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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|---|---|---|---|---|---|---|---|---|---|---|
pi.has_sum {f : ι → ∀ x, π x} {g : ∀ x, π x} :
has_sum f g ↔ ∀ x, has_sum (λ i, f i x) (g x) | by simp only [has_sum, tendsto_pi_nhds, sum_apply] | lemma | pi.has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"tendsto_pi_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.summable {f : ι → ∀ x, π x} : summable f ↔ ∀ x, summable (λ i, f i x) | by simp only [summable, pi.has_sum, skolem] | lemma | pi.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"pi.has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_apply [∀ x, t2_space (π x)] {f : ι → ∀ x, π x}{x : α} (hf : summable f) :
(∑' i, f i) x = ∑' i, f i x | (pi.has_sum.mp hf.has_sum x).tsum_eq.symm | lemma | tsum_apply | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.op (hf : has_sum f a) : has_sum (λ a, op (f a)) (op a) | (hf.map (@op_add_equiv α _) continuous_op : _) | lemma | has_sum.op | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.op (hf : summable f) : summable (op ∘ f) | hf.has_sum.op.summable | lemma | summable.op | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.unop {f : β → αᵐᵒᵖ} {a : αᵐᵒᵖ} (hf : has_sum f a) :
has_sum (λ a, unop (f a)) (unop a) | (hf.map (@op_add_equiv α _).symm continuous_unop : _) | lemma | has_sum.unop | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.unop {f : β → αᵐᵒᵖ} (hf : summable f) : summable (unop ∘ f) | hf.has_sum.unop.summable | lemma | summable.unop | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_op : has_sum (λ a, op (f a)) (op a) ↔ has_sum f a | ⟨has_sum.unop, has_sum.op⟩ | lemma | has_sum_op | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_unop {f : β → αᵐᵒᵖ} {a : αᵐᵒᵖ} :
has_sum (λ a, unop (f a)) (unop a) ↔ has_sum f a | ⟨has_sum.op, has_sum.unop⟩ | lemma | has_sum_unop | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_op : summable (λ a, op (f a)) ↔ summable f | ⟨summable.unop, summable.op⟩ | lemma | summable_op | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_unop {f : β → αᵐᵒᵖ} : summable (λ a, unop (f a)) ↔ summable f | ⟨summable.op, summable.unop⟩ | lemma | summable_unop | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_op : ∑' x, mul_opposite.op (f x) = mul_opposite.op (∑' x, f x) | begin
by_cases h : summable f,
{ exact h.has_sum.op.tsum_eq },
{ have ho := summable_op.not.mpr h,
rw [tsum_eq_zero_of_not_summable h, tsum_eq_zero_of_not_summable ho, mul_opposite.op_zero] }
end | lemma | tsum_op | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"mul_opposite.op",
"mul_opposite.op_zero",
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_unop {f : β → αᵐᵒᵖ} : ∑' x, mul_opposite.unop (f x) = mul_opposite.unop (∑' x, f x) | mul_opposite.op_injective tsum_op.symm | lemma | tsum_unop | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"mul_opposite.op_injective",
"mul_opposite.unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.star (h : has_sum f a) : has_sum (λ b, star (f b)) (star a) | by simpa only using h.map (star_add_equiv : α ≃+ α) continuous_star | lemma | has_sum.star | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"star_add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.star (hf : summable f) : summable (λ b, star (f b)) | hf.has_sum.star.summable | lemma | summable.star | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.of_star (hf : summable (λ b, star (f b))) : summable f | by simpa only [star_star] using hf.star | lemma | summable.of_star | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"star_star",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_star_iff : summable (λ b, star (f b)) ↔ summable f | ⟨summable.of_star, summable.star⟩ | lemma | summable_star_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_star_iff' : summable (star f) ↔ summable f | summable_star_iff | lemma | summable_star_iff' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"summable_star_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_star : star (∑' b, f b) = ∑' b, star (f b) | begin
by_cases hf : summable f,
{ exact hf.has_sum.star.tsum_eq.symm, },
{ rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt summable.of_star hf),
star_zero] },
end | lemma | tsum_star | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"star_zero",
"summable",
