statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
covariant_swap_div : covariant_class (filter α) (filter α) (swap (/)) (≤) | ⟨λ f g h, map₂_mono_right⟩ | instance | filter.covariant_swap_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul [has_zero α] [has_add α] : has_smul ℕ (filter α) | ⟨nsmul_rec⟩ | def | filter.has_nsmul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul"
] | Repeated pointwise addition (not the same as pointwise repeated addition!) of a `filter`. See
Note [pointwise nat action]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_npow [has_one α] [has_mul α] : has_pow (filter α) ℕ | ⟨λ s n, npow_rec n s⟩ | def | filter.has_npow | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"npow_rec"
] | Repeated pointwise multiplication (not the same as pointwise repeated multiplication!) of a
`filter`. See Note [pointwise nat action]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zsmul [has_zero α] [has_add α] [has_neg α] : has_smul ℤ (filter α) | ⟨zsmul_rec⟩ | def | filter.has_zsmul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul"
] | Repeated pointwise addition/subtraction (not the same as pointwise repeated
addition/subtraction!) of a `filter`. See Note [pointwise nat action]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zpow [has_one α] [has_mul α] [has_inv α] : has_pow (filter α) ℤ | ⟨λ s n, zpow_rec n s⟩ | def | filter.has_zpow | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"zpow_rec"
] | Repeated pointwise multiplication/division (not the same as pointwise repeated
multiplication/division!) of a `filter`. See Note [pointwise nat action]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup [semigroup α] : semigroup (filter α) | { mul := (*),
mul_assoc := λ f g h, map₂_assoc mul_assoc } | def | filter.semigroup | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"mul_assoc",
"semigroup"
] | `filter α` is a `semigroup` under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup [comm_semigroup α] : comm_semigroup (filter α) | { mul_comm := λ f g, map₂_comm mul_comm,
..filter.semigroup } | def | filter.comm_semigroup | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"comm_semigroup",
"filter",
"filter.semigroup",
"mul_comm"
] | `filter α` is a `comm_semigroup` under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_one_class : mul_one_class (filter α) | { one := 1,
mul := (*),
one_mul := map₂_left_identity one_mul,
mul_one := map₂_right_identity mul_one } | def | filter.mul_one_class | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"mul_one",
"mul_one_class",
"one_mul"
] | `filter α` is a `mul_one_class` under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_monoid_hom [monoid_hom_class F α β] (φ : F) : filter α →* filter β | { to_fun := map φ,
map_one' := filter.map_one φ,
map_mul' := λ _ _, filter.map_mul φ } | def | filter.map_monoid_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"filter.map_mul",
"filter.map_one",
"monoid_hom_class"
] | If `φ : α →* β` then `map_monoid_hom φ` is the monoid homomorphism
`filter α →* filter β` induced by `map φ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_mul_comap_le [mul_hom_class F α β] (m : F) {f g : filter β} :
f.comap m * g.comap m ≤ (f * g).comap m | λ s ⟨t, ⟨t₁, t₂, ht₁, ht₂, t₁t₂⟩, mt⟩,
⟨m ⁻¹' t₁, m ⁻¹' t₂, ⟨t₁, ht₁, subset.rfl⟩, ⟨t₂, ht₂, subset.rfl⟩,
(preimage_mul_preimage_subset _).trans $ (preimage_mono t₁t₂).trans mt⟩ | lemma | filter.comap_mul_comap_le | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.mul_mul [mul_hom_class F α β] (m : F) {f₁ g₁ : filter α} {f₂ g₂ : filter β} :
tendsto m f₁ f₂ → tendsto m g₁ g₂ → tendsto m (f₁ * g₁) (f₂ * g₂) | λ hf hg, (filter.map_mul m).trans_le $ mul_le_mul' hf hg | lemma | filter.tendsto.mul_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"filter.map_mul",
