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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|---|---|---|---|---|---|---|---|---|---|---|
mul_left_bound (x : α) :
∀ (y:α), ‖add_monoid_hom.mul_left x y‖ ≤ ‖x‖ * ‖y‖ | norm_mul_le x | lemma | mul_left_bound | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul_le"
] | In a seminormed ring, the left-multiplication `add_monoid_hom` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_bound (x : α) :
∀ (y:α), ‖add_monoid_hom.mul_right x y‖ ≤ ‖x‖ * ‖y‖ | λ y, by {rw mul_comm, convert norm_mul_le y x} | lemma | mul_right_bound | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"norm_mul_le"
] | In a seminormed ring, the right-multiplication `add_monoid_hom` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.non_unital_semi_normed_ring [non_unital_semi_normed_ring β] :
non_unital_semi_normed_ring (α × β) | { norm_mul := assume x y,
calc
‖x * y‖ = ‖(x.1*y.1, x.2*y.2)‖ : rfl
... = (max ‖x.1*y.1‖ ‖x.2*y.2‖) : rfl
... ≤ (max (‖x.1‖*‖y.1‖) (‖x.2‖*‖y.2‖)) :
max_le_max (norm_mul_le (x.1) (y.1)) (norm_mul_le (x.2) (y.2))
... = (max (‖x.1‖*‖y.1‖) (‖y.2‖*‖x.2‖)) : by simp[mul_comm]
... | instance | prod.non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"max_le_max",
"max_mul_mul_le_max_mul_max",
"mul_comm",
"non_unital_semi_normed_ring",
"norm_mul",
"norm_mul_le"
] | Non-unital seminormed ring structure on the product of two non-unital seminormed rings,
using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.non_unital_semi_normed_ring {π : ι → Type*} [fintype ι]
[Π i, non_unital_semi_normed_ring (π i)] :
non_unital_semi_normed_ring (Π i, π i) | { norm_mul := λ x y, nnreal.coe_mono $
calc finset.univ.sup (λ i, ‖x i * y i‖₊)
≤ finset.univ.sup ((λ i, ‖x i‖₊) * (λ i, ‖y i‖₊)) :
finset.sup_mono_fun $ λ b hb, norm_mul_le _ _
... ≤ finset.univ.sup (λ i, ‖x i‖₊) * finset.univ.sup (λ i, ‖y i‖₊) :
finset.sup_mul_le_mul_sup_of_no... | instance | pi.non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset.sup_mono_fun",
"finset.sup_mul_le_mul_sup_of_nonneg",
"fintype",
"nnreal.coe_mono",
"non_unital_semi_normed_ring",
"norm_mul",
"norm_mul_le"
] | Non-unital seminormed ring structure on the product of finitely many non-unital seminormed
rings, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.non_unital_semi_normed_ring : non_unital_semi_normed_ring αᵐᵒᵖ | { norm_mul := mul_opposite.rec $ λ x, mul_opposite.rec $ λ y,
(norm_mul_le y x).trans_eq (mul_comm _ _),
..mul_opposite.seminormed_add_comm_group } | instance | mul_opposite.non_unital_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"mul_opposite.rec",
"non_unital_semi_normed_ring",
"norm_mul",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.semi_normed_ring {𝕜 : Type*} {_ : comm_ring 𝕜}
{E : Type*} [semi_normed_ring E] {_ : algebra 𝕜 E} (s : subalgebra 𝕜 E) : semi_normed_ring s | { norm_mul := λ a b, norm_mul_le a.1 b.1,
..s.to_submodule.seminormed_add_comm_group } | instance | subalgebra.semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"algebra",
"comm_ring",
"norm_mul",
"norm_mul_le",
"semi_normed_ring",
"subalgebra"
] | A subalgebra of a seminormed ring is also a seminormed ring, with the restriction of the norm.
See note [implicit instance arguments]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.normed_ring {𝕜 : Type*} {_ : comm_ring 𝕜}
{E : Type*} [normed_ring E] {_ : algebra 𝕜 E} (s : subalgebra 𝕜 E) : normed_ring s | { ..s.semi_normed_ring } | instance | subalgebra.normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"algebra",
"comm_ring",
"normed_ring",
"subalgebra"
] | A subalgebra of a normed ring is also a normed ring, with the restriction of the norm.
