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 |
|---|---|---|---|---|---|---|---|---|---|---|
finite_norm_eq (x : e.finite_subspace) : ‖x‖ = (e x).to_real | rfl | lemma | enorm.finite_norm_eq | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series : formal_multilinear_series 𝕂 𝔸 𝔸 | λ n, (n!⁻¹ : 𝕂) • continuous_multilinear_map.mk_pi_algebra_fin 𝕂 n 𝔸 | def | exp_series | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"continuous_multilinear_map.mk_pi_algebra_fin",
"formal_multilinear_series"
] | `exp_series 𝕂 𝔸` is the `formal_multilinear_series` whose `n`-th term is the map
`(xᵢ) : 𝔸ⁿ ↦ (1/n! : 𝕂) • ∏ xᵢ`. Its sum is the exponential map `exp 𝕂 : 𝔸 → 𝔸`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp (x : 𝔸) : 𝔸 | (exp_series 𝕂 𝔸).sum x | def | exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series"
] | `exp 𝕂 : 𝔸 → 𝔸` is the exponential map determined by the action of `𝕂` on `𝔸`.
It is defined as the sum of the `formal_multilinear_series` `exp_series 𝕂 𝔸`.
Note that when `𝔸 = matrix n n 𝕂`, this is the **Matrix Exponential**; see
[`analysis.normed_space.matrix_exponential`](../matrix_exponential) for lemmas... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_series_apply_eq (x : 𝔸) (n : ℕ) : exp_series 𝕂 𝔸 n (λ _, x) = (n!⁻¹ : 𝕂) • x^n | by simp [exp_series] | lemma | exp_series_apply_eq | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_apply_eq' (x : 𝔸) :
(λ n, exp_series 𝕂 𝔸 n (λ _, x)) = (λ n, (n!⁻¹ : 𝕂) • x^n) | funext (exp_series_apply_eq x) | lemma | exp_series_apply_eq' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_sum_eq (x : 𝔸) : (exp_series 𝕂 𝔸).sum x = ∑' (n : ℕ), (n!⁻¹ : 𝕂) • x^n | tsum_congr (λ n, exp_series_apply_eq x n) | lemma | exp_series_sum_eq | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_eq_tsum : exp 𝕂 = (λ x : 𝔸, ∑' (n : ℕ), (n!⁻¹ : 𝕂) • x^n) | funext exp_series_sum_eq | lemma | exp_eq_tsum | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series_sum_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_apply_zero (n : ℕ) : exp_series 𝕂 𝔸 n (λ _, (0 : 𝔸)) = pi.single 0 1 n | begin
rw exp_series_apply_eq,
cases n,
{ rw [pow_zero, nat.factorial_zero, nat.cast_one, inv_one, one_smul, pi.single_eq_same], },
{ rw [zero_pow (nat.succ_pos _), smul_zero, pi.single_eq_of_ne (n.succ_ne_zero)], },
end | lemma | exp_series_apply_zero | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq",
"inv_one",
"nat.cast_one",
"nat.factorial_zero",
"one_smul",
"pow_zero",
"smul_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_zero [t2_space 𝔸] : exp 𝕂 (0 : 𝔸) = 1 | by simp_rw [exp_eq_tsum, ←exp_series_apply_eq, exp_series_apply_zero, tsum_pi_single] | lemma | exp_zero | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_eq_tsum",
"exp_series_apply_zero",
"t2_space",
"tsum_pi_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_op [t2_space 𝔸] (x : 𝔸) :
exp 𝕂 (mul_opposite.op x) = mul_opposite.op (exp 𝕂 x) | by simp_rw [exp, exp_series_sum_eq, ←mul_opposite.op_pow, ←mul_opposite.op_smul, tsum_op] | lemma | exp_op | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series_sum_eq",
"mul_opposite.op",
"t2_space",
"tsum_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_unop [t2_space 𝔸] (x : 𝔸ᵐᵒᵖ) :
exp 𝕂 (mul_opposite.unop x) = mul_opposite.unop (exp 𝕂 x) | by simp_rw [exp, exp_series_sum_eq, ←mul_opposite.unop_pow, ←mul_opposite.unop_smul, tsum_unop] | lemma | exp_unop | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series_sum_eq",
"mul_opposite.unop",
"t2_space",
"tsum_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_exp [t2_space 𝔸] [star_ring 𝔸] [has_continuous_star 𝔸] (x : 𝔸) :
star (exp 𝕂 x) = exp 𝕂 (star x) | by simp_rw [exp_eq_tsum, ←star_pow, ←star_inv_nat_cast_smul, ←tsum_star] | lemma | star_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_eq_tsum",
"has_continuous_star",
"star_ring",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint.exp [t2_space 𝔸] [star_ring 𝔸] [has_continuous_star 𝔸] {x : 𝔸}
(h : is_self_adjoint x) :
is_self_adjoint (exp 𝕂 x) | (star_exp x).trans $ h.symm ▸ rfl | lemma | is_self_adjoint.exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"has_continuous_star",
"is_self_adjoint",
"star_exp",
"star_ring",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.exp_right [t2_space 𝔸] {x y : 𝔸} (h : commute x y) : commute x (exp 𝕂 y) | begin
rw exp_eq_tsum,
exact commute.tsum_right x (λ n, (h.pow_right n).smul_right _),
end | lemma | commute.exp_right | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute",
"commute.tsum_right",
"exp",
"exp_eq_tsum",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.exp_left [t2_space 𝔸] {x y : 𝔸} (h : commute x y) : commute (exp 𝕂 x) y | (h.symm.exp_right 𝕂).symm | lemma | commute.exp_left | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute",
"exp",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.exp [t2_space 𝔸] {x y : 𝔸} (h : commute x y) : commute (exp 𝕂 x) (exp 𝕂 y) | (h.exp_left _).exp_right _ | lemma | commute.exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute",