"mul_hom_class",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_monoid_hom : α →* filter α | { ..pure_mul_hom, ..pure_one_hom } | def | filter.pure_monoid_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | `pure` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pure_monoid_hom : (pure_monoid_hom : α → filter α) = pure | rfl | lemma | filter.coe_pure_monoid_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_monoid_hom_apply (a : α) : pure_monoid_hom a = pure a | rfl | lemma | filter.pure_monoid_hom_apply | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid : monoid (filter α) | { ..filter.mul_one_class, ..filter.semigroup, ..filter.has_npow } | def | filter.monoid | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"filter.has_npow",
"filter.mul_one_class",
"filter.semigroup",
"monoid"
] | `filter α` is a `monoid` under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_mem_pow (hs : s ∈ f) : ∀ n : ℕ, s ^ n ∈ f ^ n | | 0 := by { rw pow_zero, exact one_mem_one }
| (n + 1) := by { rw pow_succ, exact mul_mem_mul hs (pow_mem_pow _) } | lemma | filter.pow_mem_pow | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_pow {n : ℕ} (hn : n ≠ 0) : (⊥ : filter α) ^ n = ⊥ | by rw [←tsub_add_cancel_of_le (nat.succ_le_of_lt $ nat.pos_of_ne_zero hn), pow_succ, bot_mul] | lemma | filter.bot_pow | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_top_of_one_le (hf : 1 ≤ f) : f * ⊤ = ⊤ | begin
refine top_le_iff.1 (λ s, _),
simp only [mem_mul, mem_top, exists_and_distrib_left, exists_eq_left],
rintro ⟨t, ht, hs⟩,
rwa [mul_univ_of_one_mem (mem_one.1 $ hf ht), univ_subset_iff] at hs,
end | lemma | filter.mul_top_of_one_le | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"exists_and_distrib_left",
"exists_eq_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul_of_one_le (hf : 1 ≤ f) : ⊤ * f = ⊤ | begin
refine top_le_iff.1 (λ s, _),
simp only [mem_mul, mem_top, exists_and_distrib_left, exists_eq_left],
rintro ⟨t, ht, hs⟩,
rwa [univ_mul_of_one_mem (mem_one.1 $ hf ht), univ_subset_iff] at hs,
end | lemma | filter.top_mul_of_one_le | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"exists_and_distrib_left",
"exists_eq_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul_top : (⊤ : filter α) * ⊤ = ⊤ | mul_top_of_one_le le_top | lemma | filter.top_mul_top | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_top {α : Type*} [add_monoid α] : ∀ {n : ℕ}, n ≠ 0 → n • (⊤ : filter α) = ⊤ | | 0 := λ h, (h rfl).elim
| 1 := λ _, one_nsmul _
| (n + 2) := λ _, by { rw [succ_nsmul, nsmul_top n.succ_ne_zero, top_add_top] } | lemma | filter.nsmul_top | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"add_monoid",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_pow : ∀ {n : ℕ}, n ≠ 0 → (⊤ : filter α) ^ n = ⊤ | | 0 := λ h, (h rfl).elim
| 1 := λ _, pow_one _
| (n + 2) := λ _, by { rw [pow_succ, top_pow n.succ_ne_zero, top_mul_top] } | lemma | filter.top_pow | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_unit.filter : is_unit a → is_unit (pure a : filter α) | is_unit.map (pure_monoid_hom : α →* filter α) | lemma | is_unit.filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"is_unit",
"is_unit.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid [comm_monoid α] : comm_monoid (filter α) | { ..filter.mul_one_class, ..filter.comm_semigroup } | def | filter.comm_monoid | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"comm_monoid",
"filter",
"filter.comm_semigroup",
"filter.mul_one_class"
] | `filter α` is a `comm_monoid` under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_one_iff : f * g = 1 ↔ ∃ a b, f = pure a ∧ g = pure b ∧ a * b = 1 | begin
refine ⟨λ hfg, _, _⟩,
{ obtain ⟨t₁, t₂, h₁, h₂, h⟩ : (1 : set α) ∈ f * g := hfg.symm.subst one_mem_one,
have hfg : (f * g).ne_bot := hfg.symm.subst one_ne_bot,
rw [(hfg.nonempty_of_mem $ mul_mem_mul h₁ h₂).subset_one_iff, set.mul_eq_one_iff] at h,
obtain ⟨a, b, rfl, rfl, h⟩ := h,
