x y) (hg x y)
... = (Kf + Kg) * edist x y : (add_mul _ _ _).symm | lemma | lipschitz_with.mul' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"edist_mul_mul_le",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :
lipschitz_with (Kf + Kg) (λ x, f x / g x) | by simpa only [div_eq_mul_inv] using hf.mul' hg.inv | lemma | lipschitz_with.div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_eq_mul_inv",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lipschitz_with (hf : antilipschitz_with Kf f) (hg : lipschitz_with Kg g)
(hK : Kg < Kf⁻¹) : antilipschitz_with (Kf⁻¹ - Kg)⁻¹ (λ x, f x * g x) | begin
letI : pseudo_metric_space α := pseudo_emetric_space.to_pseudo_metric_space hf.edist_ne_top,
refine antilipschitz_with.of_le_mul_dist (λ x y, _),
rw [nnreal.coe_inv, ← div_eq_inv_mul],
rw le_div_iff (nnreal.coe_pos.2 $ tsub_pos_iff_lt.2 hK),
rw [mul_comm, nnreal.coe_sub hK.le, sub_mul],
calc ↑Kf⁻¹ * d... | lemma | antilipschitz_with.mul_lipschitz_with | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"abs_dist_sub_le_dist_mul_mul",
"antilipschitz_with",
"div_eq_inv_mul",
"le_abs_self",
"le_div_iff",
"lipschitz_with",
"mul_comm",
"nnreal.coe_inv",
"nnreal.coe_sub",
"pseudo_emetric_space.to_pseudo_metric_space",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_lipschitz_with (hf : antilipschitz_with Kf f)
(hg : lipschitz_with Kg (g / f)) (hK : Kg < Kf⁻¹) : antilipschitz_with (Kf⁻¹ - Kg)⁻¹ g | by simpa only [pi.div_apply, mul_div_cancel'_right] using hf.mul_lipschitz_with hg hK | lemma | antilipschitz_with.mul_div_lipschitz_with | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"antilipschitz_with",
"lipschitz_with",
"mul_div_cancel'_right",
"pi.div_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_norm_div {f : E → F} (hf : antilipschitz_with K f) (x y : E) :
‖x / y‖ ≤ K * ‖f x / f y‖ | by simp [← dist_eq_norm_div, hf.le_mul_dist x y] | lemma | antilipschitz_with.le_mul_norm_div | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"antilipschitz_with",
"dist_eq_norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_comm_group.to_has_lipschitz_mul : has_lipschitz_mul E | ⟨⟨1 + 1, lipschitz_with.prod_fst.mul' lipschitz_with.prod_snd⟩⟩ | instance | seminormed_comm_group.to_has_lipschitz_mul | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"has_lipschitz_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_comm_group.to_uniform_group : uniform_group E | ⟨(lipschitz_with.prod_fst.div lipschitz_with.prod_snd).uniform_continuous⟩ | instance | seminormed_comm_group.to_uniform_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"lipschitz_with.prod_snd",
"uniform_group"
] | A seminormed group is a uniform group, i.e., multiplication and division are uniformly
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group.to_topological_group : topological_group E | infer_instance | instance | seminormed_comm_group.to_topological_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ}
(huv : ∀ n ≥ N, u n = v n) (hv : cauchy_seq (λ n, ∏ k in range (n+1), v k)) :
cauchy_seq (λ n, ∏ k in range (n + 1), u k) | begin
let d : ℕ → E := λ n, ∏ k in range (n + 1), (u k / v k),
rw show (λ n, ∏ k in range (n + 1), u k) = d * (λ n, ∏ k in range (n + 1), v k),
by { ext n, simp [d] },
suffices : ∀ n ≥ N, d n = d N,
{ exact (tendsto_at_top_of_eventually_const this).cauchy_seq.mul hv },
intros n hn,
dsimp [d],
rw event... | lemma | cauchy_seq_prod_of_eventually_eq | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"cauchy_seq",
"cauchy_seq.mul",
"tendsto_at_top_of_eventually_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_zero'' : ‖a‖ = 0 ↔ a = 1 | norm_eq_zero''' | lemma | norm_eq_zero'' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_eq_zero'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_ne_zero_iff' : ‖a‖ ≠ 0 ↔ a ≠ 1 | norm_eq_zero''.not | lemma | norm_ne_zero_iff' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pos_iff'' : 0 < ‖a‖ ↔ a ≠ 1 | norm_pos_iff''' | lemma | norm_pos_iff'' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_pos_iff'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_zero_iff'' : ‖a‖ ≤ 0 ↔ a = 1 | norm_le_zero_iff''' | lemma | norm_le_zero_iff'' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_zero_iff'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_eq_zero_iff : ‖a / b‖ = 0 ↔ a = b | by rw [norm_eq_zero'', div_eq_one] | lemma | norm_div_eq_zero_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"div_eq_one",