See note [implicit instance arguments]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.norm_cast_le : ∀ n : ℕ, ‖(n : α)‖ ≤ n * ‖(1 : α)‖ | | 0 := by simp
| (n + 1) := by { rw [n.cast_succ, n.cast_succ, add_mul, one_mul],
exact norm_add_le_of_le (nat.norm_cast_le n) le_rfl } | lemma | nat.norm_cast_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"le_rfl",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.norm_prod_le' : ∀ {l : list α}, l ≠ [] → ‖l.prod‖ ≤ (l.map norm).prod | | [] h := (h rfl).elim
| [a] _ := by simp
| (a :: b :: l) _ :=
begin
rw [list.map_cons, list.prod_cons, @list.prod_cons _ _ _ ‖a‖],
refine le_trans (norm_mul_le _ _) (mul_le_mul_of_nonneg_left _ (norm_nonneg _)),
exact list.norm_prod_le' (list.cons_ne_nil b l)
end | lemma | list.norm_prod_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"list.cons_ne_nil",
"list.prod_cons",
"mul_le_mul_of_nonneg_left",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.nnnorm_prod_le' {l : list α} (hl : l ≠ []) : ‖l.prod‖₊ ≤ (l.map nnnorm).prod | (list.norm_prod_le' hl).trans_eq $ by simp [nnreal.coe_list_prod, list.map_map] | lemma | list.nnnorm_prod_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"list.norm_prod_le'",
"nnreal.coe_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.norm_prod_le [norm_one_class α] : ∀ l : list α, ‖l.prod‖ ≤ (l.map norm).prod | | [] := by simp
| (a::l) := list.norm_prod_le' (list.cons_ne_nil a l) | lemma | list.norm_prod_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"list.cons_ne_nil",
"list.norm_prod_le'",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.nnnorm_prod_le [norm_one_class α] (l : list α) : ‖l.prod‖₊ ≤ (l.map nnnorm).prod | l.norm_prod_le.trans_eq $ by simp [nnreal.coe_list_prod, list.map_map] | lemma | list.nnnorm_prod_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnreal.coe_list_prod",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.norm_prod_le' {α : Type*} [normed_comm_ring α] (s : finset ι) (hs : s.nonempty)
(f : ι → α) :
‖∏ i in s, f i‖ ≤ ∏ i in s, ‖f i‖ | begin
rcases s with ⟨⟨l⟩, hl⟩,
have : l.map f ≠ [], by simpa using hs,
simpa using list.norm_prod_le' this
end | lemma | finset.norm_prod_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"list.norm_prod_le'",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.nnnorm_prod_le' {α : Type*} [normed_comm_ring α] (s : finset ι) (hs : s.nonempty)
(f : ι → α) :
‖∏ i in s, f i‖₊ ≤ ∏ i in s, ‖f i‖₊ | (s.norm_prod_le' hs f).trans_eq $ by simp [nnreal.coe_prod] | lemma | finset.nnnorm_prod_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"nnreal.coe_prod",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.norm_prod_le {α : Type*} [normed_comm_ring α] [norm_one_class α] (s : finset ι)
(f : ι → α) :
‖∏ i in s, f i‖ ≤ ∏ i in s, ‖f i‖ | begin
rcases s with ⟨⟨l⟩, hl⟩,
simpa using (l.map f).norm_prod_le
end | lemma | finset.norm_prod_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"norm_one_class",
"norm_prod_le",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.nnnorm_prod_le {α : Type*} [normed_comm_ring α] [norm_one_class α] (s : finset ι)
(f : ι → α) :
‖∏ i in s, f i‖₊ ≤ ∏ i in s, ‖f i‖₊ | (s.norm_prod_le f).trans_eq $ by simp [nnreal.coe_prod] | lemma | finset.nnnorm_prod_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"nnreal.coe_prod",
"norm_one_class",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_pow_le' (a : α) : ∀ {n : ℕ}, 0 < n → ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n | | 1 h := by simp only [pow_one]
| (n + 2) h := by simpa only [pow_succ _ (n + 1)] using
le_trans (nnnorm_mul_le _ _) (mul_le_mul_left' (nnnorm_pow_le' n.succ_pos) _) | lemma | nnnorm_pow_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_le_mul_left'",
"nnnorm_mul_le",
"pow_one",
"pow_succ"
] | If `α` is a seminormed ring, then `‖a ^ n‖₊ ≤ ‖a‖₊ ^ n` for `n > 0`.
See also `nnnorm_pow_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_pow_le [norm_one_class α] (a : α) (n : ℕ) : ‖a ^ n‖₊ ≤ ‖a‖₊ ^ n | nat.rec_on n (by simp only [pow_zero, nnnorm_one]) (λ k hk, nnnorm_pow_le' a k.succ_pos) | lemma | nnnorm_pow_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnnorm_one",
"nnnorm_pow_le'",
"norm_one_class",
"pow_zero"
] | If `α` is a seminormed ring with `‖1‖₊ = 1`, then `‖a ^ n‖₊ ≤ ‖a‖₊ ^ n`.