refine ⟨a, b,... | lemma | filter.mul_eq_one_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"mul_eq_one_iff",
"set.mul_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_monoid : division_monoid (filter α) | { mul_inv_rev := λ s t, map_map₂_antidistrib mul_inv_rev,
inv_eq_of_mul := λ s t h, begin
obtain ⟨a, b, rfl, rfl, hab⟩ := filter.mul_eq_one_iff.1 h,
rw [inv_pure, inv_eq_of_mul_eq_one_right hab],
end,
div_eq_mul_inv := λ f g, map_map₂_distrib_right div_eq_mul_inv,
..filter.monoid, ..filter.has_involutiv... | def | filter.division_monoid | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"div_eq_mul_inv",
"division_monoid",
"filter",
"filter.has_div",
"filter.has_involutive_inv",
"filter.has_zpow",
"filter.monoid",
"inv_eq_of_mul",
"inv_eq_of_mul_eq_one_right",
"mul_inv_rev"
] | `filter α` is a division monoid under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff : is_unit f ↔ ∃ a, f = pure a ∧ is_unit a | begin
split,
{ rintro ⟨u, rfl⟩,
obtain ⟨a, b, ha, hb, h⟩ := filter.mul_eq_one_iff.1 u.mul_inv,
refine ⟨a, ha, ⟨a, b, h, pure_injective _⟩, rfl⟩,
rw [←pure_mul_pure, ←ha, ←hb],
exact u.inv_mul },
{ rintro ⟨a, rfl, ha⟩,
exact ha.filter }
end | lemma | filter.is_unit_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_comm_monoid [division_comm_monoid α] : division_comm_monoid (filter α) | { ..filter.division_monoid, ..filter.comm_semigroup } | def | filter.division_comm_monoid | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"division_comm_monoid",
"filter",
"filter.comm_semigroup",
"filter.division_monoid"
] | `filter α` is a commutative division monoid under pointwise operations if `α` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_distrib_neg [has_mul α] [has_distrib_neg α] : has_distrib_neg (filter α) | { neg_mul := λ _ _, map₂_map_left_comm neg_mul,
mul_neg := λ _ _, map_map₂_right_comm mul_neg,
..filter.has_involutive_neg } | def | filter.has_distrib_neg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_distrib_neg",
"mul_neg",
"neg_mul"
] | `filter α` has distributive negation if `α` has. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_add_subset : f * (g + h) ≤ f * g + f * h | map₂_distrib_le_left mul_add | lemma | filter.mul_add_subset | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_subset : (f + g) * h ≤ f * h + g * h | map₂_distrib_le_right add_mul | lemma | filter.add_mul_subset | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.mul_zero_nonneg (hf : f.ne_bot) : 0 ≤ f * 0 | le_mul_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨a, ha⟩ := hf.nonempty_of_mem h₁ in ⟨_, _, ha, h₂, mul_zero _⟩ | lemma | filter.ne_bot.mul_zero_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.zero_mul_nonneg (hg : g.ne_bot) : 0 ≤ 0 * g | le_mul_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨b, hb⟩ := hg.nonempty_of_mem h₂ in ⟨_, _, h₁, hb, zero_mul _⟩ | lemma | filter.ne_bot.zero_mul_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_div_iff : 1 ≤ f / g ↔ ¬ disjoint f g | begin
refine ⟨λ h hfg, _, _⟩,
{ obtain ⟨s, hs, t, ht, hst⟩ := hfg.le_bot (mem_bot : ∅ ∈ ⊥),
exact set.one_mem_div_iff.1 (h $ div_mem_div hs ht) (disjoint_iff.2 hst.symm) },
{ rintro h s ⟨t₁, t₂, h₁, h₂, hs⟩,
exact hs (set.one_mem_div_iff.2 $ λ ht, h $ disjoint_of_disjoint_of_mem ht h₁ h₂) }
end | lemma | filter.one_le_div_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"disjoint",
"one_le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_one_le_div_iff : ¬ 1 ≤ f / g ↔ disjoint f g | filter.one_le_div_iff.not_left | lemma | filter.not_one_le_div_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.one_le_div (h : f.ne_bot) : 1 ≤ f / f | begin
rintro s ⟨t₁, t₂, h₁, h₂, hs⟩,
obtain ⟨a, ha₁, ha₂⟩ := set.not_disjoint_iff.1 (h.not_disjoint h₁ h₂),
rw [mem_one, ←div_self' a],
exact hs (set.div_mem_div ha₁ ha₂),