"summable.of_star",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action.automorphize [group α] [mul_action α β] (f : β → M) :
quotient (mul_action.orbit_rel α β) → M | @quotient.lift _ _ (mul_action.orbit_rel α β) (λ b, ∑' (a : α), f(a • b))
begin
rintros b₁ b₂ ⟨a, (rfl : a • b₂ = b₁)⟩,
simpa [mul_smul] using (equiv.mul_right a).tsum_eq (λ a', f (a' • b₂)),
end | def | mul_action.automorphize | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.mul_right",
"group",
"mul_action",
"mul_action.orbit_rel"
] | Given a group `α` acting on a type `β`, and a function `f : β → M`, we "automorphize" `f` to a
function `β ⧸ α → M` by summing over `α` orbits, `b ↦ ∑' (a : α), f(a • b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action.automorphize_smul_left [group α] [mul_action α β] (f : β → M)
(g : quotient (mul_action.orbit_rel α β) → R) :
mul_action.automorphize ((g ∘ quotient.mk') • f)
= g • (mul_action.automorphize f : quotient (mul_action.orbit_rel α β) → M) | begin
ext x,
apply quotient.induction_on' x,
intro b,
simp only [mul_action.automorphize, pi.smul_apply', function.comp_app],
set π : β → quotient (mul_action.orbit_rel α β) := quotient.mk',
have H₁ : ∀ a : α, π (a • b) = π b,
{ intro a,
rw quotient.eq_rel,
fconstructor,
exact a,
simp, },
... | lemma | mul_action.automorphize_smul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"group",
"mul_action",
"mul_action.automorphize",
"mul_action.orbit_rel",
"pi.smul_apply'",
"quotient.eq_rel",
"quotient.induction_on'",
"quotient.mk'",
"tsum_const_smul''"
] | Automorphization of a function into an `R`-`module` distributes, that is, commutes with the
`R`-scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_action.automorphize_smul_left [add_group α] [add_action α β] (f : β → M)
(g : quotient (add_action.orbit_rel α β) → R) :
add_action.automorphize ((g ∘ quotient.mk') • f)
= g • (add_action.automorphize f : quotient (add_action.orbit_rel α β) → M) | begin
ext x,
apply quotient.induction_on' x,
intro b,
simp only [add_action.automorphize, pi.smul_apply', function.comp_app],
set π : β → quotient (add_action.orbit_rel α β) := quotient.mk',
have H₁ : ∀ a : α, π (a +ᵥ b) = π b,
{ intro a,
rw quotient.eq_rel,
fconstructor,
exact a,
simp, },... | lemma | add_action.automorphize_smul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_action",
"add_group",
"pi.smul_apply'",
"quotient.eq_rel",
"quotient.induction_on'",
"quotient.mk'",
"tsum_const_smul''"
] | Automorphization of a function into an `R`-`module` distributes, that is, commutes with the
`R`-scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.automorphize (f : G → M) : G ⧸ Γ → M | mul_action.automorphize f | def | quotient_group.automorphize | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"mul_action.automorphize"
] | Given a subgroup `Γ` of a group `G`, and a function `f : G → M`, we "automorphize" `f` to a
function `G ⧸ Γ → M` by summing over `Γ` orbits, `g ↦ ∑' (γ : Γ), f(γ • g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_group.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
quotient_group.automorphize ((g ∘ quotient.mk') • f)
= g • (quotient_group.automorphize f : G ⧸ Γ → M) | mul_action.automorphize_smul_left f g | lemma | quotient_group.automorphize_smul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"mul_action.automorphize_smul_left",
"quotient.mk'",
"quotient_group.automorphize"
] | Automorphization of a function into an `R`-`module` distributes, that is, commutes with the
`R`-scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_add_group.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
quotient_add_group.automorphize ((g ∘ quotient.mk') • f)
= g • (quotient_add_group.automorphize f : G ⧸ Γ → M) | add_action.automorphize_smul_left f g | lemma | quotient_add_group.automorphize_smul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_action.automorphize_smul_left",
"quotient.mk'"
] | Automorphization of a function into an `R`-`module` distributes, that is, commutes with the `R`
-scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.smul_const {r : R} (hf : has_sum f r) (a : M) : has_sum (λ z, f z • a) (r • a) | hf.map ((smul_add_hom R M).flip a) (continuous_id.smul continuous_const) | lemma | has_sum.smul_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous_const",
"has_sum",