See also `nnnorm_pow_le'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_pow_le' (a : α) {n : ℕ} (h : 0 < n) : ‖a ^ n‖ ≤ ‖a‖ ^ n | by simpa only [nnreal.coe_pow, coe_nnnorm] using nnreal.coe_mono (nnnorm_pow_le' a h) | lemma | norm_pow_le' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnnorm_pow_le'",
"nnreal.coe_mono",
"nnreal.coe_pow"
] | If `α` is a seminormed ring, then `‖a ^ n‖ ≤ ‖a‖ ^ n` for `n > 0`. See also `norm_pow_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_pow_le [norm_one_class α] (a : α) (n : ℕ) : ‖a ^ n‖ ≤ ‖a‖ ^ n | nat.rec_on n (by simp only [pow_zero, norm_one]) (λ n hn, norm_pow_le' a n.succ_pos) | lemma | norm_pow_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_one_class",
"norm_pow_le'",
"pow_zero"
] | If `α` is a seminormed ring with `‖1‖ = 1`, then `‖a ^ n‖ ≤ ‖a‖ ^ n`. See also `norm_pow_le'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_norm_pow_le (a : α) : ∀ᶠ (n:ℕ) in at_top, ‖a ^ n‖ ≤ ‖a‖ ^ n | eventually_at_top.mpr ⟨1, λ b h, norm_pow_le' a (nat.succ_le_iff.mp h)⟩ | lemma | eventually_norm_pow_le | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_pow_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.semi_normed_ring [semi_normed_ring β] :
semi_normed_ring (α × β) | { ..prod.non_unital_semi_normed_ring,
..prod.seminormed_add_comm_group, } | instance | prod.semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"prod.non_unital_semi_normed_ring",
"semi_normed_ring"
] | Seminormed ring structure on the product of two seminormed rings,
using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.semi_normed_ring {π : ι → Type*} [fintype ι]
[Π i, semi_normed_ring (π i)] :
semi_normed_ring (Π i, π i) | { ..pi.non_unital_semi_normed_ring,
..pi.seminormed_add_comm_group, } | instance | pi.semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"fintype",
"pi.non_unital_semi_normed_ring",
"semi_normed_ring"
] | Seminormed ring structure on the product of finitely many seminormed rings,
using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.semi_normed_ring : semi_normed_ring αᵐᵒᵖ | { ..mul_opposite.non_unital_semi_normed_ring,
..mul_opposite.seminormed_add_comm_group } | instance | mul_opposite.semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_opposite.non_unital_semi_normed_ring",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.non_unital_normed_ring [non_unital_normed_ring β] :
non_unital_normed_ring (α × β) | { norm_mul := norm_mul_le,
..prod.seminormed_add_comm_group } | instance | prod.non_unital_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"norm_mul",
"norm_mul_le"
] | Non-unital normed ring structure on the product of two non-unital normed rings,
using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.non_unital_normed_ring {π : ι → Type*} [fintype ι] [Π i, non_unital_normed_ring (π i)] :
non_unital_normed_ring (Π i, π i) | { norm_mul := norm_mul_le,
..pi.normed_add_comm_group } | instance | pi.non_unital_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"fintype",
"non_unital_normed_ring",
"norm_mul",
"norm_mul_le"
] | Normed ring structure on the product of finitely many non-unital normed rings, using the sup
norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.non_unital_normed_ring : non_unital_normed_ring αᵐᵒᵖ | { norm_mul := norm_mul_le,
..mul_opposite.normed_add_comm_group } | instance | mul_opposite.non_unital_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"norm_mul",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.norm_pos [nontrivial α] (x : αˣ) : 0 < ‖(x:α)‖ | norm_pos_iff.mpr (units.ne_zero x) | lemma | units.norm_pos | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nontrivial",
"units.ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.nnnorm_pos [nontrivial α] (x : αˣ) : 0 < ‖(x:α)‖₊ | x.norm_pos | lemma | units.nnnorm_pos | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.normed_ring [normed_ring β] : normed_ring (α × β) | { norm_mul := norm_mul_le,
..prod.normed_add_comm_group } | instance | prod.normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul",
"norm_mul_le",
"normed_ring"
] | Normed ring structure on the product of two normed rings, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.normed_ring {π : ι → Type*} [fintype ι] [Π i, normed_ring (π i)] :
normed_ring (Π i, π i) | { norm_mul := norm_mul_le,
..pi.normed_add_comm_group } | instance | pi.normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"fintype",
"norm_mul",
"norm_mul_le",
"normed_ring"
] | Normed ring structure on the product of finitely many normed rings, using the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.normed_ring : normed_ring αᵐᵒᵖ | { norm_mul := norm_mul_le,
..mul_opposite.normed_add_comm_group } | instance | mul_opposite.normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul",
"norm_mul_le",
"normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semi_normed_ring_top_monoid [non_unital_semi_normed_ring α] : has_continuous_mul α | ⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