"exp",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_apply_eq_div (x : 𝔸) (n : ℕ) : exp_series 𝕂 𝔸 n (λ _, x) = x^n / n! | by rw [div_eq_mul_inv, ←(nat.cast_commute n! (x ^ n)).inv_left₀.eq, ←smul_eq_mul,
exp_series_apply_eq, inv_nat_cast_smul_eq _ _ _ _] | lemma | exp_series_apply_eq_div | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"exp_series",
"exp_series_apply_eq",
"inv_nat_cast_smul_eq",
"nat.cast_commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_apply_eq_div' (x : 𝔸) : (λ n, exp_series 𝕂 𝔸 n (λ _, x)) = (λ n, x^n / n!) | funext (exp_series_apply_eq_div x) | lemma | exp_series_apply_eq_div' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_sum_eq_div (x : 𝔸) : (exp_series 𝕂 𝔸).sum x = ∑' (n : ℕ), x^n / n! | tsum_congr (exp_series_apply_eq_div x) | lemma | exp_series_sum_eq_div | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq_div",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_eq_tsum_div : exp 𝕂 = (λ x : 𝔸, ∑' (n : ℕ), x^n / n!) | funext exp_series_sum_eq_div | lemma | exp_eq_tsum_div | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series_sum_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_summable_of_mem_ball (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
summable (λ n, ‖exp_series 𝕂 𝔸 n (λ _, x)‖) | (exp_series 𝕂 𝔸).summable_norm_apply hx | lemma | norm_exp_series_summable_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp_series",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_summable_of_mem_ball' (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
summable (λ n, ‖(n!⁻¹ : 𝕂) • x^n‖) | begin
change summable (norm ∘ _),
rw ← exp_series_apply_eq',
exact norm_exp_series_summable_of_mem_ball x hx
end | lemma | norm_exp_series_summable_of_mem_ball' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp_series",
"exp_series_apply_eq'",
"norm_exp_series_summable_of_mem_ball",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_summable_of_mem_ball (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
summable (λ n, exp_series 𝕂 𝔸 n (λ _, x)) | summable_of_summable_norm (norm_exp_series_summable_of_mem_ball x hx) | lemma | exp_series_summable_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp_series",
"norm_exp_series_summable_of_mem_ball",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_summable_of_mem_ball' (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
summable (λ n, (n!⁻¹ : 𝕂) • x^n) | summable_of_summable_norm (norm_exp_series_summable_of_mem_ball' x hx) | lemma | exp_series_summable_of_mem_ball' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp_series",
"norm_exp_series_summable_of_mem_ball'",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_has_sum_exp_of_mem_ball (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_sum (λ n, exp_series 𝕂 𝔸 n (λ _, x)) (exp 𝕂 x) | formal_multilinear_series.has_sum (exp_series 𝕂 𝔸) hx | lemma | exp_series_has_sum_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp",
"exp_series",
"formal_multilinear_series.has_sum",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_has_sum_exp_of_mem_ball' (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
has_sum (λ n, (n!⁻¹ : 𝕂) • x^n) (exp 𝕂 x) | begin
rw ← exp_series_apply_eq',
exact exp_series_has_sum_exp_of_mem_ball x hx
end | lemma | exp_series_has_sum_exp_of_mem_ball' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp",
"exp_series",
"exp_series_apply_eq'",
"exp_series_has_sum_exp_of_mem_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball_exp_of_radius_pos (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_fpower_series_on_ball (exp 𝕂) (exp_series 𝕂 𝔸) 0 (exp_series 𝕂 𝔸).radius | (exp_series 𝕂 𝔸).has_fpower_series_on_ball h | lemma | has_fpower_series_on_ball_exp_of_radius_pos | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at_exp_zero_of_radius_pos (h : 0 < (exp_series 𝕂 𝔸).radius) :
has_fpower_series_at (exp 𝕂) (exp_series 𝕂 𝔸) 0 | (has_fpower_series_on_ball_exp_of_radius_pos h).has_fpower_series_at | lemma | has_fpower_series_at_exp_zero_of_radius_pos | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series",
"has_fpower_series_at",
"has_fpower_series_on_ball_exp_of_radius_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_exp :
continuous_on (exp 𝕂 : 𝔸 → 𝔸) (emetric.ball 0 (exp_series 𝕂 𝔸).radius) | formal_multilinear_series.continuous_on | lemma | continuous_on_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"continuous_on",
"emetric.ball",
"exp",
"exp_series",
"formal_multilinear_series.continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at_exp_of_mem_ball (x : 𝔸) (hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
analytic_at 𝕂 (exp 𝕂) x | begin
by_cases h : (exp_series 𝕂 𝔸).radius = 0,
{ rw h at hx, exact (ennreal.not_lt_zero hx).elim },
{ have h := pos_iff_ne_zero.mpr h,
exact (has_fpower_series_on_ball_exp_of_radius_pos h).analytic_at_of_mem hx }
end | lemma | analytic_at_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"analytic_at",
"emetric.ball",
"ennreal.not_lt_zero",
"exp",
"exp_series",
"has_fpower_series_on_ball_exp_of_radius_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add_of_commute_of_mem_ball [char_zero 𝕂]