"smul_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.smul_const (hf : summable f) (a : M) : summable (λ z, f z • a) | (hf.has_sum.smul_const _).summable | lemma | summable.smul_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_smul_const [t2_space M] (hf : summable f) (a : M) : ∑' z, f z • a = (∑' z, f z) • a | (hf.has_sum.smul_const _).tsum_eq | lemma | tsum_smul_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_sum {f : ι → M} (φ : M →SL[σ] M₂) {x : M}
(hf : has_sum f x) :
has_sum (λ (b:ι), φ (f b)) (φ x) | by simpa only using hf.map φ.to_linear_map.to_add_monoid_hom φ.continuous | lemma | continuous_linear_map.has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"has_sum"
] | Applying a continuous linear map commutes with taking an (infinite) sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.summable {f : ι → M} (φ : M →SL[σ] M₂) (hf : summable f) :
summable (λ b:ι, φ (f b)) | (hf.has_sum.mapL φ).summable | lemma | continuous_linear_map.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.map_tsum [t2_space M₂] {f : ι → M}
(φ : M →SL[σ] M₂) (hf : summable f) : φ (∑' z, f z) = ∑' z, φ (f z) | (hf.has_sum.mapL φ).tsum_eq.symm | lemma | continuous_linear_map.map_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.has_sum {f : ι → M} (e : M ≃SL[σ] M₂) {y : M₂} :
has_sum (λ (b:ι), e (f b)) y ↔ has_sum f (e.symm y) | ⟨λ h, by simpa only [e.symm.coe_coe, e.symm_apply_apply] using h.mapL (e.symm : M₂ →SL[σ'] M),
λ h, by simpa only [e.coe_coe, e.apply_symm_apply] using (e : M →SL[σ] M₂).has_sum h⟩ | lemma | continuous_linear_equiv.has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"has_sum"
] | Applying a continuous linear map commutes with taking an (infinite) sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.has_sum' {f : ι → M} (e : M ≃SL[σ] M₂) {x : M} :
has_sum (λ (b:ι), e (f b)) (e x) ↔ has_sum f x | by rw [e.has_sum, continuous_linear_equiv.symm_apply_apply] | lemma | continuous_linear_equiv.has_sum' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"continuous_linear_equiv.symm_apply_apply",
"has_sum"
] | Applying a continuous linear map commutes with taking an (infinite) sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.summable {f : ι → M} (e : M ≃SL[σ] M₂) :
summable (λ b:ι, e (f b)) ↔ summable f | ⟨λ hf, (e.has_sum.1 hf.has_sum).summable, (e : M →SL[σ] M₂).summable⟩ | lemma | continuous_linear_equiv.summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.tsum_eq_iff [t2_space M] [t2_space M₂] {f : ι → M}
(e : M ≃SL[σ] M₂) {y : M₂} : ∑' z, e (f z) = y ↔ ∑' z, f z = e.symm y | begin
by_cases hf : summable f,
{ exact ⟨λ h, (e.has_sum.mp ((e.summable.mpr hf).has_sum_iff.mpr h)).tsum_eq,
λ h, (e.has_sum.mpr (hf.has_sum_iff.mpr h)).tsum_eq⟩ },
{ have hf' : ¬summable (λ z, e (f z)) := λ h, hf (e.summable.mp h),
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hf']... | lemma | continuous_linear_equiv.tsum_eq_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.map_tsum [t2_space M] [t2_space M₂] {f : ι → M}
(e : M ≃SL[σ] M₂) : e (∑' z, f z) = ∑' z, e (f z) | by { refine symm (e.tsum_eq_iff.mpr _), rw e.symm_apply_apply _ } | lemma | continuous_linear_equiv.map_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/module.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.module.basic"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_of_sum_range_le (hf : summable f) (h : ∀ n, ∑ i in range n, f i ≤ c) :
∑' n, f n ≤ c | let ⟨l, hl⟩ := hf in hl.tsum_eq.symm ▸ le_of_tendsto' hl.tendsto_sum_nat h | lemma | tsum_le_of_sum_range_le | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"le_of_tendsto'",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_le (h : ∀ i, f i ≤ g i) (hf : has_sum f a₁) (hg : has_sum g a₂) : a₁ ≤ a₂ | le_of_tendsto_of_tendsto' hf hg $ λ s, sum_le_sum $ λ i _, h i | lemma | has_sum_le | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"le_of_tendsto_of_tendsto'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_mono (hf : has_sum f a₁) (hg : has_sum g a₂) (h : f ≤ g) : a₁ ≤ a₂ | has_sum_le h hf hg | lemma | has_sum_mono | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_le_of_sum_le (hf : has_sum f a) (h : ∀ s, ∑ i in s, f i ≤ a₂) : a ≤ a₂ | le_of_tendsto' hf h | lemma | has_sum_le_of_sum_le | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"le_of_tendsto'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_has_sum_of_le_sum (hf : has_sum f a) (h : ∀ s, a₂ ≤ ∑ i in s, f i) : a₂ ≤ a | ge_of_tendsto' hf h | lemma | le_has_sum_of_le_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"ge_of_tendsto'",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_le_inj {g : κ → α} (e : ι → κ) (he : injective e) (hs : ∀ c ∉ set.range e, 0 ≤ g c)