begin
have : ∀ e : α × α, ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1‖ * ‖e.2 - x.2‖ + ‖e.1 - x.1‖ * ‖x.2‖,
{ intro e,
calc ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1 * (e.2 - x.2) + (e.1 - x.1) * x.2‖ :
by rw [mul_sub, sub_mul, sub_add_... | instance | semi_normed_ring_top_monoid | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_continuous_mul",
"non_unital_semi_normed_ring",
"norm_mul_le",
"squeeze_zero",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semi_normed_top_ring [non_unital_semi_normed_ring α] : topological_ring α | { } | instance | semi_normed_top_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_semi_normed_ring",
"topological_ring"
] | A seminormed ring is a topological ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mul (a b : α) : ‖a * b‖ = ‖a‖ * ‖b‖ | normed_division_ring.norm_mul' a b | lemma | norm_mul | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_division_ring.to_norm_one_class : norm_one_class α | ⟨mul_left_cancel₀ (mt norm_eq_zero.1 (one_ne_zero' α)) $
by rw [← norm_mul, mul_one, mul_one]⟩ | instance | normed_division_ring.to_norm_one_class | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_one",
"norm_mul",
"norm_one_class",
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_absolute_value_norm : is_absolute_value (norm : α → ℝ) | { abv_nonneg := norm_nonneg,
abv_eq_zero := λ _, norm_eq_zero,
abv_add := norm_add_le,
abv_mul := norm_mul } | instance | is_absolute_value_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"is_absolute_value",
"norm_eq_zero",
"norm_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_mul (a b : α) : ‖a * b‖₊ = ‖a‖₊ * ‖b‖₊ | nnreal.eq $ norm_mul a b | lemma | nnnorm_mul | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnreal.eq",
"norm_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_hom : α →*₀ ℝ | ⟨norm, norm_zero, norm_one, norm_mul⟩ | def | norm_hom | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | `norm` as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_hom : α →*₀ ℝ≥0 | ⟨nnnorm, nnnorm_zero, nnnorm_one, nnnorm_mul⟩ | def | nnnorm_hom | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnnorm_one"
] | `nnnorm` as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_pow (a : α) : ∀ (n : ℕ), ‖a ^ n‖ = ‖a‖ ^ n | (norm_hom.to_monoid_hom : α →* ℝ).map_pow a | lemma | norm_pow | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_pow (a : α) (n : ℕ) : ‖a ^ n‖₊ = ‖a‖₊ ^ n | (nnnorm_hom.to_monoid_hom : α →* ℝ≥0).map_pow a n | lemma | nnnorm_pow | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.norm_prod (l : list α) : ‖l.prod‖ = (l.map norm).prod | (norm_hom.to_monoid_hom : α →* ℝ).map_list_prod _ | lemma | list.norm_prod | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.nnnorm_prod (l : list α) : ‖l.prod‖₊ = (l.map nnnorm).prod | (nnnorm_hom.to_monoid_hom : α →* ℝ≥0).map_list_prod _ | lemma | list.nnnorm_prod | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_list_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_div (a b : α) : ‖a / b‖ = ‖a‖ / ‖b‖ | map_div₀ (norm_hom : α →*₀ ℝ) a b | lemma | norm_div | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_div₀",
"norm_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_div (a b : α) : ‖a / b‖₊ = ‖a‖₊ / ‖b‖₊ | map_div₀ (nnnorm_hom : α →*₀ ℝ≥0) a b | lemma | nnnorm_div | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_div₀",
"nnnorm_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_inv (a : α) : ‖a⁻¹‖ = ‖a‖⁻¹ | map_inv₀ (norm_hom : α →*₀ ℝ) a | lemma | norm_inv | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_inv₀",
"norm_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_inv (a : α) : ‖a⁻¹‖₊ = ‖a‖₊⁻¹ | nnreal.eq $ by simp | lemma | nnnorm_inv | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_zpow : ∀ (a : α) (n : ℤ), ‖a^n‖ = ‖a‖^n | map_zpow₀ (norm_hom : α →*₀ ℝ) | lemma | norm_zpow | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_zpow₀",
"norm_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_zpow : ∀ (a : α) (n : ℤ), ‖a ^ n‖₊ = ‖a‖₊ ^ n | map_zpow₀ (nnnorm_hom : α →*₀ ℝ≥0) | lemma | nnnorm_zpow | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_zpow₀",
"nnnorm_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_inv_inv₀ {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) :
dist z⁻¹ w⁻¹ = (dist z w) / (‖z‖ * ‖w‖) | by rw [dist_eq_norm, inv_sub_inv' hz hw, norm_mul, norm_mul, norm_inv, norm_inv, mul_comm ‖z‖⁻¹,
mul_assoc, dist_eq_norm', div_eq_mul_inv, mul_inv] | lemma | dist_inv_inv₀ | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"div_eq_mul_inv",
"inv_sub_inv'",
"mul_assoc",
"mul_comm",
"mul_inv",
"norm_inv",
"norm_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_inv_inv₀ {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) :
nndist z⁻¹ w⁻¹ = (nndist z w) / (‖z‖₊ * ‖w‖₊) | by { rw ← nnreal.coe_eq, simp [-nnreal.coe_eq, dist_inv_inv₀ hz hw], } | lemma | nndist_inv_inv₀ | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"dist_inv_inv₀",