"norm_eq_zero''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div_pos_iff : 0 < ‖a / b‖ ↔ a ≠ b | by { rw [(norm_nonneg' _).lt_iff_ne, ne_comm], exact norm_div_eq_zero_iff.not } | lemma | norm_div_pos_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"ne_comm",
"norm_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_norm_div_le_zero (h : ‖a / b‖ ≤ 0) : a = b | by rwa [←div_eq_one, ← norm_le_zero_iff''] | lemma | eq_of_norm_div_le_zero | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_zero_iff''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq_zero' : ‖a‖₊ = 0 ↔ a = 1 | by rw [← nnreal.coe_eq_zero, coe_nnnorm', norm_eq_zero''] | lemma | nnnorm_eq_zero' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"coe_nnnorm'",
"nnreal.coe_eq_zero",
"norm_eq_zero''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_ne_zero_iff' : ‖a‖₊ ≠ 0 ↔ a ≠ 1 | nnnorm_eq_zero'.not | lemma | nnnorm_ne_zero_iff' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_div_self_punctured_nhds (a : E) : tendsto (λ x, ‖x / a‖) (𝓝[≠] a) (𝓝[>] 0) | (tendsto_norm_div_self a).inf $ tendsto_principal_principal.2 $ λ x hx, norm_pos_iff''.2 $
div_ne_one.2 hx | lemma | tendsto_norm_div_self_punctured_nhds | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"tendsto_norm_div_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_nhds_within_one : tendsto (norm : E → ℝ) (𝓝[≠] 1) (𝓝[>] 0) | tendsto_norm_one.inf $ tendsto_principal_principal.2 $ λ x, norm_pos_iff''.2 | lemma | tendsto_norm_nhds_within_one | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_group_norm : group_norm E | { eq_one_of_map_eq_zero' := λ _, norm_eq_zero''.1, ..norm_group_seminorm _ } | def | norm_group_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"group_norm",
"norm_group_seminorm"
] | The norm of a normed group as a group norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm_group_norm : ⇑(norm_group_norm E) = norm | rfl | lemma | coe_norm_group_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_group_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support_norm_iff : has_compact_support (λ x, ‖f x‖) ↔ has_compact_support f | has_compact_support_comp_left $ λ x, norm_eq_zero | lemma | has_compact_support_norm_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.bounded_above_of_compact_support (hf : continuous f) (h : has_compact_support f) :
∃ C, ∀ x, ‖f x‖ ≤ C | by simpa [bdd_above_def] using hf.norm.bdd_above_range_of_has_compact_support h.norm | lemma | continuous.bounded_above_of_compact_support | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"bdd_above_def",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.exists_pos_le_norm [has_one E] (hf : has_compact_mul_support f) :
∃ (R : ℝ), (0 < R) ∧ (∀ (x : α), (R ≤ ‖x‖) → (f x = 1)) | begin
obtain ⟨K, ⟨hK1, hK2⟩⟩ := exists_compact_iff_has_compact_mul_support.mpr hf,
obtain ⟨S, hS, hS'⟩ := hK1.bounded.exists_pos_norm_le,
refine ⟨S + 1, by positivity, λ x hx, hK2 x ((mt $ hS' x) _)⟩,
contrapose! hx,
exact lt_add_of_le_of_pos hx zero_lt_one
end | lemma | has_compact_mul_support.exists_pos_le_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"has_compact_mul_support",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_def (x : ulift E) : ‖x‖ = ‖x.down‖ | rfl | lemma | ulift.norm_def | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_up (x : E) : ‖ulift.up x‖ = ‖x‖ | rfl | lemma | ulift.norm_up | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_down (x : ulift E) : ‖x.down‖ = ‖x‖ | rfl | lemma | ulift.norm_down | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_def (x : ulift E) : ‖x‖₊ = ‖x.down‖₊ | rfl | lemma | ulift.nnnorm_def | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_up (x : E) : ‖ulift.up x‖₊ = ‖x‖₊ | rfl | lemma | ulift.nnnorm_up | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_down (x : ulift E) : ‖x.down‖₊ = ‖x‖₊ | rfl | lemma | ulift.nnnorm_down | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group [seminormed_group E] : seminormed_group (ulift E) | seminormed_group.induced _ _ (⟨ulift.down, rfl, λ _ _, rfl⟩ : ulift E →* E) | instance | ulift.seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_group",
"seminormed_group.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_comm_group [seminormed_comm_group E] : seminormed_comm_group (ulift E) | seminormed_comm_group.induced _ _ (⟨ulift.down, rfl, λ _ _, rfl⟩ : ulift E →* E) | instance | ulift.seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_comm_group",
"seminormed_comm_group.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_group [normed_group E] : normed_group (ulift E) | normed_group.induced _ _ (⟨ulift.down, rfl, λ _ _, rfl⟩ : ulift E →* E) down_injective | instance | ulift.normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_group",