{x y : 𝔸} (hxy : commute x y) (hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius)
(hy : y ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
exp 𝕂 (x + y) = (exp 𝕂 x) * (exp 𝕂 y) | begin
rw [exp_eq_tsum, tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
(norm_exp_series_summable_of_mem_ball' x hx) (norm_exp_series_summable_of_mem_ball' y hy)],
dsimp only,
conv_lhs {congr, funext, rw [hxy.add_pow' _, finset.smul_sum]},
refine tsum_congr (λ n, finset.sum_congr rfl $ λ kl hkl, ... | lemma | exp_add_of_commute_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"commute",
"emetric.ball",
"exp",
"exp_eq_tsum",
"exp_series",
"finset.smul_sum",
"nat.cast_add_choose",
"norm_exp_series_summable_of_mem_ball'",
"nsmul_eq_smul_cast",
"smul_mul_smul",
"smul_smul",
"tsum_congr",
"tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm"
] | In a Banach-algebra `𝔸` over a normed field `𝕂` of characteristic zero, if `x` and `y` are
in the disk of convergence and commute, then `exp 𝕂 (x + y) = (exp 𝕂 x) * (exp 𝕂 y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_exp_of_mem_ball [char_zero 𝕂] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) : invertible (exp 𝕂 x) | { inv_of := exp 𝕂 (-x),
inv_of_mul_self := begin
have hnx : -x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius,
{ rw [emetric.mem_ball, ←neg_zero, edist_neg_neg],
exact hx },
rw [←exp_add_of_commute_of_mem_ball (commute.neg_left $ commute.refl x) hnx hx, neg_add_self,
exp_zero],
end,
mu... | def | invertible_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"commute.neg_left",
"commute.neg_right",
"commute.refl",
"emetric.ball",
"emetric.mem_ball",
"exp",
"exp_series",
"exp_zero",
"inv_of_mul_self",
"invertible",
"mul_inv_of_self"
] | `exp 𝕂 x` has explicit two-sided inverse `exp 𝕂 (-x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_exp_of_mem_ball [char_zero 𝕂] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) : is_unit (exp 𝕂 x) | @is_unit_of_invertible _ _ _ (invertible_exp_of_mem_ball hx) | lemma | is_unit_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"emetric.ball",
"exp",
"exp_series",
"invertible_exp_of_mem_ball",
"is_unit",
"is_unit_of_invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_exp_of_mem_ball [char_zero 𝕂] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) [invertible (exp 𝕂 x)] :
⅟(exp 𝕂 x) = exp 𝕂 (-x) | by { letI := invertible_exp_of_mem_ball hx, convert (rfl : ⅟(exp 𝕂 x) = _) } | lemma | inv_of_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"emetric.ball",
"exp",
"exp_series",
"invertible",
"invertible_exp_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp_of_mem_ball {F} [ring_hom_class F 𝔸 𝔹] (f : F) (hf : continuous f) (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
f (exp 𝕂 x) = exp 𝕂 (f x) | begin
rw [exp_eq_tsum, exp_eq_tsum],
refine ((exp_series_summable_of_mem_ball' _ hx).has_sum.map f hf).tsum_eq.symm.trans _,
dsimp only [function.comp],
simp_rw [one_div, map_inv_nat_cast_smul f 𝕂 𝕂, map_pow],
end | lemma | map_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"continuous",
"emetric.ball",
"exp",
"exp_eq_tsum",
"exp_series",
"exp_series_summable_of_mem_ball'",
"has_sum.map",
"map_inv_nat_cast_smul",
"map_pow",
"one_div",
"ring_hom_class"
] | Any continuous ring homomorphism commutes with `exp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_exp_comm_of_mem_ball [complete_space 𝕂] (x : 𝕂)
(hx : x ∈ emetric.ball (0 : 𝕂) (exp_series 𝕂 𝕂).radius) :
algebra_map 𝕂 𝔸 (exp 𝕂 x) = exp 𝕂 (algebra_map 𝕂 𝔸 x) | map_exp_of_mem_ball _ (continuous_algebra_map 𝕂 𝔸) _ hx | lemma | algebra_map_exp_comm_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"algebra_map",
"complete_space",
"continuous_algebra_map",
"emetric.ball",
"exp",
"exp_series",
"map_exp_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_div_summable_of_mem_ball (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
summable (λ n, ‖x^n / n!‖) | begin
change summable (norm ∘ _),
rw ← exp_series_apply_eq_div' x,
exact norm_exp_series_summable_of_mem_ball x hx
end | lemma | norm_exp_series_div_summable_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"emetric.ball",
"exp_series",
"exp_series_apply_eq_div'",
"norm_exp_series_summable_of_mem_ball",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_div_summable_of_mem_ball [complete_space 𝔸] (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) : summable (λ n, x^n / n!) | summable_of_summable_norm (norm_exp_series_div_summable_of_mem_ball 𝕂 x hx) | lemma | exp_series_div_summable_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"emetric.ball",
"exp_series",
"norm_exp_series_div_summable_of_mem_ball",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_div_has_sum_exp_of_mem_ball [complete_space 𝔸] (x : 𝔸)
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) : has_sum (λ n, x^n / n!) (exp 𝕂 x) | begin
rw ← exp_series_apply_eq_div' x,
exact exp_series_has_sum_exp_of_mem_ball x hx
end | lemma | exp_series_div_has_sum_exp_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"emetric.ball",