"nnreal.coe_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_mul_left_cobounded {a : α} (ha : a ≠ 0) :
tendsto ((*) a) (comap norm at_top) (comap norm at_top) | by simpa only [tendsto_comap_iff, (∘), norm_mul]
using tendsto_const_nhds.mul_at_top (norm_pos_iff.2 ha) tendsto_comap | lemma | filter.tendsto_mul_left_cobounded | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul"
] | Multiplication on the left by a nonzero element of a normed division ring tends to infinity at
infinity. TODO: use `bornology.cobounded` instead of `filter.comap has_norm.norm filter.at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto_mul_right_cobounded {a : α} (ha : a ≠ 0) :
tendsto (λ x, x * a) (comap norm at_top) (comap norm at_top) | by simpa only [tendsto_comap_iff, (∘), norm_mul]
using tendsto_comap.at_top_mul (norm_pos_iff.2 ha) tendsto_const_nhds | lemma | filter.tendsto_mul_right_cobounded | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul",
"tendsto_const_nhds"
] | Multiplication on the right by a nonzero element of a normed division ring tends to infinity at
infinity. TODO: use `bornology.cobounded` instead of `filter.comap has_norm.norm filter.at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_division_ring.to_has_continuous_inv₀ : has_continuous_inv₀ α | begin
refine ⟨λ r r0, tendsto_iff_norm_tendsto_zero.2 _⟩,
have r0' : 0 < ‖r‖ := norm_pos_iff.2 r0,
rcases exists_between r0' with ⟨ε, ε0, εr⟩,
have : ∀ᶠ e in 𝓝 r, ‖e⁻¹ - r⁻¹‖ ≤ ‖r - e‖ / ‖r‖ / ε,
{ filter_upwards [(is_open_lt continuous_const continuous_norm).eventually_mem εr] with e he,
have e0 : e ≠ 0... | instance | normed_division_ring.to_has_continuous_inv₀ | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"continuous_const",
"continuous_id",
"div_le_div_of_le_left",
"div_nonneg",
"exists_between",
"has_continuous_inv₀",
"inv_mul_cancel",
"is_open_lt",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_one",
"one_mul",
"squeeze_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_division_ring.to_topological_division_ring : topological_division_ring α | { } | instance | normed_division_ring.to_topological_division_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"topological_division_ring"
] | A normed division ring is a topological division ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_map_one_of_pow_eq_one [monoid β] (φ : β →* α) {x : β} {k : ℕ+}
(h : x ^ (k : ℕ) = 1) :
‖φ x‖ = 1 | begin
rw [← pow_left_inj, ← norm_pow, ← map_pow, h, map_one, norm_one, one_pow],
exacts [norm_nonneg _, zero_le_one, k.pos],
end | lemma | norm_map_one_of_pow_eq_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_one",
"map_pow",
"monoid",
"norm_pow",
"one_pow",
"pow_left_inj",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_one_of_pow_eq_one {x : α} {k : ℕ+} (h : x ^ (k : ℕ) = 1) :
‖x‖ = 1 | norm_map_one_of_pow_eq_one (monoid_hom.id α) h | lemma | norm_one_of_pow_eq_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"monoid_hom.id",
"norm_map_one_of_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_field (α : Type*) extends has_norm α, field α, metric_space α | (dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul' : ∀ a b, norm (a * b) = norm a * norm b) | class | normed_field | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"field",
"has_norm",
"metric_space"
] | A normed field is a field with a norm satisfying ‖x y‖ = ‖x‖ ‖y‖. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nontrivially_normed_field (α : Type*) extends normed_field α | (non_trivial : ∃ x : α, 1 < ‖x‖) | class | nontrivially_normed_field | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_field"
] | A nontrivially normed field is a normed field in which there is an element of norm different
from `0` and `1`. This makes it possible to bring any element arbitrarily close to `0` by
multiplication by the powers of any element, and thus to relate algebra and topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
densely_normed_field (α : Type*) extends normed_field α | (lt_norm_lt : ∀ x y : ℝ, 0 ≤ x → x < y → ∃ a : α, x < ‖a‖ ∧ ‖a‖ < y) | class | densely_normed_field | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_field"
] | A densely normed field is a normed field for which the image of the norm is dense in `ℝ≥0`,
which means it is also nontrivially normed. However, not all nontrivally normed fields are densely
normed; in particular, the `padic`s exhibit this fact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
densely_normed_field.to_nontrivially_normed_field [densely_normed_field α] :
nontrivially_normed_field α | { non_trivial := let ⟨a, h, _⟩ := densely_normed_field.lt_norm_lt 1 2 zero_le_one one_lt_two in
⟨a, h⟩ } | instance | densely_normed_field.to_nontrivially_normed_field | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"densely_normed_field",
"nontrivially_normed_field",
"one_lt_two",
"zero_le_one"
] | A densely normed field is always a nontrivially normed field.