"normed_group.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group [normed_comm_group E] : normed_comm_group (ulift E) | normed_comm_group.induced _ _ (⟨ulift.down, rfl, λ _ _, rfl⟩ : ulift E →* E) down_injective | instance | ulift.normed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group",
"normed_comm_group.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_mul (x) : ‖(to_mul x : E)‖ = ‖x‖ | rfl | lemma | norm_to_mul | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_mul (x : E) : ‖of_mul x‖ = ‖x‖ | rfl | lemma | norm_of_mul | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_add (x) : ‖(to_add x : E)‖ = ‖x‖ | rfl | lemma | norm_to_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_add (x : E) : ‖of_add x‖ = ‖x‖ | rfl | lemma | norm_of_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_to_mul (x) : ‖(to_mul x : E)‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_to_mul | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_mul (x : E) : ‖of_mul x‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_of_mul | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_to_add (x) : ‖(to_add x : E)‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_to_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_add (x : E) : ‖of_add x‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_of_add | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_dual (x : E) : ‖to_dual x‖ = ‖x‖ | rfl | lemma | norm_to_dual | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_dual (x : Eᵒᵈ) : ‖of_dual x‖ = ‖x‖ | rfl | lemma | norm_of_dual | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_to_dual (x : E) : ‖to_dual x‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_to_dual | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_dual (x : Eᵒᵈ) : ‖of_dual x‖₊ = ‖x‖₊ | rfl | lemma | nnnorm_of_dual | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.norm_def (x : E × F) : ‖x‖ = (max ‖x.1‖ ‖x.2‖) | rfl | lemma | prod.norm_def | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fst_le (x : E × F) : ‖x.1‖ ≤ ‖x‖ | le_max_left _ _ | lemma | norm_fst_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_snd_le (x : E × F) : ‖x.2‖ ≤ ‖x‖ | le_max_right _ _ | lemma | norm_snd_le | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_prod_le_iff : ‖x‖ ≤ r ↔ ‖x.1‖ ≤ r ∧ ‖x.2‖ ≤ r | max_le_iff | lemma | norm_prod_le_iff | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"max_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.nnorm_def (x : E × F) : ‖x‖₊ = (max ‖x.1‖₊ ‖x.2‖₊) | rfl | lemma | prod.nnorm_def | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.norm_def' : ‖f‖ = ↑(finset.univ.sup (λ b, ‖f b‖₊)) | rfl | lemma | pi.norm_def' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.nnnorm_def' : ‖f‖₊ = finset.univ.sup (λ b, ‖f b‖₊) | subtype.eta _ _ | lemma | pi.nnnorm_def' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_norm_le_iff_of_nonneg' (hr : 0 ≤ r) : ‖x‖ ≤ r ↔ ∀ i, ‖x i‖ ≤ r | by simp only [←dist_one_right, dist_pi_le_iff hr, pi.one_apply] | lemma | pi_norm_le_iff_of_nonneg' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_pi_le_iff",
"pi.one_apply"
] | The seminorm of an element in a product space is `≤ r` if and only if the norm of each
component is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_nnnorm_le_iff' {r : ℝ≥0} : ‖x‖₊ ≤ r ↔ ∀ i, ‖x i‖₊ ≤ r | pi_norm_le_iff_of_nonneg' r.coe_nonneg | lemma | pi_nnnorm_le_iff' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"pi_norm_le_iff_of_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_norm_le_iff_of_nonempty' [nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r | begin
by_cases hr : 0 ≤ r,
{ exact pi_norm_le_iff_of_nonneg' hr },
{ exact iff_of_false (λ h, hr $ (norm_nonneg' _).trans h)
(λ h, hr $ (norm_nonneg' _).trans $ h $ classical.arbitrary _) }
end | lemma | pi_norm_le_iff_of_nonempty' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"classical.arbitrary",
"iff_of_false",
"norm_nonneg'",
"pi_norm_le_iff_of_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_norm_lt_iff' (hr : 0 < r) : ‖x‖ < r ↔ ∀ i, ‖x i‖ < r | by simp only [←dist_one_right, dist_pi_lt_iff hr, pi.one_apply] | lemma | pi_norm_lt_iff' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_pi_lt_iff",
"pi.one_apply"
] | The seminorm of an element in a product space is `< r` if and only if the norm of each