"exp",
"exp_series",
"exp_series_apply_eq_div'",
"exp_series_has_sum_exp_of_mem_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_neg_of_mem_ball [char_zero 𝕂] [complete_space 𝔸] {x : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
exp 𝕂 (-x) = (exp 𝕂 x)⁻¹ | begin
letI := invertible_exp_of_mem_ball hx,
exact inv_of_eq_inv (exp 𝕂 x),
end | lemma | exp_neg_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"complete_space",
"emetric.ball",
"exp",
"exp_series",
"inv_of_eq_inv",
"invertible_exp_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add_of_mem_ball [char_zero 𝕂] {x y : 𝔸}
(hx : x ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius)
(hy : y ∈ emetric.ball (0 : 𝔸) (exp_series 𝕂 𝔸).radius) :
exp 𝕂 (x + y) = (exp 𝕂 x) * (exp 𝕂 y) | exp_add_of_commute_of_mem_ball (commute.all x y) hx hy | lemma | exp_add_of_mem_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"char_zero",
"commute.all",
"emetric.ball",
"exp",
"exp_add_of_commute_of_mem_ball",
"exp_series"
] | In a commutative Banach-algebra `𝔸` over a normed field `𝕂` of characteristic zero,
`exp 𝕂 (x+y) = (exp 𝕂 x) * (exp 𝕂 y)` for all `x`, `y` in the disk of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_series_radius_eq_top : (exp_series 𝕂 𝔸).radius = ∞ | begin
refine (exp_series 𝕂 𝔸).radius_eq_top_of_summable_norm (λ r, _),
refine summable_of_norm_bounded_eventually _ (real.summable_pow_div_factorial r) _,
filter_upwards [eventually_cofinite_ne 0] with n hn,
rw [norm_mul, norm_norm (exp_series 𝕂 𝔸 n), exp_series, norm_smul, norm_inv, norm_pow,
nnreal.... | lemma | exp_series_radius_eq_top | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"div_nonneg",
"exp_series",
"mul_comm",
"mul_le_of_le_one_right",
"nnreal.norm_eq",
"norm_inv",
"norm_mul",
"norm_norm",
"norm_pow",
"norm_smul",
"pow_nonneg",
"real.summable_pow_div_factorial",
"summable_of_norm_bounded_eventually"
] | In a normed algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ`, the series defining the exponential map
has an infinite radius of convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_series_radius_pos : 0 < (exp_series 𝕂 𝔸).radius | begin
rw exp_series_radius_eq_top,
exact with_top.zero_lt_top
end | lemma | exp_series_radius_pos | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_radius_eq_top",
"with_top.zero_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_summable (x : 𝔸) : summable (λ n, ‖exp_series 𝕂 𝔸 n (λ _, x)‖) | norm_exp_series_summable_of_mem_ball x ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | norm_exp_series_summable | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp_series_radius_eq_top",
"norm_exp_series_summable_of_mem_ball",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_summable' (x : 𝔸) : summable (λ n, ‖(n!⁻¹ : 𝕂) • x^n‖) | norm_exp_series_summable_of_mem_ball' x ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | norm_exp_series_summable' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp_series_radius_eq_top",
"norm_exp_series_summable_of_mem_ball'",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_summable (x : 𝔸) : summable (λ n, exp_series 𝕂 𝔸 n (λ _, x)) | summable_of_summable_norm (norm_exp_series_summable x) | lemma | exp_series_summable | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"norm_exp_series_summable",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_summable' (x : 𝔸) : summable (λ n, (n!⁻¹ : 𝕂) • x^n) | summable_of_summable_norm (norm_exp_series_summable' x) | lemma | exp_series_summable' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"norm_exp_series_summable'",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_has_sum_exp (x : 𝔸) : has_sum (λ n, exp_series 𝕂 𝔸 n (λ _, x)) (exp 𝕂 x) | exp_series_has_sum_exp_of_mem_ball x ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_series_has_sum_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series",
"exp_series_has_sum_exp_of_mem_ball",
"exp_series_radius_eq_top",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_has_sum_exp' (x : 𝔸) : has_sum (λ n, (n!⁻¹ : 𝕂) • x^n) (exp 𝕂 x) | exp_series_has_sum_exp_of_mem_ball' x ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_series_has_sum_exp' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series_has_sum_exp_of_mem_ball'",
"exp_series_radius_eq_top",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_has_fpower_series_on_ball :
has_fpower_series_on_ball (exp 𝕂) (exp_series 𝕂 𝔸) 0 ∞ | exp_series_radius_eq_top 𝕂 𝔸 ▸
has_fpower_series_on_ball_exp_of_radius_pos (exp_series_radius_pos _ _) | lemma | exp_has_fpower_series_on_ball | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series",
"exp_series_radius_eq_top",
"exp_series_radius_pos",
"has_fpower_series_on_ball",
"has_fpower_series_on_ball_exp_of_radius_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_has_fpower_series_at_zero :
has_fpower_series_at (exp 𝕂) (exp_series 𝕂 𝔸) 0 | exp_has_fpower_series_on_ball.has_fpower_series_at | lemma | exp_has_fpower_series_at_zero | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_continuous : continuous (exp 𝕂 : 𝔸 → 𝔸) | begin
