statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
dist_eq_re_im (z w : ℂ) : dist z w = real.sqrt ((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) | by { rw [sq, sq], refl } | lemma | complex.dist_eq_re_im | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"real.sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mk (x₁ y₁ x₂ y₂ : ℝ) :
dist (mk x₁ y₁) (mk x₂ y₂) = real.sqrt ((x₁ - x₂) ^ 2 + (y₁ - y₂) ^ 2) | dist_eq_re_im _ _ | lemma | complex.dist_mk | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"real.sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_re_eq {z w : ℂ} (h : z.re = w.re) : dist z w = dist z.im w.im | by rw [dist_eq_re_im, h, sub_self, zero_pow two_pos, zero_add, real.sqrt_sq_eq_abs, real.dist_eq] | lemma | complex.dist_of_re_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"real.dist_eq",
"real.sqrt_sq_eq_abs",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_of_re_eq {z w : ℂ} (h : z.re = w.re) : nndist z w = nndist z.im w.im | nnreal.eq $ dist_of_re_eq h | lemma | complex.nndist_of_re_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_of_re_eq {z w : ℂ} (h : z.re = w.re) : edist z w = edist z.im w.im | by rw [edist_nndist, edist_nndist, nndist_of_re_eq h] | lemma | complex.edist_of_re_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"edist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_of_im_eq {z w : ℂ} (h : z.im = w.im) : dist z w = dist z.re w.re | by rw [dist_eq_re_im, h, sub_self, zero_pow two_pos, add_zero, real.sqrt_sq_eq_abs, real.dist_eq] | lemma | complex.dist_of_im_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"real.dist_eq",
"real.sqrt_sq_eq_abs",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_of_im_eq {z w : ℂ} (h : z.im = w.im) : nndist z w = nndist z.re w.re | nnreal.eq $ dist_of_im_eq h | lemma | complex.nndist_of_im_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_of_im_eq {z w : ℂ} (h : z.im = w.im) : edist z w = edist z.re w.re | by rw [edist_nndist, edist_nndist, nndist_of_im_eq h] | lemma | complex.edist_of_im_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"edist_nndist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_conj_self (z : ℂ) : dist (conj z) z = 2 * |z.im| | by rw [dist_of_re_eq (conj_re z), conj_im, dist_comm, real.dist_eq, sub_neg_eq_add, ← two_mul,
_root_.abs_mul, abs_of_pos (zero_lt_two' ℝ)] | lemma | complex.dist_conj_self | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"abs_of_pos",
"dist_comm",
"real.dist_eq",
"two_mul",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_conj_self (z : ℂ) : nndist (conj z) z = 2 * real.nnabs z.im | nnreal.eq $ by rw [← dist_nndist, nnreal.coe_mul, nnreal.coe_two, real.coe_nnabs, dist_conj_self] | lemma | complex.nndist_conj_self | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"dist_nndist",
"nnreal.coe_mul",
"nnreal.coe_two",
"nnreal.eq",
"real.coe_nnabs",
"real.nnabs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_self_conj (z : ℂ) : dist z (conj z) = 2 * |z.im| | by rw [dist_comm, dist_conj_self] | lemma | complex.dist_self_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * real.nnabs z.im | by rw [nndist_comm, nndist_conj_self] | lemma | complex.nndist_self_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"nndist_comm",
"real.nnabs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_abs_nhds_zero : filter.comap abs (𝓝 0) = 𝓝 0 | comap_norm_nhds_zero | lemma | complex.comap_abs_nhds_zero | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"filter.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖ | abs_of_real _ | lemma | complex.norm_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_rat (r : ℚ) : ‖(r : ℂ)‖ = |(r : ℝ)| | by { rw ← of_real_rat_cast, exact norm_real _ } | lemma | complex.norm_rat | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_nat (n : ℕ) : ‖(n : ℂ)‖ = n | abs_of_nat _ | lemma | complex.norm_nat | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_int {n : ℤ} : ‖(n : ℂ)‖ = |n| | by simp [← rat.cast_coe_int] {single_pass := tt} | lemma | complex.norm_int | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"rat.cast_coe_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖(n : ℂ)‖ = n | by simp [hn] | lemma | complex.norm_int_of_nonneg | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_abs : continuous abs | continuous_norm | lemma | complex.continuous_abs | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous",