(h : ∀ i, f i ≤ g (e i)) (hf : has_sum f a₁) (hg : has_sum g a₂) : a₁ ≤ a₂ | have has_sum (λ c, (partial_inv e c).cases_on' 0 f) a₁,
begin
refine (has_sum_iff_has_sum_of_ne_zero_bij (e ∘ coe) (λ c₁ c₂ hc, he hc) (λ c hc, _) _).2 hf,
{ rw mem_support at hc,
cases eq : partial_inv e c with i; rw eq at hc,
{ contradiction },
{ rw [partial_inv_of_injective he] at eq,
exact ⟨⟨i... | lemma | has_sum_le_inj | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"em",
"has_sum",
"has_sum_iff_has_sum_of_ne_zero_bij",
"has_sum_le",
"option.cases_on'",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_tsum_of_inj {g : κ → α} (e : ι → κ) (he : injective e)
(hs : ∀ c ∉ set.range e, 0 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : summable f) (hg : summable g) :
tsum f ≤ tsum g | has_sum_le_inj _ he hs h hf.has_sum hg.has_sum | lemma | tsum_le_tsum_of_inj | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum_le_inj",
"set.range",
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_has_sum (s : finset ι) (hs : ∀ i ∉ s, 0 ≤ f i) (hf : has_sum f a) :
∑ i in s, f i ≤ a | ge_of_tendsto hf (eventually_at_top.2 ⟨s, λ t hst,
sum_le_sum_of_subset_of_nonneg hst $ λ i hbt hbs, hs i hbs⟩) | lemma | sum_le_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"finset",
"ge_of_tendsto",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_has_sum (h : ∀ i, 0 ≤ f i) (hf : has_sum f a) :
is_lub (set.range $ λ s, ∑ i in s, f i) a | is_lub_of_tendsto_at_top (finset.sum_mono_set_of_nonneg h) hf | lemma | is_lub_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"is_lub",
"is_lub_of_tendsto_at_top",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_has_sum (hf : has_sum f a) (i : ι) (hb : ∀ b' ≠ i, 0 ≤ f b') : f i ≤ a | calc f i = ∑ i in {i}, f i : finset.sum_singleton.symm
... ≤ a : sum_le_has_sum _ (by { convert hb, simp }) hf | lemma | le_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"sum_le_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_le_tsum {f : ι → α} (s : finset ι) (hs : ∀ i ∉ s, 0 ≤ f i) (hf : summable f) :
∑ i in s, f i ≤ ∑' i, f i | sum_le_has_sum s hs hf.has_sum | lemma | sum_le_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"finset",
"sum_le_has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsum (hf : summable f) (i : ι) (hb : ∀ b' ≠ i, 0 ≤ f b') : f i ≤ ∑' i, f i | le_has_sum (summable.has_sum hf) i hb | lemma | le_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"le_has_sum",
"summable",
"summable.has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_tsum (h : ∀ i, f i ≤ g i) (hf : summable f) (hg : summable g) :
∑' i, f i ≤ ∑' i, g i | has_sum_le h hf.has_sum hg.has_sum | lemma | tsum_le_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum_le",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mono (hf : summable f) (hg : summable g) (h : f ≤ g) :
∑' n, f n ≤ ∑' n, g n | tsum_le_tsum h hf hg | lemma | tsum_mono | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"tsum_le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_of_sum_le (hf : summable f) (h : ∀ s, ∑ i in s, f i ≤ a₂) : ∑' i, f i ≤ a₂ | has_sum_le_of_sum_le hf.has_sum h | lemma | tsum_le_of_sum_le | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum_le_of_sum_le",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_le_of_sum_le' (ha₂ : 0 ≤ a₂) (h : ∀ s, ∑ i in s, f i ≤ a₂) : ∑' i, f i ≤ a₂ | begin
by_cases hf : summable f,
{ exact tsum_le_of_sum_le hf h },
{ rw tsum_eq_zero_of_not_summable hf,
exact ha₂ }
end | lemma | tsum_le_of_sum_le' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"tsum_eq_zero_of_not_summable",
"tsum_le_of_sum_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.nonneg (h : ∀ i, 0 ≤ g i) (ha : has_sum g a) : 0 ≤ a | has_sum_le h has_sum_zero ha | lemma | has_sum.nonneg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_le",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.nonpos (h : ∀ i, g i ≤ 0) (ha : has_sum g a) : a ≤ 0 | has_sum_le h ha has_sum_zero | lemma | has_sum.nonpos | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_le",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_nonneg (h : ∀ i, 0 ≤ g i) : 0 ≤ ∑' i, g i | begin
by_cases hg : summable g,
{ exact hg.has_sum.nonneg h },
{ simp [tsum_eq_zero_of_not_summable hg] }