rw [continuous_iff_continuous_on_univ, ← metric.eball_top_eq_univ (0 : 𝔸),
← exp_series_radius_eq_top 𝕂 𝔸],
exact continuous_on_exp
end | lemma | exp_continuous | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_exp",
"exp",
"exp_series_radius_eq_top",
"metric.eball_top_eq_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_analytic (x : 𝔸) :
analytic_at 𝕂 (exp 𝕂) x | analytic_at_exp_of_mem_ball x ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_analytic | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"analytic_at",
"analytic_at_exp_of_mem_ball",
"edist_lt_top",
"exp",
"exp_series_radius_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add_of_commute
{x y : 𝔸} (hxy : commute x y) :
exp 𝕂 (x + y) = (exp 𝕂 x) * (exp 𝕂 y) | exp_add_of_commute_of_mem_ball hxy ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _)
((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_add_of_commute | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute",
"edist_lt_top",
"exp",
"exp_add_of_commute_of_mem_ball",
"exp_series_radius_eq_top"
] | In a Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ`, if `x` and `y` commute, then
`exp 𝕂 (x+y) = (exp 𝕂 x) * (exp 𝕂 y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_exp (x : 𝔸) : invertible (exp 𝕂 x) | invertible_exp_of_mem_ball $ (exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | def | invertible_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series_radius_eq_top",
"invertible",
"invertible_exp_of_mem_ball"
] | `exp 𝕂 x` has explicit two-sided inverse `exp 𝕂 (-x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_exp (x : 𝔸) : is_unit (exp 𝕂 x) | is_unit_exp_of_mem_ball $ (exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | is_unit_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series_radius_eq_top",
"is_unit",
"is_unit_exp_of_mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_exp (x : 𝔸) [invertible (exp 𝕂 x)] :
⅟(exp 𝕂 x) = exp 𝕂 (-x) | inv_of_exp_of_mem_ball $ (exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | inv_of_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series_radius_eq_top",
"inv_of_exp_of_mem_ball",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.inverse_exp (x : 𝔸) : ring.inverse (exp 𝕂 x) = exp 𝕂 (-x) | begin
letI := invertible_exp 𝕂 x,
exact ring.inverse_invertible _,
end | lemma | ring.inverse_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"invertible_exp",
"ring.inverse",
"ring.inverse_invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_mem_unitary_of_mem_skew_adjoint [star_ring 𝔸] [has_continuous_star 𝔸] {x : 𝔸}
(h : x ∈ skew_adjoint 𝔸) :
exp 𝕂 x ∈ unitary 𝔸 | by rw [unitary.mem_iff, star_exp, skew_adjoint.mem_iff.mp h,
←exp_add_of_commute (commute.refl x).neg_left, ←exp_add_of_commute (commute.refl x).neg_right,
add_left_neg, add_right_neg, exp_zero, and_self] | lemma | exp_mem_unitary_of_mem_skew_adjoint | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute.refl",
"exp",
"exp_zero",
"has_continuous_star",
"skew_adjoint",
"star_exp",
"star_ring",
"unitary",
"unitary.mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_sum_of_commute {ι} (s : finset ι) (f : ι → 𝔸)
(h : (s : set ι).pairwise $ λ i j, commute (f i) (f j)) :
exp 𝕂 (∑ i in s, f i) = s.noncomm_prod (λ i, exp 𝕂 (f i))
(λ i hi j hj _, (h.of_refl hi hj).exp 𝕂) | begin
classical,
induction s using finset.induction_on with a s ha ih,
{ simp },
rw [finset.noncomm_prod_insert_of_not_mem _ _ _ _ ha, finset.sum_insert ha,
exp_add_of_commute, ih (h.mono $ finset.subset_insert _ _)],
refine commute.sum_right _ _ _ (λ i hi, _),
exact h.of_refl (finset.mem_insert_self ... | lemma | exp_sum_of_commute | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute",
"commute.sum_right",
"exp",
"exp_add_of_commute",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.noncomm_prod_insert_of_not_mem",
"finset.subset_insert",
"ih",
"pairwise"
] | In a Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ`, if a family of elements `f i` mutually
commute then `exp 𝕂 (∑ i, f i) = ∏ i, exp 𝕂 (f i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_nsmul (n : ℕ) (x : 𝔸) :
exp 𝕂 (n • x) = exp 𝕂 x ^ n | begin
induction n with n ih,
{ rw [zero_smul, pow_zero, exp_zero], },
{ rw [succ_nsmul, pow_succ, exp_add_of_commute ((commute.refl x).smul_right n), ih] }
end | lemma | exp_nsmul | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute.refl",
"exp",
"exp_add_of_commute",
"exp_zero",
"ih",
"pow_succ",
"pow_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp {F} [ring_hom_class F 𝔸 𝔹] (f : F) (hf : continuous f) (x : 𝔸) :
f (exp 𝕂 x) = exp 𝕂 (f x) | map_exp_of_mem_ball f hf x $ (exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | map_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"continuous",
"edist_lt_top",
"exp",
"exp_series_radius_eq_top",
"map_exp_of_mem_ball",
"ring_hom_class"
] | Any continuous ring homomorphism commutes with `exp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_smul {G} [monoid G] [mul_semiring_action G 𝔸] [has_continuous_const_smul G 𝔸]
