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
value |
|---|---|---|---|---|---|---|---|---|---|---|
has_sum.mul (hf : has_sum f s) (hg : has_sum g t)
(hfg : summable (λ (x : ι × κ), f x.1 * g x.2)) :
has_sum (λ (x : ι × κ), f x.1 * g x.2) (s * t) | let ⟨u, hu⟩ := hfg in
(hf.mul_eq hg hu).symm ▸ hu | lemma | has_sum.mul | 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",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum (hf : summable f) (hg : summable g)
(hfg : summable (λ (x : ι × κ), f x.1 * g x.2)) :
(∑' x, f x) * (∑' y, g y) = (∑' z : ι × κ, f z.1 * g z.2) | hf.has_sum.mul_eq hg.has_sum hfg.has_sum | lemma | tsum_mul_tsum | 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"
] | Product of two infinites sums indexed by arbitrary types.
See also `tsum_mul_tsum_of_summable_norm` if `f` and `g` are abolutely summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_mul_prod_iff_summable_mul_sigma_antidiagonal :
summable (λ x : ℕ × ℕ, f x.1 * g x.2) ↔
summable (λ x : (Σ (n : ℕ), nat.antidiagonal n), f (x.2 : ℕ × ℕ).1 * g (x.2 : ℕ × ℕ).2) | nat.sigma_antidiagonal_equiv_prod.summable_iff.symm | lemma | summable_mul_prod_iff_summable_mul_sigma_antidiagonal | 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_sum_mul_antidiagonal_of_summable_mul (h : summable (λ x : ℕ × ℕ, f x.1 * g x.2)) :
summable (λ n, ∑ kl in nat.antidiagonal n, f kl.1 * g kl.2) | begin
rw summable_mul_prod_iff_summable_mul_sigma_antidiagonal at h,
conv {congr, funext, rw [← finset.sum_finset_coe, ← tsum_fintype]},
exact h.sigma' (λ n, (has_sum_fintype _).summable),
end | lemma | summable_sum_mul_antidiagonal_of_summable_mul | 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_fintype",
"summable",
"summable_mul_prod_iff_summable_mul_sigma_antidiagonal",
"tsum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum_eq_tsum_sum_antidiagonal (hf : summable f) (hg : summable g)
(hfg : summable (λ (x : ℕ × ℕ), f x.1 * g x.2)) :
(∑' n, f n) * (∑' n, g n) = (∑' n, ∑ kl in nat.antidiagonal n, f kl.1 * g kl.2) | begin
conv_rhs {congr, funext, rw [← finset.sum_finset_coe, ← tsum_fintype]},
rw [tsum_mul_tsum hf hg hfg, ← nat.sigma_antidiagonal_equiv_prod.tsum_eq (_ : ℕ × ℕ → α)],
exact tsum_sigma' (λ n, (has_sum_fintype _).summable)
(summable_mul_prod_iff_summable_mul_sigma_antidiagonal.mp hfg)
end | lemma | tsum_mul_tsum_eq_tsum_sum_antidiagonal | 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_fintype",
"summable",
"tsum_fintype",
"tsum_mul_tsum",
"tsum_sigma'"
] | The **Cauchy product formula** for the product of two infinites sums indexed by `ℕ`, expressed
by summing on `finset.nat.antidiagonal`.
See also `tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm` if `f` and `g` are absolutely
summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_sum_mul_range_of_summable_mul (h : summable (λ x : ℕ × ℕ, f x.1 * g x.2)) :
summable (λ n, ∑ k in range (n+1), f k * g (n - k)) | begin
simp_rw ← nat.sum_antidiagonal_eq_sum_range_succ (λ k l, f k * g l),
exact summable_sum_mul_antidiagonal_of_summable_mul h
end | lemma | summable_sum_mul_range_of_summable_mul | 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_sum_mul_antidiagonal_of_summable_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_mul_tsum_eq_tsum_sum_range (hf : summable f) (hg : summable g)
(hfg : summable (λ (x : ℕ × ℕ), f x.1 * g x.2)) :
(∑' n, f n) * (∑' n, g n) = ∑' n, ∑ k in range (n + 1), f k * g (n - k) | begin
simp_rw ← nat.sum_antidiagonal_eq_sum_range_succ (λ k l, f k * g l),
exact tsum_mul_tsum_eq_tsum_sum_antidiagonal hf hg hfg
end | lemma | tsum_mul_tsum_eq_tsum_sum_range | 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",
"tsum_mul_tsum_eq_tsum_sum_antidiagonal"
] | The **Cauchy product formula** for the product of two infinites sums indexed by `ℕ`, expressed
by summing on `finset.range`.