end | lemma | tsum_nonneg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_nonpos (h : ∀ i, f i ≤ 0) : ∑' i, f i ≤ 0 | begin
by_cases hf : summable f,
{ exact hf.has_sum.nonpos h },
{ simp [tsum_eq_zero_of_not_summable hf] }
end | lemma | tsum_nonpos | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_lt (h : f ≤ g) (hi : f i < g i) (hf : has_sum f a₁) (hg : has_sum g a₂) : a₁ < a₂ | have update f i 0 ≤ update g i 0 := update_le_update_iff.mpr ⟨rfl.le, λ i _, h i⟩,
have 0 - f i + a₁ ≤ 0 - g i + a₂ := has_sum_le this (hf.update i 0) (hg.update i 0),
by simpa only [zero_sub, add_neg_cancel_left] using add_lt_add_of_lt_of_le hi this | lemma | has_sum_lt | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_le",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_strict_mono (hf : has_sum f a₁) (hg : has_sum g a₂) (h : f < g) : a₁ < a₂ | let ⟨hle, i, hi⟩ := pi.lt_def.mp h in has_sum_lt hle hi hf hg | lemma | has_sum_strict_mono | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_lt_tsum (h : f ≤ g) (hi : f i < g i) (hf : summable f) (hg : summable g) :
∑' n, f n < ∑' n, g n | has_sum_lt h hi hf.has_sum hg.has_sum | lemma | tsum_lt_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum_lt",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_strict_mono (hf : summable f) (hg : summable g) (h : f < g) :
∑' n, f n < ∑' n, g n | let ⟨hle, i, hi⟩ := pi.lt_def.mp h in tsum_lt_tsum hle hi hf hg | lemma | tsum_strict_mono | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"tsum_lt_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_pos (hsum : summable g) (hg : ∀ i, 0 ≤ g i) (i : ι) (hi : 0 < g i) : 0 < ∑' i, g i | by { rw ←tsum_zero, exact tsum_lt_tsum hg hi summable_zero hsum } | lemma | tsum_pos | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable",
"summable_zero",
"tsum_lt_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_zero_iff_of_nonneg (hf : ∀ i, 0 ≤ f i) : has_sum f 0 ↔ f = 0 | begin
refine ⟨λ hf', _, _⟩,
{ ext i,
refine (hf i).eq_of_not_gt (λ hi, _),
simpa using has_sum_lt hf hi has_sum_zero hf' },
{ rintro rfl,
exact has_sum_zero }
end | lemma | has_sum_zero_iff_of_nonneg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_lt",
"has_sum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_has_sum' (hf : has_sum f a) (i : ι) : f i ≤ a | le_has_sum hf i $ λ _ _, zero_le _ | lemma | le_has_sum' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"le_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsum' (hf : summable f) (i : ι) : f i ≤ ∑' i, f i | le_tsum hf i $ λ _ _, zero_le _ | lemma | le_tsum' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"le_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_zero_iff : has_sum f 0 ↔ ∀ x, f x = 0 | begin
refine ⟨_, λ h, _⟩,
{ contrapose!,
exact λ ⟨x, hx⟩ h, hx (nonpos_iff_eq_zero.1$ le_has_sum' h x) },
{ convert has_sum_zero,
exact funext h }
end | lemma | has_sum_zero_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"has_sum_zero",
"le_has_sum'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_zero_iff (hf : summable f) : ∑' i, f i = 0 ↔ ∀ x, f x = 0 | by rw [←has_sum_zero_iff, hf.has_sum_iff] | lemma | tsum_eq_zero_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_ne_zero_iff (hf : summable f) : ∑' i, f i ≠ 0 ↔ ∃ x, f x ≠ 0 | by rw [ne.def, tsum_eq_zero_iff hf, not_forall] | lemma | tsum_ne_zero_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"not_forall",
"summable",
"tsum_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_has_sum' (hf : has_sum f a) : is_lub (set.range $ λ s, ∑ i in s, f i) a | is_lub_of_tendsto_at_top (finset.sum_mono_set f) hf | lemma | is_lub_has_sum' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"is_lub",
"is_lub_of_tendsto_at_top",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_of_is_lub_of_nonneg [linear_ordered_add_comm_monoid α] [topological_space α]
[order_topology α] {f : ι → α} (i : α) (h : ∀ i, 0 ≤ f i)
(hf : is_lub (set.range $ λ s, ∑ i in s, f i) i) :
has_sum f i | tendsto_at_top_is_lub (finset.sum_mono_set_of_nonneg h) hf | lemma | has_sum_of_is_lub_of_nonneg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"has_sum",
"is_lub",
"linear_ordered_add_comm_monoid",
"order_topology",
"set.range",
"tendsto_at_top_is_lub",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_of_is_lub [canonically_linear_ordered_add_monoid α] [topological_space α]