(g : G) (x : 𝔸) :
exp 𝕂 (g • x) = g • exp 𝕂 x | (map_exp 𝕂 (mul_semiring_action.to_ring_hom G 𝔸 g) (continuous_const_smul _) x).symm | lemma | exp_smul | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"has_continuous_const_smul",
"map_exp",
"monoid",
"mul_semiring_action",
"mul_semiring_action.to_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_units_conj (y : 𝔸ˣ) (x : 𝔸) :
exp 𝕂 (y * x * ↑(y⁻¹) : 𝔸) = y * exp 𝕂 x * ↑(y⁻¹) | exp_smul _ (conj_act.to_conj_act y) x | lemma | exp_units_conj | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"conj_act.to_conj_act",
"exp",
"exp_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_units_conj' (y : 𝔸ˣ) (x : 𝔸) :
exp 𝕂 (↑(y⁻¹) * x * y) = ↑(y⁻¹) * exp 𝕂 x * y | exp_units_conj _ _ _ | lemma | exp_units_conj' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_units_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.fst_exp [complete_space 𝔹] (x : 𝔸 × 𝔹) : (exp 𝕂 x).fst = exp 𝕂 x.fst | map_exp _ (ring_hom.fst 𝔸 𝔹) continuous_fst x | lemma | prod.fst_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"continuous_fst",
"exp",
"map_exp",
"ring_hom.fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.snd_exp [complete_space 𝔹] (x : 𝔸 × 𝔹) : (exp 𝕂 x).snd = exp 𝕂 x.snd | map_exp _ (ring_hom.snd 𝔸 𝔹) continuous_snd x | lemma | prod.snd_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"continuous_snd",
"exp",
"map_exp",
"ring_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.exp_apply {ι : Type*} {𝔸 : ι → Type*} [fintype ι]
[Π i, normed_ring (𝔸 i)] [Π i, normed_algebra 𝕂 (𝔸 i)] [Π i, complete_space (𝔸 i)]
(x : Π i, 𝔸 i) (i : ι) :
exp 𝕂 x i = exp 𝕂 (x i) | begin
-- Lean struggles to infer this instance due to it wanting `[Π i, semi_normed_ring (𝔸 i)]`
letI : normed_algebra 𝕂 (Π i, 𝔸 i) := pi.normed_algebra _,
exact map_exp _ (pi.eval_ring_hom 𝔸 i) (continuous_apply _) x
end | lemma | pi.exp_apply | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"continuous_apply",
"exp",
"fintype",
"map_exp",
"normed_algebra",
"normed_ring",
"pi.eval_ring_hom",
"pi.normed_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.exp_def {ι : Type*} {𝔸 : ι → Type*} [fintype ι]
[Π i, normed_ring (𝔸 i)] [Π i, normed_algebra 𝕂 (𝔸 i)] [Π i, complete_space (𝔸 i)]
(x : Π i, 𝔸 i) :
exp 𝕂 x = λ i, exp 𝕂 (x i) | funext $ pi.exp_apply 𝕂 x | lemma | pi.exp_def | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"exp",
"fintype",
"normed_algebra",
"normed_ring",
"pi.exp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.update_exp {ι : Type*} {𝔸 : ι → Type*} [fintype ι] [decidable_eq ι]
[Π i, normed_ring (𝔸 i)] [Π i, normed_algebra 𝕂 (𝔸 i)] [Π i, complete_space (𝔸 i)]
(x : Π i, 𝔸 i) (j : ι) (xj : 𝔸 j) :
function.update (exp 𝕂 x) j (exp 𝕂 xj) = exp 𝕂 (function.update x j xj) | begin
ext i,
simp_rw [pi.exp_def],
exact (function.apply_update (λ i, exp 𝕂) x j xj i).symm,
end | lemma | function.update_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"complete_space",
"exp",
"fintype",
"normed_algebra",
"normed_ring",
"pi.exp_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_exp_comm (x : 𝕂) :
algebra_map 𝕂 𝔸 (exp 𝕂 x) = exp 𝕂 (algebra_map 𝕂 𝔸 x) | algebra_map_exp_comm_of_mem_ball x $
(exp_series_radius_eq_top 𝕂 𝕂).symm ▸ edist_lt_top _ _ | lemma | algebra_map_exp_comm | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"algebra_map",
"algebra_map_exp_comm_of_mem_ball",
"edist_lt_top",
"exp",
"exp_series_radius_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_series_div_summable (x : 𝔸) : summable (λ n, ‖x^n / n!‖) | norm_exp_series_div_summable_of_mem_ball 𝕂 x
((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | norm_exp_series_div_summable | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp_series_radius_eq_top",
"norm_exp_series_div_summable_of_mem_ball",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_div_summable (x : 𝔸) : summable (λ n, x^n / n!) | summable_of_summable_norm (norm_exp_series_div_summable 𝕂 x) | lemma | exp_series_div_summable | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"norm_exp_series_div_summable",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_series_div_has_sum_exp (x : 𝔸) : has_sum (λ n, x^n / n!) (exp 𝕂 x) | exp_series_div_has_sum_exp_of_mem_ball 𝕂 x
((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_series_div_has_sum_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_series_div_has_sum_exp_of_mem_ball",
"exp_series_radius_eq_top",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_neg (x : 𝔸) : exp 𝕂 (-x) = (exp 𝕂 x)⁻¹ | exp_neg_of_mem_ball $ (exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _ | lemma | exp_neg | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_neg_of_mem_ball",
"exp_series_radius_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_zsmul (z : ℤ) (x : 𝔸) : exp 𝕂 (z • x) = (exp 𝕂 x) ^ z | begin
obtain ⟨n, rfl | rfl⟩ := z.eq_coe_or_neg,
{ rw [zpow_coe_nat, coe_nat_zsmul, exp_nsmul] },
{ rw [zpow_neg, zpow_coe_nat, neg_smul, exp_neg, coe_nat_zsmul, exp_nsmul] },
end | lemma | exp_zsmul | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_neg",