See also `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm` if `f` and `g` are absolutely summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_smul.of_nhds_zero [topological_ring R] [topological_add_group M]
(hmul : tendsto (λ p : R × M, p.1 • p.2) (𝓝 0 ×ᶠ (𝓝 0)) (𝓝 0))
(hmulleft : ∀ m : M, tendsto (λ a : R, a • m) (𝓝 0) (𝓝 0))
(hmulright : ∀ a : R, tendsto (λ m : M, a • m) (𝓝 0) (𝓝 0)) : has_continuous_smul R M | ⟨begin
rw continuous_iff_continuous_at,
rintros ⟨a₀, m₀⟩,
have key : ∀ p : R × M,
p.1 • p.2 = a₀ • m₀ + ((p.1 - a₀) • m₀ + a₀ • (p.2 - m₀) + (p.1 - a₀) • (p.2 - m₀)),
{ rintro ⟨a, m⟩,
simp [sub_smul, smul_sub],
abel },
rw funext key, clear key,
refine tendsto_const_nhds.add (tendsto.add (tendsto... | lemma | has_continuous_smul.of_nhds_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_iff_continuous_at",
"has_continuous_smul",
"nhds_prod_eq",
"smul_sub",
"smul_zero",
"sub_smul",
"tendsto_const_nhds",
"topological_add_group",
"topological_ring",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.eq_top_of_nonempty_interior'
[ne_bot (𝓝[{x : R | is_unit x}] 0)]
(s : submodule R M) (hs : (interior (s:set M)).nonempty) :
s = ⊤ | begin
rcases hs with ⟨y, hy⟩,
refine (submodule.eq_top_iff'.2 $ λ x, _),
rw [mem_interior_iff_mem_nhds] at hy,
have : tendsto (λ c:R, y + c • x) (𝓝[{x : R | is_unit x}] 0) (𝓝 (y + (0:R) • x)),
from tendsto_const_nhds.add ((tendsto_nhds_within_of_tendsto_nhds tendsto_id).smul
tendsto_const_nhds),
r... | lemma | submodule.eq_top_of_nonempty_interior' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"interior",
"is_unit",
"mem_interior_iff_mem_nhds",
"mem_of_mem_nhds",
"self_mem_nhds_within",
"submodule",
"tendsto_const_nhds",
"tendsto_nhds_within_of_tendsto_nhds",
"zero_smul"
] | If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then
`⊤` is the only submodule of `M` with a nonempty interior.
This is the case, e.g., if `R` is a nontrivially normed field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.punctured_nhds_ne_bot [nontrivial M] [ne_bot (𝓝[≠] (0 : R))]
[no_zero_smul_divisors R M] (x : M) :
ne_bot (𝓝[≠] x) | begin
rcases exists_ne (0 : M) with ⟨y, hy⟩,
suffices : tendsto (λ c : R, x + c • y) (𝓝[≠] 0) (𝓝[≠] x), from this.ne_bot,
refine tendsto.inf _ (tendsto_principal_principal.2 $ _),
{ convert tendsto_const_nhds.add ((@tendsto_id R _).smul_const y),
rw [zero_smul, add_zero] },
{ intros c hc,
simpa [hy]... | lemma | module.punctured_nhds_ne_bot | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"exists_ne",
"no_zero_smul_divisors",
"nontrivial",
"zero_smul"
] | Let `R` be a topological ring such that zero is not an isolated point (e.g., a nontrivially
normed field, see `normed_field.punctured_nhds_ne_bot`). Let `M` be a nontrivial module over `R`
such that `c • x = 0` implies `c = 0 ∨ x = 0`. Then `M` has no isolated points. We formulate this
using `ne_bot (𝓝[≠] x)`.