[order_topology α] {f : ι → α} (b : α) (hf : is_lub (set.range $ λ s, ∑ i in s, f i) b) :
has_sum f b | tendsto_at_top_is_lub (finset.sum_mono_set f) hf | lemma | has_sum_of_is_lub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"canonically_linear_ordered_add_monoid",
"has_sum",
"is_lub",
"order_topology",
"set.range",
"tendsto_at_top_is_lub",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_abs_iff [linear_ordered_add_comm_group α] [uniform_space α] [uniform_add_group α]
[complete_space α] {f : ι → α} :
summable (λ x, |f x|) ↔ summable f | have h1 : ∀ x : {x | 0 ≤ f x}, |f x| = f x := λ x, abs_of_nonneg x.2,
have h2 : ∀ x : {x | 0 ≤ f x}ᶜ, |f x| = -f x := λ x, abs_of_neg (not_le.1 x.2),
calc summable (λ x, |f x|) ↔
summable (λ x : {x | 0 ≤ f x}, |f x|) ∧ summable (λ x : {x | 0 ≤ f x}ᶜ, |f x|) :
summable_subtype_and_compl.symm
... ↔ summable (λ x : {x... | lemma | summable_abs_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"abs_of_neg",
"abs_of_nonneg",
"complete_space",
"linear_ordered_add_comm_group",
"summable",
"summable_neg_iff",
"summable_subtype_and_compl",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_summable_const [linear_ordered_add_comm_group α] [topological_space α]
[archimedean α] [order_closed_topology α] {b : α} (hb : 0 < b) (hf : summable (λ i : ι, b)) :
(set.univ : set ι).finite | begin
have H : ∀ s : finset ι, s.card • b ≤ ∑' i : ι, b,
{ intros s,
simpa using sum_le_has_sum s (λ a ha, hb.le) hf.has_sum },
obtain ⟨n, hn⟩ := archimedean.arch (∑' i : ι, b) hb,
have : ∀ s : finset ι, s.card ≤ n,
{ intros s,
simpa [nsmul_le_nsmul_iff hb] using (H s).trans hn },
haveI : fintype ι ... | lemma | finite_of_summable_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"archimedean",
"finite",
"finset",
"fintype",
"fintype_of_finset_card_le",
"linear_ordered_add_comm_group",
"order_closed_topology",
"set.finite_univ",
"sum_le_has_sum",
"summable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.tendsto_top_of_pos [linear_ordered_field α] [topological_space α] [order_topology α]
{f : ℕ → α} (hf : summable f⁻¹) (hf' : ∀ n, 0 < f n) : tendsto f at_top at_top | begin
rw ←inv_inv f,
apply filter.tendsto.inv_tendsto_zero,
apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
(summable.tendsto_at_top_zero hf),
rw eventually_iff_exists_mem,
refine ⟨set.Ioi 0, Ioi_mem_at_top _, λ _ _, _⟩,
rw [set.mem_Ioi, inv_eq_one_div, one_div, pi.inv_apply, _root_.inv... | lemma | summable.tendsto_top_of_pos | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/order.lean | [
"algebra.order.archimedean",
"topology.algebra.infinite_sum.basic",
"topology.algebra.order.field",
"topology.algebra.order.monotone_convergence"
] | [
"filter.tendsto.inv_tendsto_zero",
"inv_eq_one_div",
"linear_ordered_field",
"one_div",
"order_topology",
"pi.inv_apply",
"set.mem_Ioi",
"summable",
"summable.tendsto_at_top_zero",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_of_edist_le_of_summable [pseudo_emetric_space α] {f : ℕ → α} (d : ℕ → ℝ≥0)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : summable d) : cauchy_seq f | begin
refine emetric.cauchy_seq_iff_nnreal.2 (λ ε εpos, _),
-- Actually we need partial sums of `d` to be a Cauchy sequence
replace hd : cauchy_seq (λ (n : ℕ), ∑ x in range n, d x) :=
let ⟨_, H⟩ := hd in H.tendsto_sum_nat.cauchy_seq,
-- Now we take the same `N` as in one of the definitions of a Cauchy seque... | lemma | cauchy_seq_of_edist_le_of_summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"add_tsub_cancel_left",
"cauchy_seq",
"dist_nndist",
"edist_le_Ico_sum_of_edist_le",
"nnreal.nndist_eq",
"pseudo_emetric_space",
"summable"
] | If the extended distance between consecutive points of a sequence is estimated
by a summable series of `nnreal`s, then the original sequence is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : summable d) : cauchy_seq f | begin
refine metric.cauchy_seq_iff'.2 (λε εpos, _),
replace hd : cauchy_seq (λ (n : ℕ), ∑ x in range n, d x) :=
let ⟨_, H⟩ := hd in H.tendsto_sum_nat.cauchy_seq,
refine (metric.cauchy_seq_iff'.1 hd ε εpos).imp (λ N hN n hn, _),
have hsum := hN n hn,
rw [real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum,
calc... | lemma | cauchy_seq_of_dist_le_of_summable | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"cauchy_seq",
"dist_comm",
"dist_le_Ico_sum_of_dist_le",
"le_abs_self",
"real.dist_eq",
"summable"
] | If the distance between consecutive points of a sequence is estimated by a summable series,