See note [lower instance priority]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_field.to_normed_division_ring : normed_division_ring α | { ..‹normed_field α› } | instance | normed_field.to_normed_division_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_division_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_field.to_normed_comm_ring : normed_comm_ring α | { norm_mul := λ a b, (norm_mul a b).le, ..‹normed_field α› } | instance | normed_field.to_normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_mul",
"normed_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_prod (s : finset β) (f : β → α) :
‖∏ b in s, f b‖ = ∏ b in s, ‖f b‖ | (norm_hom.to_monoid_hom : α →* ℝ).map_prod f s | lemma | norm_prod | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_prod (s : finset β) (f : β → α) :
‖∏ b in s, f b‖₊ = ∏ b in s, ‖f b‖₊ | (nnnorm_hom.to_monoid_hom : α →* ℝ≥0).map_prod f s | lemma | nnnorm_prod | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_one_lt_norm : ∃x : α, 1 < ‖x‖ | ‹nontrivially_normed_field α›.non_trivial | lemma | normed_field.exists_one_lt_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_norm (r : ℝ) : ∃ x : α, r < ‖x‖ | let ⟨w, hw⟩ := exists_one_lt_norm α in
let ⟨n, hn⟩ := pow_unbounded_of_one_lt r hw in
⟨w^n, by rwa norm_pow⟩ | lemma | normed_field.exists_lt_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"pow_unbounded_of_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_norm_lt {r : ℝ} (hr : 0 < r) : ∃ x : α, 0 < ‖x‖ ∧ ‖x‖ < r | let ⟨w, hw⟩ := exists_lt_norm α r⁻¹ in
⟨w⁻¹, by rwa [← set.mem_Ioo, norm_inv, ← set.mem_inv, set.inv_Ioo_0_left hr]⟩ | lemma | normed_field.exists_norm_lt | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"norm_inv",
"set.inv_Ioo_0_left",
"set.mem_Ioo",
"set.mem_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_norm_lt_one : ∃x : α, 0 < ‖x‖ ∧ ‖x‖ < 1 | exists_norm_lt α one_pos | lemma | normed_field.exists_norm_lt_one | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
punctured_nhds_ne_bot (x : α) : ne_bot (𝓝[≠] x) | begin
rw [← mem_closure_iff_nhds_within_ne_bot, metric.mem_closure_iff],
rintros ε ε0,
rcases exists_norm_lt α ε0 with ⟨b, hb0, hbε⟩,
refine ⟨x + b, mt (set.mem_singleton_iff.trans add_right_eq_self).1 $ norm_pos_iff.1 hb0, _⟩,
rwa [dist_comm, dist_eq_norm, add_sub_cancel'],
end | lemma | normed_field.punctured_nhds_ne_bot | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"dist_comm",
"mem_closure_iff_nhds_within_ne_bot",
"metric.mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_is_unit_ne_bot : ne_bot (𝓝[{x : α | is_unit x}] 0) | by simpa only [is_unit_iff_ne_zero] using punctured_nhds_ne_bot (0:α) | lemma | normed_field.nhds_within_is_unit_ne_bot | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"is_unit",
"is_unit_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_norm_lt {r₁ r₂ : ℝ} (h₀ : 0 ≤ r₁) (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖ ∧ ‖x‖ < r₂ | densely_normed_field.lt_norm_lt r₁ r₂ h₀ h | lemma | normed_field.exists_lt_norm_lt | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_nnnorm_lt {r₁ r₂ : ℝ≥0} (h : r₁ < r₂) : ∃ x : α, r₁ < ‖x‖₊ ∧ ‖x‖₊ < r₂ | by exact_mod_cast exists_lt_norm_lt α r₁.prop h | lemma | normed_field.exists_lt_nnnorm_lt | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
densely_ordered_range_norm : densely_ordered (set.range (norm : α → ℝ)) | { dense :=
begin
rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy,
exact let ⟨z, h⟩ := exists_lt_norm_lt α (norm_nonneg _) hxy in ⟨⟨‖z‖, z, rfl⟩, h⟩,
end } | instance | normed_field.densely_ordered_range_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"dense",
"densely_ordered",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
densely_ordered_range_nnnorm : densely_ordered (set.range (nnnorm : α → ℝ≥0)) | { dense :=
begin
rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy,
exact let ⟨z, h⟩ := exists_lt_nnnorm_lt α hxy in ⟨⟨‖z‖₊, z, rfl⟩, h⟩,
end } | instance | normed_field.densely_ordered_range_nnnorm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"dense",
"densely_ordered",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_nnnorm : dense_range (nnnorm : α → ℝ≥0) | dense_of_exists_between $ λ _ _ hr, let ⟨x, h⟩ := exists_lt_nnnorm_lt α hr in ⟨‖x‖₊, ⟨x, rfl⟩, h⟩ | lemma | normed_field.dense_range_nnnorm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"dense_of_exists_between",
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nnreal_mul_nnnorm {x : ℝ} (y : ℝ) (hx : 0 ≤ x) : x.to_nnreal * ‖y‖₊ = ‖x * y‖₊ | by simp [real.to_nnreal_of_nonneg, nnnorm, norm_of_nonneg, hx] | lemma | real.to_nnreal_mul_nnnorm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"real.to_nnreal_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_mul_to_nnreal (x : ℝ) {y : ℝ} (hy : 0 ≤ y) : ‖x‖₊ * y.to_nnreal = ‖x * y‖₊ | by simp [real.to_nnreal_of_nonneg, nnnorm, norm_of_nonneg, hy] | lemma | real.nnnorm_mul_to_nnreal | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"real.to_nnreal_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq (x : ℝ≥0) : ‖(x : ℝ)‖ = x | by rw [real.norm_eq_abs, x.abs_eq] | lemma | nnreal.norm_eq | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq (x : ℝ≥0) : ‖(x : ℝ)‖₊ = x | nnreal.eq $ real.norm_of_nonneg x.2 | lemma | nnreal.nnnorm_eq | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"nnreal.eq",