This l... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_smul_induced :
@has_continuous_smul R M₁ _ u (t.induced f) | { continuous_smul :=
begin
letI : topological_space M₁ := t.induced f,
refine continuous_induced_rng.2 _,
simp_rw [function.comp, f.map_smul],
refine continuous_fst.smul (continuous_induced_dom.comp continuous_snd)
end } | lemma | has_continuous_smul_induced | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_snd",
"has_continuous_smul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.closure_smul_self_subset (s : submodule R M) :
(λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s) ⊆ closure s | calc
(λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s)
= (λ p : R × M, p.1 • p.2) '' closure (set.univ ×ˢ s) :
by simp [closure_prod_eq]
... ⊆ closure ((λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ s)) :
image_closure_subset_closure_image continuous_smul
... = closure s : begin
congr,
ext x,
refine ⟨_, λ hx,... | lemma | submodule.closure_smul_self_subset | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"closure_prod_eq",
"image_closure_subset_closure_image",
"one_smul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.closure_smul_self_eq (s : submodule R M) :
(λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s) = closure s | s.closure_smul_self_subset.antisymm $ λ x hx, ⟨⟨1, x⟩, ⟨set.mem_univ _, hx⟩, one_smul R _⟩ | lemma | submodule.closure_smul_self_eq | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"one_smul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.topological_closure (s : submodule R M) : submodule R M | { carrier := closure (s : set M),
smul_mem' := λ c x hx, s.closure_smul_self_subset ⟨⟨c, x⟩, ⟨set.mem_univ _, hx⟩, rfl⟩,
..s.to_add_submonoid.topological_closure } | def | submodule.topological_closure | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"submodule"
] | The (topological-space) closure of a submodule of a topological `R`-module `M` is itself
a submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.topological_closure_coe (s : submodule R M) :
(s.topological_closure : set M) = closure (s : set M) | rfl | lemma | submodule.topological_closure_coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.le_topological_closure (s : submodule R M) :
s ≤ s.topological_closure | subset_closure | lemma | submodule.le_topological_closure | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.is_closed_topological_closure (s : submodule R M) :
is_closed (s.topological_closure : set M) | by convert is_closed_closure | lemma | submodule.is_closed_topological_closure | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"is_closed_closure",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.topological_closure_minimal
(s : submodule R M) {t : submodule R M} (h : s ≤ t) (ht : is_closed (t : set M)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | submodule.topological_closure_minimal | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure_minimal",
"is_closed",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.topological_closure_mono {s : submodule R M} {t : submodule R M} (h : s ≤ t) :
s.topological_closure ≤ t.topological_closure | s.topological_closure_minimal (h.trans t.le_topological_closure)
t.is_closed_topological_closure | lemma | submodule.topological_closure_mono | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.submodule_topological_closure_eq {s : submodule R M} (hs : is_closed (s : set M)) :
s.topological_closure = s | le_antisymm (s.topological_closure_minimal rfl.le hs) s.le_topological_closure | lemma | is_closed.submodule_topological_closure_eq | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"submodule"
] | The topological closure of a closed submodule `s` is equal to `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.dense_iff_topological_closure_eq_top {s : submodule R M} :
dense (s : set M) ↔ s.topological_closure = ⊤ | by { rw [←set_like.coe_set_eq, dense_iff_closure_eq], simp } | lemma | submodule.dense_iff_topological_closure_eq_top | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"dense",
"dense_iff_closure_eq",
"submodule"
] | A subspace is dense iff its topological closure is the entire space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.is_closed_or_dense_of_is_coatom (s : submodule R M) (hs : is_coatom s) :
is_closed (s : set M) ∨ dense (s : set M) | (hs.le_iff.mp s.le_topological_closure).swap.imp (is_closed_of_closure_subset ∘ eq.le)
submodule.dense_iff_topological_closure_eq_top.mpr | lemma | submodule.is_closed_or_dense_of_is_coatom | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"dense",
"is_closed",
"is_closed_of_closure_subset",
"is_coatom",
"submodule"
] | A maximal proper subspace of a topological module (i.e a `submodule` satisfying `is_coatom`)
is either closed or dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [finite ι] [semiring R]
[topological_space R] [add_comm_monoid M] [module R M] [topological_space M]
[has_continuous_add M] [has_continuous_smul R M] (f : (ι → R) →ₗ[R] M) :
continuous f | begin
casesI nonempty_fintype ι,
classical,
-- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous
-- function.