component is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_nnnorm_lt_iff' {r : ℝ≥0} (hr : 0 < r) : ‖x‖₊ < r ↔ ∀ i, ‖x i‖₊ < r | pi_norm_lt_iff' hr | lemma | pi_nnnorm_lt_iff' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"pi_norm_lt_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_pi_norm' (i : ι) : ‖f i‖ ≤ ‖f‖ | (pi_norm_le_iff_of_nonneg' $ norm_nonneg' _).1 le_rfl i | lemma | norm_le_pi_norm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"le_rfl",
"norm_nonneg'",
"pi_norm_le_iff_of_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_le_pi_nnnorm' (i : ι) : ‖f i‖₊ ≤ ‖f‖₊ | norm_le_pi_norm' _ i | lemma | nnnorm_le_pi_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"norm_le_pi_norm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_norm_const_le' (a : E) : ‖(λ _ : ι, a)‖ ≤ ‖a‖ | (pi_norm_le_iff_of_nonneg' $ norm_nonneg' _).2 $ λ _, le_rfl | lemma | pi_norm_const_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"le_rfl",
"norm_nonneg'",
"pi_norm_le_iff_of_nonneg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_nnnorm_const_le' (a : E) : ‖(λ _ : ι, a)‖₊ ≤ ‖a‖₊ | pi_norm_const_le' _ | lemma | pi_nnnorm_const_le' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"pi_norm_const_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_norm_const' [nonempty ι] (a : E) : ‖(λ i : ι, a)‖ = ‖a‖ | by simpa only [←dist_one_right] using dist_pi_const a 1 | lemma | pi_norm_const' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"dist_pi_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_nnnorm_const' [nonempty ι] (a : E) : ‖(λ i : ι, a)‖₊ = ‖a‖₊ | nnreal.eq $ pi_norm_const' a | lemma | pi_nnnorm_const' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"nnreal.eq",
"pi_norm_const'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ fintype.card ι • ‖f‖ | finset.sum_le_card_nsmul _ _ _ $ λ i hi, norm_le_pi_norm' _ i | lemma | pi.sum_norm_apply_le_norm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"fintype.card",
"norm_le_pi_norm'"
] | The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ fintype.card ι • ‖f‖₊ | nnreal.coe_sum.trans_le $ pi.sum_norm_apply_le_norm' _ | lemma | pi.sum_nnnorm_apply_le_nnnorm' | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"fintype.card",
"pi.sum_norm_apply_le_norm'"
] | The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.seminormed_comm_group [Π i, seminormed_comm_group (π i)] :
seminormed_comm_group (Π i, π i) | { ..pi.seminormed_group } | instance | pi.seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_comm_group"
] | Finite product of seminormed groups, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.normed_group [Π i, normed_group (π i)] : normed_group (Π i, π i) | { ..pi.seminormed_group } | instance | pi.normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_group"
] | Finite product of normed groups, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.normed_comm_group [Π i, normed_comm_group (π i)] : normed_comm_group (Π i, π i) | { ..pi.seminormed_group } | instance | pi.normed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group"
] | Finite product of normed groups, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_op [seminormed_add_group E] (a : E) : ‖mul_opposite.op a‖ = ‖a‖ | rfl | lemma | mul_opposite.norm_op | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_unop [seminormed_add_group E] (a : Eᵐᵒᵖ) : ‖mul_opposite.unop a‖ = ‖a‖ | rfl | lemma | mul_opposite.norm_unop | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_op [seminormed_add_group E] (a : E) : ‖mul_opposite.op a‖₊ = ‖a‖₊ | rfl | lemma | mul_opposite.nnnorm_op | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_unop [seminormed_add_group E] (a : Eᵐᵒᵖ) : ‖mul_opposite.unop a‖₊ = ‖a‖₊ | rfl | lemma | mul_opposite.nnnorm_unop | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_group : seminormed_group s | seminormed_group.induced _ _ s.subtype | instance | subgroup.seminormed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_group",
"seminormed_group.induced"
] | A subgroup of a seminormed group is also a seminormed group,
with the restriction of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm (x : s) : ‖x‖ = ‖(x : E)‖ | rfl | lemma | subgroup.coe_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [] | If `x` is an element of a subgroup `s` of a seminormed group `E`, its norm in `s` is equal to
its norm in `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_coe {s : subgroup E} (x : s) : ‖(x : E)‖ = ‖x‖ | rfl | lemma | subgroup.norm_coe | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"subgroup"
] | If `x` is an element of a subgroup `s` of a seminormed group `E`, its norm in `s` is equal to
its norm in `E`.