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_norm [seminormed_add_comm_group α] (x : α) : ‖‖x‖‖ = ‖x‖ | real.norm_of_nonneg (norm_nonneg _) | lemma | norm_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"real.norm_of_nonneg",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_norm [seminormed_add_comm_group α] (a : α) : ‖‖a‖‖₊ = ‖a‖₊ | by simpa [real.nnnorm_of_nonneg (norm_nonneg a)] | lemma | nnnorm_norm | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"real.nnnorm_of_nonneg",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group.tendsto_at_top [nonempty α] [semilattice_sup α] {β : Type*}
[seminormed_add_comm_group β] {f : α → β} {b : β} :
tendsto f at_top (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N ≤ n → ‖f n - b‖ < ε | (at_top_basis.tendsto_iff metric.nhds_basis_ball).trans (by simp [dist_eq_norm]) | lemma | normed_add_comm_group.tendsto_at_top | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"metric.nhds_basis_ball",
"semilattice_sup",
"seminormed_add_comm_group"
] | A restatement of `metric_space.tendsto_at_top` in terms of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_comm_group.tendsto_at_top' [nonempty α] [semilattice_sup α] [no_max_order α]
{β : Type*} [seminormed_add_comm_group β]
{f : α → β} {b : β} :
tendsto f at_top (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N < n → ‖f n - b‖ < ε | (at_top_basis_Ioi.tendsto_iff metric.nhds_basis_ball).trans (by simp [dist_eq_norm]) | lemma | normed_add_comm_group.tendsto_at_top' | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"metric.nhds_basis_ball",
"no_max_order",
"semilattice_sup",
"seminormed_add_comm_group"
] | A variant of `normed_add_comm_group.tendsto_at_top` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_isometric [semiring R₁] [semiring R₂] [has_norm R₁] [has_norm R₂]
(σ : R₁ →+* R₂) : Prop | (is_iso : ∀ {x : R₁}, ‖σ x‖ = ‖x‖) | class | ring_hom_isometric | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"has_norm",
"semiring"
] | This class states that a ring homomorphism is isometric. This is a sufficient assumption
for a continuous semilinear map to be bounded and this is the main use for this typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_isometric.ids : ring_hom_isometric (ring_hom.id R₁) | ⟨λ x, rfl⟩ | instance | ring_hom_isometric.ids | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"ring_hom.id",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semi_normed_ring.induced [non_unital_ring R] [non_unital_semi_normed_ring S]
[non_unital_ring_hom_class F R S] (f : F) : non_unital_semi_normed_ring R | { norm_mul := λ x y, by { unfold norm, exact (map_mul f x y).symm ▸ norm_mul_le (f x) (f y) },
.. seminormed_add_comm_group.induced R S f } | def | non_unital_semi_normed_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_mul",
"non_unital_ring",
"non_unital_ring_hom_class",
"non_unital_semi_normed_ring",
"norm_mul",
"norm_mul_le"
] | A non-unital ring homomorphism from an `non_unital_ring` to a `non_unital_semi_normed_ring`
induces a `non_unital_semi_normed_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_normed_ring.induced [non_unital_ring R] [non_unital_normed_ring S]
[non_unital_ring_hom_class F R S] (f : F) (hf : function.injective f) :
non_unital_normed_ring R | { .. non_unital_semi_normed_ring.induced R S f,
.. normed_add_comm_group.induced R S f hf } | def | non_unital_normed_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_normed_ring",
"non_unital_ring",
"non_unital_ring_hom_class",
"non_unital_semi_normed_ring.induced"
] | An injective non-unital ring homomorphism from an `non_unital_ring` to a
`non_unital_normed_ring` induces a `non_unital_normed_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semi_normed_ring.induced [ring R] [semi_normed_ring S] [non_unital_ring_hom_class F R S]
(f : F) : semi_normed_ring R | { .. non_unital_semi_normed_ring.induced R S f,
.. seminormed_add_comm_group.induced R S f } | def | semi_normed_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_ring_hom_class",
"non_unital_semi_normed_ring.induced",
"ring",
"semi_normed_ring"
] | A non-unital ring homomorphism from an `ring` to a `semi_normed_ring` induces a
`semi_normed_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ring.induced [ring R] [normed_ring S] [non_unital_ring_hom_class F R S] (f : F)
(hf : function.injective f) : normed_ring R | { .. non_unital_semi_normed_ring.induced R S f,
.. normed_add_comm_group.induced R S f hf } | def | normed_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"non_unital_ring_hom_class",