have : (f : (ι → R) → M) =
(λx, ∑ i : ι, x i • (f (λ j, if i = j then 1 else 0))),
by { ext x, exact f.pi_apply_eq_sum_univ x },
rw this,
re... | lemma | linear_map.continuous_on_pi | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous",
"continuous_apply",
"continuous_const",
"finite",
"has_continuous_add",
"has_continuous_smul",
"module",
"nonempty_fintype",
"semiring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map
{R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S)
(M : Type*) [topological_space M] [add_comm_monoid M]
(M₂ : Type*) [topological_space M₂] [add_comm_monoid M₂]
[module R M] [module S M₂]
extends M →ₛₗ[σ] M₂ | (cont : continuous to_fun . tactic.interactive.continuity') | structure | continuous_linear_map | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"cont",
"continuous",
"module",
"semiring",
"tactic.interactive.continuity'",
"topological_space"
] | Continuous linear maps between modules. We only put the type classes that are necessary for the
definition, although in applications `M` and `M₂` will be topological modules over the topological
ring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_semilinear_map_class (F : Type*) {R S : out_param Type*} [semiring R] [semiring S]
(σ : out_param $ R →+* S) (M : out_param Type*) [topological_space M] [add_comm_monoid M]
(M₂ : out_param Type*) [topological_space M₂] [add_comm_monoid M₂] [module R M] [module S M₂]
extends semilinear_map_class F σ M M... | class | continuous_semilinear_map_class | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous_map_class",
"module",
"semilinear_map_class",
"semiring",
"topological_space"
] | `continuous_semilinear_map_class F σ M M₂` asserts `F` is a type of bundled continuous
`σ`-semilinear maps `M → M₂`. See also `continuous_linear_map_class F R M M₂` for the case where
`σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring
homomorphism `σ : R →+* S` is semilinear... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map_class (F : Type*)
(R : out_param Type*) [semiring R]
(M : out_param Type*) [topological_space M] [add_comm_monoid M]
(M₂ : out_param Type*) [topological_space M₂] [add_comm_monoid M₂]
[module R M] [module R M₂] | continuous_semilinear_map_class F (ring_hom.id R) M M₂ | abbreviation | continuous_linear_map_class | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous_semilinear_map_class",
"module",
"ring_hom.id",
"semiring",
"topological_space"
] | `continuous_linear_map_class F R M M₂` asserts `F` is a type of bundled continuous
`R`-linear maps `M → M₂`. This is an abbreviation for
`continuous_semilinear_map_class F (ring_hom.id R) M M₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv
{R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S)
{σ' : S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ]
(M : Type*) [topological_space M] [add_comm_monoid M]
(M₂ : Type*) [topological_space M₂] [add_comm_monoid M₂]
[module R M] [module S M₂]
extends M ≃ₛₗ[σ] ... | (continuous_to_fun : continuous to_fun . tactic.interactive.continuity')
(continuous_inv_fun : continuous inv_fun . tactic.interactive.continuity') | structure | continuous_linear_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous",
"inv_fun",
"module",
"ring_hom_inv_pair",
"semiring",
"tactic.interactive.continuity'",
"topological_space"
] | Continuous linear equivalences between modules. We only put the type classes that are necessary
for the definition, although in applications `M` and `M₂` will be topological modules over the
topological semiring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_semilinear_equiv_class (F : Type*)
{R : out_param Type*} {S : out_param Type*} [semiring R] [semiring S] (σ : out_param $ R →+* S)
{σ' : out_param $ S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ]
(M : out_param Type*) [topological_space M] [add_comm_monoid M]
(M₂ : out_param Type*) [topol... | (map_continuous : ∀ (f : F), continuous f . tactic.interactive.continuity')
(inv_continuous : ∀ (f : F), continuous (inv f) . tactic.interactive.continuity') | class | continuous_semilinear_equiv_class | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous",
"module",
"ring_hom_inv_pair",
"semilinear_equiv_class",
"semiring",
"tactic.interactive.continuity'",
"topological_space"
] | `continuous_semilinear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous
`σ`-semilinear equivs `M → M₂`. See also `continuous_linear_equiv_class F R M M₂` for the case
where `σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring
homomorphism `σ : R →+* S` is semi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv_class (F : Type*)
(R : out_param Type*) [semiring R]
(M : out_param Type*) [topological_space M] [add_comm_monoid M]
(M₂ : out_param Type*) [topological_space M₂] [add_comm_monoid M₂]
[module R M] [module R M₂] | continuous_semilinear_equiv_class F (ring_hom.id R) M M₂ | abbreviation | continuous_linear_equiv_class | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"continuous_semilinear_equiv_class",
"module",
"ring_hom.id",
"semiring",
"topological_space"
] | `continuous_linear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous
`R`-linear equivs `M → M₂`. This is an abbreviation for
`continuous_semilinear_equiv_class F (ring_hom.id) M M₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_set_of_map_smul : is_closed {f : M₁ → M₂ | ∀ c x, f (c • x) = σ c • f x} | begin
simp only [set.set_of_forall],
exact is_closed_Inter (λ c, is_closed_Inter (λ x, is_closed_eq (continuous_apply _)
((continuous_apply _).const_smul _)))
end | lemma | is_closed_set_of_map_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_apply",
"is_closed",
"is_closed_Inter",
"is_closed_eq",
"set.set_of_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map_of_mem_closure_range_coe (f : M₁ → M₂)
(hf : f ∈ closure (set.range (coe_fn : (M₁ →ₛₗ[σ] M₂) → (M₁ → M₂)))) :
M₁ →ₛₗ[σ] M₂ | { to_fun := f,
map_smul' := (is_closed_set_of_map_smul M₁ M₂ σ).closure_subset_iff.2
(set.range_subset_iff.2 linear_map.map_smulₛₗ) hf,
.. add_monoid_hom_of_mem_closure_range_coe f hf } | def | linear_map_of_mem_closure_range_coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"is_closed_set_of_map_smul",
"linear_map.map_smulₛₗ",
"set.range"
] | Constructs a bundled linear map from a function and a proof that this function belongs to the
closure of the set of linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_of_tendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.ne_bot]
(h : tendsto (λ a x, g a x) l (𝓝 f)) : M₁ →ₛₗ[σ] M₂ | linear_map_of_mem_closure_range_coe f $ mem_closure_of_tendsto h $
eventually_of_forall $ λ a, set.mem_range_self _ | def | linear_map_of_tendsto | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map_of_mem_closure_range_coe",
"mem_closure_of_tendsto",
"set.mem_range_self"
] | Construct a bundled linear map from a pointwise limit of linear maps | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.is_closed_range_coe :
is_closed (set.range (coe_fn : (M₁ →ₛₗ[σ] M₂) → (M₁ → M₂))) | is_closed_of_closure_subset $ λ f hf, ⟨linear_map_of_mem_closure_range_coe f hf, rfl⟩ | lemma | linear_map.is_closed_range_coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"is_closed_of_closure_subset",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_eq_coe (f : M₁ →SL[σ₁₂] M₂) : f.to_linear_map = f | rfl | lemma | continuous_linear_map.to_linear_map_eq_coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : (M₁ →SL[σ₁₂] M₂) → (M₁ →ₛₗ[σ₁₂] M₂)) | by { intros f g H, cases f, cases g, congr' } | theorem | continuous_linear_map.coe_injective | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun : has_coe_to_fun (M₁ →SL[σ₁₂] M₂) (λ _, M₁ → M₂) | ⟨λ f, f.to_fun⟩ | instance | continuous_linear_map.to_fun | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ →ₛₗ[σ₁₂] M₂) = f | rfl | lemma | continuous_linear_map.coe_mk | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk' (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ → M₂) = f | rfl | lemma | continuous_linear_map.coe_mk' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (f : M₁ →SL[σ₁₂] M₂) : continuous f | f.2 | lemma | continuous_linear_map.continuous | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous {E₁ E₂ : Type*} [uniform_space E₁] [uniform_space E₂]
[add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]
[uniform_add_group E₁] [uniform_add_group E₂] (f : E₁ →SL[σ₁₂] E₂) :
uniform_continuous f | uniform_continuous_add_monoid_hom_of_continuous f.continuous | lemma | continuous_linear_map.uniform_continuous | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_group",
"module",
"uniform_add_group",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {f g : M₁ →SL[σ₁₂] M₂} :
(f : M₁ →ₛₗ[σ₁₂] M₂) = g ↔ f = g | coe_injective.eq_iff | lemma | continuous_linear_map.coe_inj | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @function.injective (M₁ →SL[σ₁₂] M₂) (M₁ → M₂) coe_fn | fun_like.coe_injective | theorem | continuous_linear_map.coe_fn_injective | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (h : M₁ →SL[σ₁₂] M₂) : M₁ → M₂ | h | def | continuous_linear_map.simps.apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.coe (h : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂ | h