end | lemma | filter.ne_bot.one_le_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"set.div_mem_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_pure (a : α) : is_unit (pure a : filter α) | (group.is_unit a).filter | lemma | filter.is_unit_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"group.is_unit",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_singleton : is_unit f ↔ ∃ a, f = pure a | by simp only [is_unit_iff, group.is_unit, and_true] | lemma | filter.is_unit_iff_singleton | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"group.is_unit",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv' : f⁻¹.map m = (f.map m)⁻¹ | semiconj.filter_map (map_inv m) f | lemma | filter.map_inv' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.inv_inv : tendsto m f₁ f₂ → tendsto m f₁⁻¹ f₂⁻¹ | λ hf, (filter.map_inv' m).trans_le $ filter.inv_le_inv hf | lemma | filter.tendsto.inv_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter.inv_le_inv",
"filter.map_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div : (f / g).map m = f.map m / g.map m | map_map₂_distrib $ map_div m | lemma | filter.map_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.div_div : tendsto m f₁ f₂ → tendsto m g₁ g₂ → tendsto m (f₁ / g₁) (f₂ / g₂) | λ hf hg, (filter.map_div m).trans_le $ filter.div_le_div hf hg | lemma | filter.tendsto.div_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter.div_le_div",
"filter.map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.div_zero_nonneg (hf : f.ne_bot) : 0 ≤ f / 0 | filter.le_div_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨a, ha⟩ := hf.nonempty_of_mem h₁ in
⟨_, _, ha, h₂, div_zero _⟩ | lemma | filter.ne_bot.div_zero_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"div_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.zero_div_nonneg (hg : g.ne_bot) : 0 ≤ 0 / g | filter.le_div_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨b, hb⟩ := hg.nonempty_of_mem h₂ in
⟨_, _, h₁, hb, zero_div _⟩ | lemma | filter.ne_bot.zero_div_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul : has_smul (filter α) (filter β) | /- This is defeq to `map₂ (•) f g`, but the hypothesis unfolds to `t₁ • t₂ ⊆ s` rather than all the
way to `set.image2 (•) t₁ t₂ ⊆ s`. -/
⟨λ f g, { sets := {s | ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ • t₂ ⊆ s}, ..map₂ (•) f g }⟩ | def | filter.has_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul"
] | The filter `f • g` is generated by `{s • t | s ∈ f, t ∈ g}` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_smul : map₂ (•) f g = f • g | rfl | lemma | filter.map₂_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul : t ∈ f • g ↔ ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ • t₂ ⊆ t | iff.rfl | lemma | filter.mem_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_smul : s ∈ f → t ∈ g → s • t ∈ f • g | image2_mem_map₂ | lemma | filter.smul_mem_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_smul : (⊥ : filter α) • g = ⊥ | map₂_bot_left | lemma | filter.bot_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_bot : f • (⊥ : filter β) = ⊥ | map₂_bot_right | lemma | filter.smul_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_bot_iff : f • g = ⊥ ↔ f = ⊥ ∨ g = ⊥ | map₂_eq_bot_iff | lemma | filter.smul_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ne_bot_iff : (f • g).ne_bot ↔ f.ne_bot ∧ g.ne_bot | map₂_ne_bot_iff | lemma | filter.smul_ne_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.smul : ne_bot f → ne_bot g → ne_bot (f • g) | ne_bot.map₂ | lemma | filter.ne_bot.smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_smul_left : (f • g).ne_bot → f.ne_bot | ne_bot.of_map₂_left | lemma | filter.ne_bot.of_smul_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_smul_right : (f • g).ne_bot → g.ne_bot | ne_bot.of_map₂_right | lemma | filter.ne_bot.of_smul_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_smul : (pure a : filter α) • g = g.map ((•) a) | map₂_pure_left | lemma | filter.pure_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pure : f • pure b = f.map (• b) | map₂_pure_right | lemma | filter.smul_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_smul_pure :