"non_unital_semi_normed_ring.induced",
"normed_ring",
"ring"
] | An injective non-unital ring homomorphism from an `ring` to a `normed_ring` induces a
`normed_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semi_normed_comm_ring.induced [comm_ring R] [semi_normed_ring S]
[non_unital_ring_hom_class F R S] (f : F) : semi_normed_comm_ring R | { mul_comm := mul_comm,
.. non_unital_semi_normed_ring.induced R S f,
.. seminormed_add_comm_group.induced R S f } | def | semi_normed_comm_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"comm_ring",
"mul_comm",
"non_unital_ring_hom_class",
"non_unital_semi_normed_ring.induced",
"semi_normed_comm_ring",
"semi_normed_ring"
] | A non-unital ring homomorphism from a `comm_ring` to a `semi_normed_ring` induces a
`semi_normed_comm_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_comm_ring.induced [comm_ring R] [normed_ring S] [non_unital_ring_hom_class F R S] (f : F)
(hf : function.injective f) : normed_comm_ring R | { .. semi_normed_comm_ring.induced R S f,
.. normed_add_comm_group.induced R S f hf } | def | normed_comm_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"comm_ring",
"non_unital_ring_hom_class",
"normed_comm_ring",
"normed_ring",
"semi_normed_comm_ring.induced"
] | An injective non-unital ring homomorphism from an `comm_ring` to a `normed_ring` induces a
`normed_comm_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_division_ring.induced [division_ring R] [normed_division_ring S]
[non_unital_ring_hom_class F R S] (f : F) (hf : function.injective f) : normed_division_ring R | { norm_mul' := λ x y, by { unfold norm, exact (map_mul f x y).symm ▸ norm_mul (f x) (f y) },
.. normed_add_comm_group.induced R S f hf } | def | normed_division_ring.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"division_ring",
"map_mul",
"non_unital_ring_hom_class",
"norm_mul",
"normed_division_ring"
] | An injective non-unital ring homomorphism from an `division_ring` to a `normed_ring` induces a
`normed_division_ring` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_field.induced [field R] [normed_field S]
[non_unital_ring_hom_class F R S] (f : F) (hf : function.injective f) : normed_field R | { .. normed_division_ring.induced R S f hf } | def | normed_field.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"field",
"non_unital_ring_hom_class",
"normed_division_ring.induced",
"normed_field"
] | An injective non-unital ring homomorphism from an `field` to a `normed_ring` induces a
`normed_field` structure on the domain.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_one_class.induced {F : Type*} (R S : Type*) [ring R] [semi_normed_ring S]
[norm_one_class S] [ring_hom_class F R S] (f : F) :
@norm_one_class R (semi_normed_ring.induced R S f).to_has_norm _ | { norm_one := (congr_arg norm (map_one f)).trans norm_one } | lemma | norm_one_class.induced | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"map_one",
"norm_one_class",
"ring",
"ring_hom_class",
"semi_normed_ring",
"semi_normed_ring.induced"
] | A ring homomorphism from a `ring R` to a `semi_normed_ring S` which induces the norm structure
`semi_normed_ring.induced` makes `R` satisfy `‖(1 : R)‖ = 1` whenever `‖(1 : S)‖ = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_semi_normed_ring [semi_normed_ring R] [subring_class S R] (s : S) :
semi_normed_ring s | semi_normed_ring.induced s R (subring_class.subtype s) | instance | subring_class.to_semi_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"semi_normed_ring",
"semi_normed_ring.induced",
"subring_class",
"subring_class.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_ring [normed_ring R] [subring_class S R] (s : S) :
normed_ring s | normed_ring.induced s R (subring_class.subtype s) subtype.val_injective | instance | subring_class.to_normed_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"normed_ring",
"normed_ring.induced",
"subring_class",
"subring_class.subtype",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_semi_normed_comm_ring [semi_normed_comm_ring R] [h : subring_class S R] (s : S) :
semi_normed_comm_ring s | { mul_comm := mul_comm, .. subring_class.to_semi_normed_ring s } | instance | subring_class.to_semi_normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"semi_normed_comm_ring",
"subring_class",
"subring_class.to_semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_comm_ring [normed_comm_ring R] [subring_class S R] (s : S) :
normed_comm_ring s | { mul_comm := mul_comm, .. subring_class.to_normed_ring s } | instance | subring_class.to_normed_comm_ring | analysis.normed.field | src/analysis/normed/field/basic.lean | [
"algebra.algebra.subalgebra.basic",
"analysis.normed.group.basic",
"topology.instances.ennreal"
] | [
"mul_comm",
"normed_comm_ring",
"subring_class",
"subring_class.to_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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