initialize_simps_projections continuous_linear_map
(to_linear_map_to_fun → apply, to_linear_map → coe) | def | continuous_linear_map.simps.coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : M₁ →SL[σ₁₂] M₂} (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | theorem | continuous_linear_map.ext | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : M₁ →SL[σ₁₂] M₂} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | continuous_linear_map.ext_iff | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : M₁ →SL[σ₁₂] M₂ | { to_linear_map := f.to_linear_map.copy f' h,
cont := show continuous f', from h.symm ▸ f.continuous } | def | continuous_linear_map.copy | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"cont",
"continuous"
] | Copy of a `continuous_linear_map` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' | rfl | lemma | continuous_linear_map.coe_copy | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : f.copy f' h = f | fun_like.ext' h | lemma | continuous_linear_map.copy_eq | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (f : M₁ →SL[σ₁₂] M₂) : f (0 : M₁) = 0 | map_zero f | lemma | continuous_linear_map.map_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : M₁ →SL[σ₁₂] M₂) (x y : M₁) : f (x + y) = f x + f y | map_add f x y | lemma | continuous_linear_map.map_add | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) :
f (c • x) = (σ₁₂ c) • f x | (to_linear_map _).map_smulₛₗ _ _ | lemma | continuous_linear_map.map_smulₛₗ | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul [module R₁ M₂] (f : M₁ →L[R₁] M₂)(c : R₁) (x : M₁) : f (c • x) = c • f x | by simp only [ring_hom.id_apply, continuous_linear_map.map_smulₛₗ] | lemma | continuous_linear_map.map_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.map_smulₛₗ",
"module",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul_of_tower {R S : Type*} [semiring S] [has_smul R M₁]
[module S M₁] [has_smul R M₂] [module S M₂]
[linear_map.compatible_smul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) :
f (c • x) = c • f x | linear_map.compatible_smul.map_smul f c x | lemma | continuous_linear_map.map_smul_of_tower | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_smul",
"linear_map.compatible_smul",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum {ι : Type*} (f : M₁ →SL[σ₁₂] M₂) (s : finset ι) (g : ι → M₁) :
f (∑ i in s, g i) = ∑ i in s, f (g i) | f.to_linear_map.map_sum | lemma | continuous_linear_map.map_sum | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe (f : M₁ →SL[σ₁₂] M₂) : ⇑(f : M₁ →ₛₗ[σ₁₂] M₂) = f | rfl | lemma | continuous_linear_map.coe_coe | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ring [topological_space R₁] {f g : R₁ →L[R₁] M₁} (h : f 1 = g 1) : f = g | coe_inj.1 $ linear_map.ext_ring h | theorem | continuous_linear_map.ext_ring | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.ext_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ring_iff [topological_space R₁] {f g : R₁ →L[R₁] M₁} : f = g ↔ f 1 = g 1 | ⟨λ h, h ▸ rfl, ext_ring⟩ | theorem | continuous_linear_map.ext_ring_iff | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_closure_span [t2_space M₂] {s : set M₁} {f g : M₁ →SL[σ₁₂] M₂} (h : set.eq_on f g s) :
set.eq_on f g (closure (submodule.span R₁ s : set M₁)) | (linear_map.eq_on_span' h).closure f.continuous g.continuous | lemma | continuous_linear_map.eq_on_closure_span | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"closure",
"linear_map.eq_on_span'",
"set.eq_on",
"submodule.span",
"t2_space"
] | If two continuous linear maps are equal on a set `s`, then they are equal on the closure
of the `submodule.span` of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_on [t2_space M₂] {s : set M₁} (hs : dense (submodule.span R₁ s : set M₁))
{f g : M₁ →SL[σ₁₂] M₂} (h : set.eq_on f g s) :
f = g | ext $ λ x, eq_on_closure_span h (hs x) | lemma | continuous_linear_map.ext_on | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"dense",
"set.eq_on",
"submodule.span",
"t2_space"
] | If the submodule generated by a set `s` is dense in the ambient module, then two continuous
linear maps equal on `s` are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.submodule.topological_closure_map [ring_hom_surjective σ₁₂] [topological_space R₁]