"exp_nsmul",
"neg_smul",
"zpow_coe_nat",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_conj (y : 𝔸) (x : 𝔸) (hy : y ≠ 0) :
exp 𝕂 (y * x * y⁻¹) = y * exp 𝕂 x * y⁻¹ | exp_units_conj _ (units.mk0 y hy) x | lemma | exp_conj | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_units_conj",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_conj' (y : 𝔸) (x : 𝔸) (hy : y ≠ 0) :
exp 𝕂 (y⁻¹ * x * y) = y⁻¹ * exp 𝕂 x * y | exp_units_conj' _ (units.mk0 y hy) x | lemma | exp_conj' | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_units_conj'",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add {x y : 𝔸} : exp 𝕂 (x + y) = (exp 𝕂 x) * (exp 𝕂 y) | exp_add_of_mem_ball ((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _)
((exp_series_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | lemma | exp_add | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"edist_lt_top",
"exp",
"exp_add_of_mem_ball",
"exp_series_radius_eq_top"
] | In a commutative Banach-algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ`,
`exp 𝕂 (x+y) = (exp 𝕂 x) * (exp 𝕂 y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_sum {ι} (s : finset ι) (f : ι → 𝔸) :
exp 𝕂 (∑ i in s, f i) = ∏ i in s, exp 𝕂 (f i) | begin
rw [exp_sum_of_commute, finset.noncomm_prod_eq_prod],
exact λ i hi j hj _, commute.all _ _,
end | lemma | exp_sum | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"commute.all",
"exp",
"exp_sum_of_commute",
"finset",
"finset.noncomm_prod_eq_prod"
] | A version of `exp_sum_of_commute` for a commutative Banach-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_series_eq_exp_series (n : ℕ) (x : 𝔸) :
(exp_series 𝕂 𝔸 n (λ _, x)) = (exp_series 𝕂' 𝔸 n (λ _, x)) | by rw [exp_series_apply_eq, exp_series_apply_eq, inv_nat_cast_smul_eq 𝕂 𝕂'] | lemma | exp_series_eq_exp_series | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp_series",
"exp_series_apply_eq",
"inv_nat_cast_smul_eq"
] | If a normed ring `𝔸` is a normed algebra over two fields, then they define the same
`exp_series` on `𝔸`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_eq_exp : (exp 𝕂 : 𝔸 → 𝔸) = exp 𝕂' | begin
ext,
rw [exp, exp],
refine tsum_congr (λ n, _),
rw exp_series_eq_exp_series 𝕂 𝕂' 𝔸 n x
end | lemma | exp_eq_exp | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_series_eq_exp_series",
"tsum_congr"
] | If a normed ring `𝔸` is a normed algebra over two fields, then they define the same
exponential function on `𝔸`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_ℝ_ℂ_eq_exp_ℂ_ℂ : (exp ℝ : ℂ → ℂ) = exp ℂ | exp_eq_exp ℝ ℂ ℂ | lemma | exp_ℝ_ℂ_eq_exp_ℂ_ℂ | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"exp",
"exp_eq_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_exp_ℝ_ℝ (r : ℝ) : ↑(exp ℝ r) = exp ℂ (r : ℂ) | (map_exp ℝ (algebra_map ℝ ℂ) (continuous_algebra_map _ _) r).trans
(congr_fun exp_ℝ_ℂ_eq_exp_ℂ_ℂ _) | lemma | of_real_exp_ℝ_ℝ | analysis.normed_space | src/analysis/normed_space/exponential.lean | [
"analysis.analytic.basic",
"analysis.complex.basic",
"analysis.normed.field.infinite_sum",
"data.nat.choose.cast",
"data.finset.noncomm_prod",
"topology.algebra.algebra"
] | [
"algebra_map",
"continuous_algebra_map",
"exp",
"exp_ℝ_ℂ_eq_exp_ℂ_ℂ",
"map_exp"
] | A version of `complex.of_real_exp` for `exp` instead of `complex.exp` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_to_𝕜' (fr : F →ₗ[ℝ] ℝ) : F →ₗ[𝕜] 𝕜 | begin
let fc : F → 𝕜 := λ x, (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x)),
have add : ∀ x y : F, fc (x + y) = fc x + fc y,
{ assume x y,
simp only [fc],
simp only [smul_add, linear_map.map_add, of_real_add],
rw mul_add,
abel, },
have A : ∀ (c : ℝ) (x : F), (fr ((c : 𝕜) • x) : 𝕜) = (c : 𝕜) * ... | def | linear_map.extend_to_𝕜' | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"add_smul",
"algebra.id.smul_eq_mul",
"is_R_or_C.of_real_alg",
"linear_map.map_add",
"linear_map.map_neg",
"mul_assoc",
"mul_comm",
"mul_neg",
"neg_mul",
"neg_smul",
"one_mul",
"one_smul",
"ring",
"smul_add",
"smul_assoc",
"smul_smul"
] | Extend `fr : F →ₗ[ℝ] ℝ` to `F →ₗ[𝕜] 𝕜` in a way that will also be continuous and have its norm
bounded by `‖fr‖` if `fr` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_to_𝕜'_apply (fr : F →ₗ[ℝ] ℝ) (x : F) :
fr.extend_to_𝕜' x = (fr x : 𝕜) - (I : 𝕜) * fr ((I : 𝕜) • x) | rfl | lemma | linear_map.extend_to_𝕜'_apply | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_to_𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extend_to_𝕜' x : 𝕜) = fr x | by simp only [extend_to_𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero] with is_R_or_C_simps | lemma | linear_map.extend_to_𝕜'_apply_re | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_extend_to_𝕜'_apply_sq (f : F →ₗ[ℝ] ℝ) (x : F) :
‖(f.extend_to_𝕜' x : 𝕜)‖ ^ 2 = f (conj (f.extend_to_𝕜' x : 𝕜) • x) | calc ‖(f.extend_to_𝕜' x : 𝕜)‖ ^ 2 = re (conj (f.extend_to_𝕜' x) * f.extend_to_𝕜' x : 𝕜) :
by rw [is_R_or_C.conj_mul, norm_sq_eq_def', of_real_re]
... = f (conj (f.extend_to_𝕜' x : 𝕜) • x) :