then the original sequence is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_summable_dist (h : summable (λ n, dist (f n) (f n.succ))) : cauchy_seq f | cauchy_seq_of_dist_le_of_summable _ (λ _, le_rfl) h | lemma | cauchy_seq_of_summable_dist | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"cauchy_seq",
"cauchy_seq_of_dist_le_of_summable",
"le_rfl",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : summable d) {a : α} (ha : tendsto f at_top (𝓝 a)) (n : ℕ) :
dist (f n) a ≤ ∑' m, d (n + m) | begin
refine le_of_tendsto (tendsto_const_nhds.dist ha)
(eventually_at_top.2 ⟨n, λ m hnm, _⟩),
refine le_trans (dist_le_Ico_sum_of_dist_le hnm (λ k _ _, hf k)) _,
rw [sum_Ico_eq_sum_range],
refine sum_le_tsum (range _) (λ _ _, le_trans dist_nonneg (hf _)) _,
exact hd.comp_injective (add_right_injective n)... | lemma | dist_le_tsum_of_dist_le_of_tendsto | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"dist_le_Ico_sum_of_dist_le",
"dist_nonneg",
"le_of_tendsto",
"sum_le_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : summable d) (ha : tendsto f at_top (𝓝 a)) :
dist (f 0) a ≤ tsum d | by simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0 | lemma | dist_le_tsum_of_dist_le_of_tendsto₀ | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"dist_le_tsum_of_dist_le_of_tendsto",
"summable",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_tsum_dist_of_tendsto (h : summable (λ n, dist (f n) (f n.succ)))
(ha : tendsto f at_top (𝓝 a)) (n) :
dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) | show dist (f n) a ≤ ∑' m, (λx, dist (f x) (f x.succ)) (n + m), from
dist_le_tsum_of_dist_le_of_tendsto (λ n, dist (f n) (f n.succ)) (λ _, le_rfl) h ha n | lemma | dist_le_tsum_dist_of_tendsto | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"dist_le_tsum_of_dist_le_of_tendsto",
"le_rfl",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_tsum_dist_of_tendsto₀ (h : summable (λ n, dist (f n) (f n.succ)))
(ha : tendsto f at_top (𝓝 a)) :
dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) | by simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0 | lemma | dist_le_tsum_dist_of_tendsto₀ | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/real.lean | [
"algebra.big_operators.intervals",
"topology.algebra.infinite_sum.order",
"topology.instances.real"
] | [
"dist_le_tsum_dist_of_tendsto",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.mul_left (a₂) (h : has_sum f a₁) : has_sum (λ i, a₂ * f i) (a₂ * a₁) | by simpa only using h.map (add_monoid_hom.mul_left a₂) (continuous_const.mul continuous_id) | lemma | has_sum.mul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"add_monoid_hom.mul_left",
"continuous_id",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.mul_right (a₂) (hf : has_sum f a₁) : has_sum (λ i, f i * a₂) (a₁ * a₂) | by simpa only using hf.map (add_monoid_hom.mul_right a₂) (continuous_id.mul continuous_const) | lemma | has_sum.mul_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"add_monoid_hom.mul_right",
"continuous_const",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.mul_left (a) (hf : summable f) : summable (λ i, a * f i) | (hf.has_sum.mul_left _).summable | lemma | summable.mul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.mul_right (a) (hf : summable f) : summable (λ i, f i * a) | (hf.has_sum.mul_right _).summable | lemma | summable.mul_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.tsum_mul_left (a) (hf : summable f) : ∑' i, a * f i = a * ∑' i, f i | (hf.has_sum.mul_left _).tsum_eq | lemma | summable.tsum_mul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.tsum_mul_right (a) (hf : summable f) : ∑' i, f i * a = (∑' i, f i) * a | (hf.has_sum.mul_right _).tsum_eq | lemma | summable.tsum_mul_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.tsum_right (a) (h : ∀ i, commute a (f i)) : commute a (∑' i, f i) | if hf : summable f then
(hf.tsum_mul_left a).symm.trans ((congr_arg _ $ funext h).trans (hf.tsum_mul_right a))
else
(tsum_eq_zero_of_not_summable hf).symm ▸ commute.zero_right _ | lemma | commute.tsum_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"commute",
"commute.zero_right",
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.tsum_left (a) (h : ∀ i, commute (f i) a) : commute (∑' i, f i) a | (commute.tsum_right _ $ λ i, (h i).symm).symm | lemma | commute.tsum_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"commute",