[topological_space R₂] [has_continuous_smul R₁ M₁] [has_continuous_add M₁]
[has_continuous_smul R₂ M₂] [has_continuous_add M₂] (f : M₁ →SL[σ₁₂] M₂) (s : submodule R₁ M₁) :
(s.topological_closure.map (f : M₁ →ₛₗ[σ₁₂] M₂))
≤... | image_closure_subset_closure_image f.continuous | lemma | submodule.topological_closure_map | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add",
"has_continuous_smul",
"image_closure_subset_closure_image",
"ring_hom_surjective",
"submodule",
"topological_space"
] | Under a continuous linear map, the image of the `topological_closure` of a submodule is
contained in the `topological_closure` of its image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.dense_range.topological_closure_map_submodule [ring_hom_surjective σ₁₂]
[topological_space R₁] [topological_space R₂] [has_continuous_smul R₁ M₁] [has_continuous_add M₁]
[has_continuous_smul R₂ M₂] [has_continuous_add M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : dense_range f)
{s : submodule R₁ M₁} (hs : s.topological_... | begin
rw set_like.ext'_iff at hs ⊢,
simp only [submodule.topological_closure_coe, submodule.top_coe, ← dense_iff_closure_eq] at hs ⊢,
exact hf'.dense_image f.continuous hs
end | lemma | dense_range.topological_closure_map_submodule | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"dense_iff_closure_eq",
"dense_range",
"has_continuous_add",
"has_continuous_smul",
"ring_hom_surjective",
"set_like.ext'_iff",
"submodule",
"submodule.top_coe",
"submodule.topological_closure_coe",
"topological_space"
] | Under a dense continuous linear map, a submodule whose `topological_closure` is `⊤` is sent to
another such submodule. That is, the image of a dense set under a map with dense range is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_apply (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (c • f) x = c • (f x) | rfl | lemma | continuous_linear_map.smul_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : (↑(c • f) : M₁ →ₛₗ[σ₁₂] M₂) = c • f | rfl | lemma | continuous_linear_map.coe_smul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul' (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : ⇑(c • f) = c • f | rfl | lemma | continuous_linear_map.coe_smul' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
default_def : (default : M₁ →SL[σ₁₂] M₂) = 0 | rfl | lemma | continuous_linear_map.default_def | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : M₁) : (0 : M₁ →SL[σ₁₂] M₂) x = 0 | rfl | lemma | continuous_linear_map.zero_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂) = 0 | rfl | lemma | continuous_linear_map.coe_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero' : ⇑(0 : M₁ →SL[σ₁₂] M₂) = 0 | rfl | lemma | continuous_linear_map.coe_zero' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_of_left [subsingleton M₁] : unique (M₁ →SL[σ₁₂] M₂) | coe_injective.unique | instance | continuous_linear_map.unique_of_left | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_of_right [subsingleton M₂] : unique (M₁ →SL[σ₁₂] M₂) | coe_injective.unique | instance | continuous_linear_map.unique_of_right | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ne_zero {f : M₁ →SL[σ₁₂] M₂} (hf : f ≠ 0) : ∃ x, f x ≠ 0 | by { by_contra' h, exact hf (continuous_linear_map.ext h) } | lemma | continuous_linear_map.exists_ne_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : M₁ →L[R₁] M₁ | ⟨linear_map.id, continuous_id⟩ | def | continuous_linear_map.id | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | the identity map as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_def : (1 : M₁ →L[R₁] M₁) = id R₁ M₁ | rfl | lemma | continuous_linear_map.one_def | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : M₁) : id R₁ M₁ x = x | rfl | lemma | continuous_linear_map.id_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id : (id R₁ M₁ : M₁ →ₗ[R₁] M₁) = linear_map.id | rfl | lemma | continuous_linear_map.coe_id | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id' : ⇑(id R₁ M₁) = _root_.id | rfl | lemma | continuous_linear_map.coe_id' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_id {f : M₁ →L[R₁] M₁} :