(pure a : filter α) • (pure b : filter β) = pure (a • b) | map₂_pure | lemma | filter.pure_smul_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul : f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ • g₁ ≤ f₂ • g₂ | map₂_mono | lemma | filter.smul_le_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul_left : g₁ ≤ g₂ → f • g₁ ≤ f • g₂ | map₂_mono_left | lemma | filter.smul_le_smul_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul_right : f₁ ≤ f₂ → f₁ • g ≤ f₂ • g | map₂_mono_right | lemma | filter.smul_le_smul_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_smul_iff : h ≤ f • g ↔ ∀ ⦃s⦄, s ∈ f → ∀ ⦃t⦄, t ∈ g → s • t ∈ h | le_map₂_iff | lemma | filter.le_smul_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_smul : covariant_class (filter α) (filter β) (•) (≤) | ⟨λ f g h, map₂_mono_left⟩ | instance | filter.covariant_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_vsub : has_vsub (filter α) (filter β) | /- This is defeq to `map₂ (-ᵥ) f g`, but the hypothesis unfolds to `t₁ -ᵥ t₂ ⊆ s` rather than all
the way to `set.image2 (-ᵥ) t₁ t₂ ⊆ s`. -/
⟨λ f g, { sets := {s | ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ -ᵥ t₂ ⊆ s}, ..map₂ (-ᵥ) f g }⟩ | def | filter.has_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_vsub"
] | The filter `f -ᵥ g` is generated by `{s -ᵥ t | s ∈ f, t ∈ g}` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_vsub : map₂ (-ᵥ) f g = f -ᵥ g | rfl | lemma | filter.map₂_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vsub {s : set α} : s ∈ f -ᵥ g ↔ ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ -ᵥ t₂ ⊆ s | iff.rfl | lemma | filter.mem_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_mem_vsub : s ∈ f → t ∈ g → s -ᵥ t ∈ f -ᵥ g | image2_mem_map₂ | lemma | filter.vsub_mem_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_vsub : (⊥ : filter β) -ᵥ g = ⊥ | map₂_bot_left | lemma | filter.bot_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_bot : f -ᵥ (⊥ : filter β) = ⊥ | map₂_bot_right | lemma | filter.vsub_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_eq_bot_iff : f -ᵥ g = ⊥ ↔ f = ⊥ ∨ g = ⊥ | map₂_eq_bot_iff | lemma | filter.vsub_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_ne_bot_iff : (f -ᵥ g : filter α).ne_bot ↔ f.ne_bot ∧ g.ne_bot | map₂_ne_bot_iff | lemma | filter.vsub_ne_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.vsub : ne_bot f → ne_bot g → ne_bot (f -ᵥ g) | ne_bot.map₂ | lemma | filter.ne_bot.vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_vsub_left : (f -ᵥ g : filter α).ne_bot → f.ne_bot | ne_bot.of_map₂_left | lemma | filter.ne_bot.of_vsub_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_vsub_right : (f -ᵥ g : filter α).ne_bot → g.ne_bot | ne_bot.of_map₂_right | lemma | filter.ne_bot.of_vsub_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_vsub : (pure a : filter β) -ᵥ g = g.map ((-ᵥ) a) | map₂_pure_left | lemma | filter.pure_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_pure : f -ᵥ pure b = f.map (-ᵥ b) | map₂_pure_right | lemma | filter.vsub_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_vsub_pure : (pure a : filter β) -ᵥ pure b = (pure (a -ᵥ b) : filter α) | map₂_pure | lemma | filter.pure_vsub_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_le_vsub : f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ -ᵥ g₁ ≤ f₂ -ᵥ g₂ | map₂_mono | lemma | filter.vsub_le_vsub | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_le_vsub_left : g₁ ≤ g₂ → f -ᵥ g₁ ≤ f -ᵥ g₂ | map₂_mono_left | lemma | filter.vsub_le_vsub_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vsub_le_vsub_right : f₁ ≤ f₂ → f₁ -ᵥ g ≤ f₂ -ᵥ g | map₂_mono_right | lemma | filter.vsub_le_vsub_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_vsub_iff : h ≤ f -ᵥ g ↔ ∀ ⦃s⦄, s ∈ f → ∀ ⦃t⦄, t ∈ g → s -ᵥ t ∈ h | le_map₂_iff | lemma | filter.le_vsub_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul_filter : has_smul α (filter β) | ⟨λ a, map ((•) a)⟩ | def | filter.has_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul"