This is a reversed version of the `simp` lemma `subgroup.coe_norm` for use by `norm_cast`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_comm_group [seminormed_comm_group E] {s : subgroup E} :
seminormed_comm_group s | seminormed_comm_group.induced _ _ s.subtype | instance | subgroup.seminormed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"seminormed_comm_group",
"seminormed_comm_group.induced",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_group [normed_group E] {s : subgroup E} : normed_group s | normed_group.induced _ _ s.subtype subtype.coe_injective | instance | subgroup.normed_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_group",
"normed_group.induced",
"subgroup",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_comm_group [normed_comm_group E] {s : subgroup E} : normed_comm_group s | normed_comm_group.induced _ _ s.subtype subtype.coe_injective | instance | subgroup.normed_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"normed_comm_group",
"normed_comm_group.induced",
"subgroup",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seminormed_add_comm_group {_ : ring 𝕜} [seminormed_add_comm_group E] {_ : module 𝕜 E}
(s : submodule 𝕜 E) :
seminormed_add_comm_group s | seminormed_add_comm_group.induced _ _ s.subtype.to_add_monoid_hom | instance | submodule.seminormed_add_comm_group | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"module",
"ring",
"seminormed_add_comm_group",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_norm {_ : ring 𝕜} [seminormed_add_comm_group E] {_ : module 𝕜 E}
{s : submodule 𝕜 E} (x : s) :
‖x‖ = ‖(x : E)‖ | rfl | lemma | submodule.coe_norm | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"module",
"ring",
"seminormed_add_comm_group",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe {_ : ring 𝕜} [seminormed_add_comm_group E] {_ : module 𝕜 E}
{s : submodule 𝕜 E} (x : s) :
‖(x : E)‖ = ‖x‖ | rfl | lemma | submodule.norm_coe | analysis.normed.group | src/analysis/normed/group/basic.lean | [
"analysis.normed.group.seminorm",
"order.liminf_limsup",
"topology.algebra.uniform_group",
"topology.instances.rat",
"topology.metric_space.algebra",
"topology.metric_space.isometric_smul",
"topology.sequences"
] | [
"module",
"ring",
"seminormed_add_comm_group",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe {E} [seminormed_add_comm_group E] (x : E) :
‖(x : completion E)‖ = ‖x‖ | completion.extension_coe uniform_continuous_norm x | lemma | uniform_space.completion.norm_coe | analysis.normed.group | src/analysis/normed/group/completion.lean | [
"analysis.normed.group.basic",
"topology.algebra.group_completion",
"topology.metric_space.completion"
] | [
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
controlled_closure_of_complete {f : normed_add_group_hom G H} {K : add_subgroup H}
{C ε : ℝ} (hC : 0 < C) (hε : 0 < ε) (hyp : f.surjective_on_with K C) :
f.surjective_on_with K.topological_closure (C + ε) | begin
rintros (h : H) (h_in : h ∈ K.topological_closure),
/- We first get rid of the easy case where `h = 0`.-/
by_cases hyp_h : h = 0,
{ rw hyp_h,
use 0,
simp },
/- The desired preimage will be constructed as the sum of a series. Convergence of
the series will be guaranteed by completeness of `G`. ... | lemma | controlled_closure_of_complete | analysis.normed.group | src/analysis/normed/group/controlled_closure.lean | [
"analysis.normed.group.hom",
"analysis.specific_limits.normed"
] | [
"add_subgroup",
"cauchy_seq",
"cauchy_seq_tendsto_of_complete",
"div_pos",
"le_of_tendsto'",
"mul_comm",
"mul_div_cancel'",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"norm_sum_le",
"normed_add_comm_group.cauchy_series_of_le_geometric''",
"normed_add_group_hom",