by rw [← smul_eq_mul, ← map_smul, extend_to_𝕜'_apply_re] | lemma | linear_map.norm_extend_to_𝕜'_apply_sq | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"is_R_or_C.conj_mul",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_extend_to_𝕜'_bound (fr : F →L[ℝ] ℝ) (x : F) :
‖(fr.to_linear_map.extend_to_𝕜' x : 𝕜)‖ ≤ ‖fr‖ * ‖x‖ | begin
set lm : F →ₗ[𝕜] 𝕜 := fr.to_linear_map.extend_to_𝕜',
classical,
by_cases h : lm x = 0,
{ rw [h, norm_zero],
apply mul_nonneg; exact norm_nonneg _ },
rw [← mul_le_mul_left (norm_pos_iff.2 h), ← sq],
calc ‖lm x‖ ^ 2 = fr (conj (lm x : 𝕜) • x) : fr.to_linear_map.norm_extend_to_𝕜'_apply_sq x
..... | lemma | continuous_linear_map.norm_extend_to_𝕜'_bound | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"le_abs_self",
"mul_le_mul_left",
"mul_left_comm",
"norm_smul"
] | The norm of the extension is bounded by `‖fr‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_to_𝕜' (fr : F →L[ℝ] ℝ) : F →L[𝕜] 𝕜 | linear_map.mk_continuous _ (‖fr‖) fr.norm_extend_to_𝕜'_bound | def | continuous_linear_map.extend_to_𝕜' | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"linear_map.mk_continuous"
] | Extend `fr : F →L[ℝ] ℝ` to `F →L[𝕜] 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_to_𝕜'_apply (fr : F →L[ℝ] ℝ) (x : F) :
fr.extend_to_𝕜' x = (fr x : 𝕜) - (I : 𝕜) * fr ((I : 𝕜) • x) | rfl | lemma | continuous_linear_map.extend_to_𝕜'_apply | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_extend_to_𝕜' (fr : F →L[ℝ] ℝ) : ‖(fr.extend_to_𝕜' : F →L[𝕜] 𝕜)‖ = ‖fr‖ | le_antisymm (linear_map.mk_continuous_norm_le _ (norm_nonneg _) _) $
op_norm_le_bound _ (norm_nonneg _) $ λ x,
calc ‖fr x‖ = ‖re (fr.extend_to_𝕜' x : 𝕜)‖ : congr_arg norm (fr.extend_to_𝕜'_apply_re x).symm
... ≤ ‖(fr.extend_to_𝕜' x : 𝕜)‖ : abs_re_le_norm _
... ≤ ‖(fr.extend_to_𝕜' : F →L[𝕜] 𝕜)‖ * ‖x... | lemma | continuous_linear_map.norm_extend_to_𝕜' | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"linear_map.mk_continuous_norm_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.extend_to_𝕜 (fr : (restrict_scalars ℝ 𝕜 F) →ₗ[ℝ] ℝ) : F →ₗ[𝕜] 𝕜 | fr.extend_to_𝕜' | def | linear_map.extend_to_𝕜 | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"restrict_scalars"
] | Extend `fr : restrict_scalars ℝ 𝕜 F →ₗ[ℝ] ℝ` to `F →ₗ[𝕜] 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.extend_to_𝕜_apply (fr : (restrict_scalars ℝ 𝕜 F) →ₗ[ℝ] ℝ) (x : F) :
fr.extend_to_𝕜 x = (fr x : 𝕜) - (I : 𝕜) * fr ((I : 𝕜) • x : _) | rfl | lemma | linear_map.extend_to_𝕜_apply | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.extend_to_𝕜 (fr : (restrict_scalars ℝ 𝕜 F) →L[ℝ] ℝ) :
F →L[𝕜] 𝕜 | fr.extend_to_𝕜' | def | continuous_linear_map.extend_to_𝕜 | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"restrict_scalars"
] | Extend `fr : restrict_scalars ℝ 𝕜 F →L[ℝ] ℝ` to `F →L[𝕜] 𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.extend_to_𝕜_apply (fr : (restrict_scalars ℝ 𝕜 F) →L[ℝ] ℝ) (x : F) :
fr.extend_to_𝕜 x = (fr x : 𝕜) - (I : 𝕜) * fr ((I : 𝕜) • x : _) | rfl | lemma | continuous_linear_map.extend_to_𝕜_apply | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.norm_extend_to_𝕜 (fr : (restrict_scalars ℝ 𝕜 F) →L[ℝ] ℝ) :
‖fr.extend_to_𝕜‖ = ‖fr‖ | fr.norm_extend_to_𝕜' | lemma | continuous_linear_map.norm_extend_to_𝕜 | analysis.normed_space | src/analysis/normed_space/extend.lean | [
"analysis.normed_space.operator_norm",
"algebra.algebra.restrict_scalars",
"data.is_R_or_C.basic"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.norm_add_same_ray (h : is_max_filter (norm ∘ f) l c) (hy : same_ray ℝ (f c) y) :
is_max_filter (λ x, ‖f x + y‖) l c | h.mono $ λ x hx,
calc ‖f x + y‖ ≤ ‖f x‖ + ‖y‖ : norm_add_le _ _
... ≤ ‖f c‖ + ‖y‖ : add_le_add_right hx _
... = ‖f c + y‖ : hy.norm_add.symm | lemma | is_max_filter.norm_add_same_ray | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_max_filter",
"same_ray"
] | If `f : α → E` is a function such that `norm ∘ f` has a maximum along a filter `l` at a point
`c` and `y` is a vector on the same ray as `f c`, then the function `λ x, ‖f x + y‖` has a maximul
along `l` at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_filter.norm_add_self (h : is_max_filter (norm ∘ f) l c) :
is_max_filter (λ x, ‖f x + f c‖) l c | h.norm_add_same_ray same_ray.rfl | lemma | is_max_filter.norm_add_self | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_max_filter",
"same_ray.rfl"
] | If `f : α → E` is a function such that `norm ∘ f` has a maximum along a filter `l` at a point
`c`, then the function `λ x, ‖f x + f c‖` has a maximul along `l` at `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_max_on.norm_add_same_ray (h : is_max_on (norm ∘ f) s c) (hy : same_ray ℝ (f c) y) :
is_max_on (λ x, ‖f x + y‖) s c | h.norm_add_same_ray hy | lemma | is_max_on.norm_add_same_ray | analysis.normed_space | src/analysis/normed_space/extr.lean | [
"analysis.normed_space.ray",
"topology.local_extr"
] | [
"is_max_on",
"same_ray"
] | If `f : α → E` is a function such that `norm ∘ f` has a maximum on a set `s` at a point `c` and
`y` is a vector on the same ray as `f c`, then the function `λ x, ‖f x + y‖` has a maximul on `s` at
`c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.