"commute.tsum_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.div_const (h : has_sum f a) (b : α) : has_sum (λ i, f i / b) (a / b) | by simp only [div_eq_mul_inv, h.mul_right b⁻¹] | lemma | has_sum.div_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"div_eq_mul_inv",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.div_const (h : summable f) (b : α) : summable (λ i, f i / b) | (h.has_sum.div_const _).summable | lemma | summable.div_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_mul_left_iff (h : a₂ ≠ 0) : has_sum (λ i, a₂ * f i) (a₂ * a₁) ↔ has_sum f a₁ | ⟨λ H, by simpa only [inv_mul_cancel_left₀ h] using H.mul_left a₂⁻¹, has_sum.mul_left _⟩ | lemma | has_sum_mul_left_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"has_sum",
"has_sum.mul_left",
"inv_mul_cancel_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_mul_right_iff (h : a₂ ≠ 0) : has_sum (λ i, f i * a₂) (a₁ * a₂) ↔ has_sum f a₁ | ⟨λ H, by simpa only [mul_inv_cancel_right₀ h] using H.mul_right a₂⁻¹, has_sum.mul_right _⟩ | lemma | has_sum_mul_right_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"has_sum",
"has_sum.mul_right",
"mul_inv_cancel_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_div_const_iff (h : a₂ ≠ 0) : has_sum (λ i, f i / a₂) (a₁ / a₂) ↔ has_sum f a₁ | by simpa only [div_eq_mul_inv] using has_sum_mul_right_iff (inv_ne_zero h) | lemma | has_sum_div_const_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"div_eq_mul_inv",
"has_sum",
"has_sum_mul_right_iff",
"inv_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_mul_left_iff (h : a ≠ 0) : summable (λ i, a * f i) ↔ summable f | ⟨λ H, by simpa only [inv_mul_cancel_left₀ h] using H.mul_left a⁻¹, λ H, H.mul_left _⟩ | lemma | summable_mul_left_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"inv_mul_cancel_left₀",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_mul_right_iff (h : a ≠ 0) : summable (λ i, f i * a) ↔ summable f | ⟨λ H, by simpa only [mul_inv_cancel_right₀ h] using H.mul_right a⁻¹, λ H, H.mul_right _⟩ | lemma | summable_mul_right_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"mul_inv_cancel_right₀",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_div_const_iff (h : a ≠ 0) : summable (λ i, f i / a) ↔ summable f | by simpa only [div_eq_mul_inv] using summable_mul_right_iff (inv_ne_zero h) | lemma | summable_div_const_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"div_eq_mul_inv",
"inv_ne_zero",
"summable",
"summable_mul_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_left [t2_space α] : (∑' x, a * f x) = a * ∑' x, f x | if hf : summable f then hf.tsum_mul_left a
else if ha : a = 0 then by simp [ha]
else by rw [tsum_eq_zero_of_not_summable hf,
tsum_eq_zero_of_not_summable (mt (summable_mul_left_iff ha).mp hf), mul_zero] | lemma | tsum_mul_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"mul_zero",
"summable",
"summable_mul_left_iff",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_right [t2_space α] : (∑' x, f x * a) = (∑' x, f x) * a | if hf : summable f then hf.tsum_mul_right a
else if ha : a = 0 then by simp [ha]
else by rw [tsum_eq_zero_of_not_summable hf,
tsum_eq_zero_of_not_summable (mt (summable_mul_right_iff ha).mp hf), zero_mul] | lemma | tsum_mul_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"summable",
"summable_mul_right_iff",
"t2_space",
"tsum_eq_zero_of_not_summable",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_div_const [t2_space α] : (∑' x, f x / a) = (∑' x, f x) / a | by simpa only [div_eq_mul_inv] using tsum_mul_right | lemma | tsum_div_const | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"div_eq_mul_inv",
"t2_space",
"tsum_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.mul_eq (hf : has_sum f s) (hg : has_sum g t)
(hfg : has_sum (λ (x : ι × κ), f x.1 * g x.2) u) :
s * t = u | have key₁ : has_sum (λ i, f i * t) (s * t),
from hf.mul_right t,
have this : ∀ i : ι, has_sum (λ c : κ, f i * g c) (f i * t),
from λ i, hg.mul_left (f i),
have key₂ : has_sum (λ i, f i * t) u,
from has_sum.prod_fiberwise hfg this,
key₁.unique key₂ | lemma | has_sum.mul_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/ring.lean | [
"algebra.big_operators.nat_antidiagonal",
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic"
] | [
"has_sum",
"has_sum.prod_fiberwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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