"one_half_lt_... | Given `f : normed_add_group_hom G H` for some complete `G` and a subgroup `K` of `H`, if every
element `x` of `K` has a preimage under `f` whose norm is at most `C*‖x‖` then the same holds for
elements of the (topological) closure of `K` with constant `C+ε` instead of `C`, for any
positive `ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
controlled_closure_range_of_complete {f : normed_add_group_hom G H}
{K : Type*} [seminormed_add_comm_group K] {j : normed_add_group_hom K H} (hj : ∀ x, ‖j x‖ = ‖x‖)
{C ε : ℝ} (hC : 0 < C) (hε : 0 < ε) (hyp : ∀ k, ∃ g, f g = j k ∧ ‖g‖ ≤ C*‖k‖) :
f.surjective_on_with j.range.topological_closure (C + ε) | begin
replace hyp : ∀ h ∈ j.range, ∃ g, f g = h ∧ ‖g‖ ≤ C*‖h‖,
{ intros h h_in,
rcases (j.mem_range _).mp h_in with ⟨k, rfl⟩,
rw hj,
exact hyp k },
exact controlled_closure_of_complete hC hε hyp
end | lemma | controlled_closure_range_of_complete | analysis.normed.group | src/analysis/normed/group/controlled_closure.lean | [
"analysis.normed.group.hom",
"analysis.specific_limits.normed"
] | [
"controlled_closure_of_complete",
"normed_add_group_hom",
"seminormed_add_comm_group"
] | Given `f : normed_add_group_hom G H` for some complete `G`, if every element `x` of the image of
an isometric immersion `j : normed_add_group_hom K H` has a preimage under `f` whose norm is at most
`C*‖x‖` then the same holds for elements of the (topological) closure of this image with constant
`C+ε` instead of `C`, fo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_group_hom (V W : Type*) [seminormed_add_comm_group V]
[seminormed_add_comm_group W] | (to_fun : V → W)
(map_add' : ∀ v₁ v₂, to_fun (v₁ + v₂) = to_fun v₁ + to_fun v₂)
(bound' : ∃ C, ∀ v, ‖to_fun v‖ ≤ C * ‖v‖) | structure | normed_add_group_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"seminormed_add_comm_group"
] | A morphism of seminormed abelian groups is a bounded group homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_normed_add_group_hom (f : V →+ W)
(C : ℝ) (h : ∀ v, ‖f v‖ ≤ C * ‖v‖) : normed_add_group_hom V W | { bound' := ⟨C, h⟩, ..f } | def | add_monoid_hom.mk_normed_add_group_hom | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"normed_add_group_hom"
] | Associate to a group homomorphism a bounded group homomorphism under a norm control condition.
See `add_monoid_hom.mk_normed_add_group_hom'` for a version that uses `ℝ≥0` for the bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_normed_add_group_hom' (f : V →+ W) (C : ℝ≥0) (hC : ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊) :
normed_add_group_hom V W | { bound' := ⟨C, hC⟩ .. f} | def | add_monoid_hom.mk_normed_add_group_hom' | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"normed_add_group_hom"
] | Associate to a group homomorphism a bounded group homomorphism under a norm control condition.
See `add_monoid_hom.mk_normed_add_group_hom` for a version that uses `ℝ` for the bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_pos_bound_of_bound {V W : Type*} [seminormed_add_comm_group V]
[seminormed_add_comm_group W]
{f : V → W} (M : ℝ) (h : ∀x, ‖f x‖ ≤ M * ‖x‖) :
∃ N, 0 < N ∧ ∀x, ‖f x‖ ≤ N * ‖x‖ | ⟨max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _), λx, calc
‖f x‖ ≤ M * ‖x‖ : h x
... ≤ max M 1 * ‖x‖ : mul_le_mul_of_nonneg_right (le_max_left _ _) (norm_nonneg _) ⟩ | lemma | exists_pos_bound_of_bound | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"mul_le_mul_of_nonneg_right",
"seminormed_add_comm_group",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj (H : (f : V₁ → V₂) = g) : f = g | by cases f; cases g; congr'; exact funext H | lemma | normed_add_group_hom.coe_inj | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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