(f : M₁ →ₗ[R₁] M₁) = linear_map.id ↔ f = id _ _ | by rw [← coe_id, coe_inj] | lemma | continuous_linear_map.coe_eq_id | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : M₁) : (1 : M₁ →L[R₁] M₁) x = x | rfl | lemma | continuous_linear_map.one_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : M₁ →SL[σ₁₂] M₂) (x : M₁) : (f + g) x = f x + g x | rfl | lemma | continuous_linear_map.add_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (f g : M₁ →SL[σ₁₂] M₂) : (↑(f + g) : M₁ →ₛₗ[σ₁₂] M₂) = f + g | rfl | lemma | continuous_linear_map.coe_add | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add' (f g : M₁ →SL[σ₁₂] M₂) : ⇑(f + g) = f + g | rfl | lemma | continuous_linear_map.coe_add' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sum {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :
↑(∑ d in t, f d) = (∑ d in t, f d : M₁ →ₛₗ[σ₁₂] M₂) | (add_monoid_hom.mk (coe : (M₁ →SL[σ₁₂] M₂) → (M₁ →ₛₗ[σ₁₂] M₂)) rfl (λ _ _, rfl)).map_sum _ _ | lemma | continuous_linear_map.coe_sum | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sum' {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :
⇑(∑ d in t, f d) = ∑ d in t, f d | by simp only [← coe_coe, coe_sum, linear_map.coe_fn_sum] | lemma | continuous_linear_map.coe_sum' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"coe_coe",
"finset",
"linear_map.coe_fn_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_apply {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) (b : M₁) :
(∑ d in t, f d) b = ∑ d in t, f d b | by simp only [coe_sum', finset.sum_apply] | lemma | continuous_linear_map.sum_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : M₁ →SL[σ₁₃] M₃ | ⟨(g : M₂ →ₛₗ[σ₂₃] M₃).comp ↑f, g.2.comp f.2⟩ | def | continuous_linear_map.comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | Composition of bounded linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(h.comp f : M₁ →ₛₗ[σ₁₃] M₃) = (h : M₂ →ₛₗ[σ₂₃] M₃).comp (f : M₁ →ₛₗ[σ₁₂] M₂) | rfl | lemma | continuous_linear_map.coe_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp' (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
⇑(h.comp f) = h ∘ f | rfl | lemma | continuous_linear_map.coe_comp' | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (g.comp f) x = g (f x) | rfl | lemma | continuous_linear_map.comp_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : M₁ →SL[σ₁₂] M₂) : f.comp (id R₁ M₁) = f | ext $ λ x, rfl | theorem | continuous_linear_map.comp_id | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : M₁ →SL[σ₁₂] M₂) : (id R₂ M₂).comp f = f | ext $ λ x, rfl | theorem | continuous_linear_map.id_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁₂] M₂) = 0 | by { ext, simp } | theorem | continuous_linear_map.comp_zero | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (f : M₁ →SL[σ₁₂] M₂) : (0 : M₂ →SL[σ₂₃] M₃).comp f = 0 | by { ext, simp } | theorem | continuous_linear_map.zero_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_add [has_continuous_add M₂] [has_continuous_add M₃]
(g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M₁ →SL[σ₁₂] M₂) :
g.comp (f₁ + f₂) = g.comp f₁ + g.comp f₂ | by { ext, simp } | lemma | continuous_linear_map.comp_add | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp [has_continuous_add M₃]
(g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
(g₁ + g₂).comp f = g₁.comp f + g₂.comp f | by { ext, simp } | lemma | continuous_linear_map.add_comp | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc {R₄ : Type*} [semiring R₄] [module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄}
{σ₃₄ : R₃ →+* R₄} [ring_hom_comp_triple σ₁₃ σ₃₄ σ₁₄] [ring_hom_comp_triple σ₂₃ σ₃₄ σ₂₄]
[ring_hom_comp_triple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃)
(f : M₁ →SL[σ₁₂] M₂) :
(h.comp g).comp f = h.comp (g.comp ... | rfl | theorem | continuous_linear_map.comp_assoc | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"module",
"ring_hom_comp_triple",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (f g : M₁ →L[R₁] M₁) : f * g = f.comp g | rfl | lemma | continuous_linear_map.mul_def | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : M₁ →L[R₁] M₁) : ⇑(f * g) = f ∘ g | rfl | lemma | continuous_linear_map.coe_mul | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : M₁ →L[R₁] M₁) (x : M₁) : (f * g) x = f (g x) | rfl | lemma | continuous_linear_map.mul_apply | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_ring_hom [has_continuous_add M₁] : (M₁ →L[R₁] M₁) →+* (M₁ →ₗ[R₁] M₁) | { to_fun := to_linear_map,
map_zero' := rfl,
map_one' := rfl,
map_add' := λ _ _, rfl,
map_mul' := λ _ _, rfl } | def | continuous_linear_map.to_linear_map_ring_hom | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"has_continuous_add"
] | `continuous_linear_map.to_linear_map` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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