] | `a • f` is the map of `f` under `a •` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul : map (λ b, a • b) f = a • f | rfl | lemma | filter.map_smul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_filter : s ∈ a • f ↔ (•) a ⁻¹' s ∈ f | iff.rfl | lemma | filter.mem_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_set_mem_smul_filter : s ∈ f → a • s ∈ a • f | image_mem_map | lemma | filter.smul_set_mem_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_filter_bot : a • (⊥ : filter β) = ⊥ | map_bot | lemma | filter.smul_filter_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_filter_eq_bot_iff : a • f = ⊥ ↔ f = ⊥ | map_eq_bot_iff | lemma | filter.smul_filter_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_filter_ne_bot_iff : (a • f).ne_bot ↔ f.ne_bot | map_ne_bot_iff _ | lemma | filter.smul_filter_ne_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.smul_filter : f.ne_bot → (a • f).ne_bot | λ h, h.map _ | lemma | filter.ne_bot.smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_smul_filter : (a • f).ne_bot → f.ne_bot | ne_bot.of_map | lemma | filter.ne_bot.of_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_filter_le_smul_filter (hf : f₁ ≤ f₂) : a • f₁ ≤ a • f₂ | map_mono hf | lemma | filter.smul_filter_le_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_smul_filter : covariant_class α (filter β) (•) (≤) | ⟨λ f, map_mono⟩ | instance | filter.covariant_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_filter [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
smul_comm_class α β (filter γ) | ⟨λ _ _ _, map_comm (funext $ smul_comm _ _) _⟩ | instance | filter.smul_comm_class_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_filter' [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
smul_comm_class α (filter β) (filter γ) | ⟨λ a f g, map_map₂_distrib_right $ smul_comm a⟩ | instance | filter.smul_comm_class_filter' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_filter'' [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
smul_comm_class (filter α) β (filter γ) | by haveI := smul_comm_class.symm α β γ; exact smul_comm_class.symm _ _ _ | instance | filter.smul_comm_class_filter'' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class [has_smul α γ] [has_smul β γ] [smul_comm_class α β γ] :
smul_comm_class (filter α) (filter β) (filter γ) | ⟨λ f g h, map₂_left_comm smul_comm⟩ | instance | filter.smul_comm_class | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
is_scalar_tower α β (filter γ) | ⟨λ a b f, by simp only [←map_smul, map_map, smul_assoc]⟩ | instance | filter.is_scalar_tower | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
is_scalar_tower α (filter β) (filter γ) | ⟨λ a f g, by { refine (map_map₂_distrib_left $ λ _ _, _).symm, exact (smul_assoc a _ _).symm }⟩ | instance | filter.is_scalar_tower' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] :
is_scalar_tower (filter α) (filter β) (filter γ) | ⟨λ f g h, map₂_assoc smul_assoc⟩ | instance | filter.is_scalar_tower'' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_central_scalar [has_smul α β] [has_smul αᵐᵒᵖ β] [is_central_scalar α β] :
is_central_scalar α (filter β) | ⟨λ a f, congr_arg (λ m, map m f) $ by exact funext (λ _, op_smul_eq_smul _ _)⟩ | instance | filter.is_central_scalar | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_smul",
"is_central_scalar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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