"continuous_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_norm_sq : continuous norm_sq | by simpa [← norm_sq_eq_abs] using continuous_abs.pow 2 | lemma | complex.continuous_norm_sq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_real (r : ℝ) : ‖(r : ℂ)‖₊ = ‖r‖₊ | subtype.ext $ norm_real r | lemma | complex.nnnorm_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_nat (n : ℕ) : ‖(n : ℂ)‖₊ = n | subtype.ext $ by simp | lemma | complex.nnnorm_nat | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_int (n : ℤ) : ‖(n : ℂ)‖₊ = ‖n‖₊ | subtype.ext $ by simp only [coe_nnnorm, norm_int, int.norm_eq_abs] | lemma | complex.nnnorm_int | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"int.norm_eq_abs",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) :
‖ζ‖₊ = 1 | begin
refine (@pow_left_inj nnreal _ _ _ _ zero_le' zero_le' hn.bot_lt).mp _,
rw [←nnnorm_pow, h, nnnorm_one, one_pow],
end | lemma | complex.nnnorm_eq_one_of_pow_eq_one | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"nnnorm_one",
"nnreal",
"one_pow",
"pow_left_inj",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) :
‖ζ‖ = 1 | congr_arg coe (nnnorm_eq_one_of_pow_eq_one h hn) | lemma | complex.norm_eq_one_of_pow_eq_one | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_real_prod_apply_le (z : ℂ) : ‖equiv_real_prod z‖ ≤ abs z | by simp [prod.norm_def, abs_re_le_abs, abs_im_le_abs] | lemma | complex.equiv_real_prod_apply_le | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"prod.norm_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_real_prod_apply_le' (z : ℂ) : ‖equiv_real_prod z‖ ≤ 1 * abs z | by simpa using equiv_real_prod_apply_le z | lemma | complex.equiv_real_prod_apply_le' | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_equiv_real_prod : lipschitz_with 1 equiv_real_prod | by simpa using
add_monoid_hom_class.lipschitz_of_bound equiv_real_prod_lm 1 equiv_real_prod_apply_le' | lemma | complex.lipschitz_equiv_real_prod | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_equiv_real_prod : antilipschitz_with (nnreal.sqrt 2) equiv_real_prod | by simpa using
add_monoid_hom_class.antilipschitz_of_bound equiv_real_prod_lm abs_le_sqrt_two_mul_max | lemma | complex.antilipschitz_equiv_real_prod | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"antilipschitz_with",
"nnreal.sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_equiv_real_prod : uniform_embedding equiv_real_prod | antilipschitz_equiv_real_prod.uniform_embedding lipschitz_equiv_real_prod.uniform_continuous | lemma | complex.uniform_embedding_equiv_real_prod | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_real_prod_clm : ℂ ≃L[ℝ] ℝ × ℝ | equiv_real_prod_lm.to_continuous_linear_equiv_of_bounds 1 (real.sqrt 2)
equiv_real_prod_apply_le'
(λ p, abs_le_sqrt_two_mul_max (equiv_real_prod.symm p)) | def | complex.equiv_real_prod_clm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"real.sqrt"
] | The natural `continuous_linear_equiv` from `ℂ` to `ℝ × ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_abs_cocompact_at_top : filter.tendsto abs (filter.cocompact ℂ) filter.at_top | tendsto_norm_cocompact_at_top | lemma | complex.tendsto_abs_cocompact_at_top | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"filter.at_top",
"filter.cocompact",
"filter.tendsto"
] | The `abs` function on `ℂ` is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_norm_sq_cocompact_at_top :
filter.tendsto norm_sq (filter.cocompact ℂ) filter.at_top | by simpa [mul_self_abs] using
tendsto_abs_cocompact_at_top.at_top_mul_at_top tendsto_abs_cocompact_at_top | lemma | complex.tendsto_norm_sq_cocompact_at_top | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"filter.at_top",
"filter.cocompact",
"filter.tendsto"
] | The `norm_sq` function on `ℂ` is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_clm : ℂ →L[ℝ] ℝ | re_lm.mk_continuous 1 (λ x, by simp [abs_re_le_abs]) | def | complex.re_clm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | Continuous linear map version of the real part function, from `ℂ` to `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_re : continuous re | re_clm.continuous | lemma | complex.continuous_re | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_clm_coe : (coe (re_clm) : ℂ →ₗ[ℝ] ℝ) = re_lm | rfl | lemma | complex.re_clm_coe | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_clm_apply (z : ℂ) : (re_clm : ℂ → ℝ) z = z.re | rfl | lemma | complex.re_clm_apply | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_clm : ℂ →L[ℝ] ℝ | im_lm.mk_continuous 1 (λ x, by simp [abs_im_le_abs]) | def | complex.im_clm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | Continuous linear map version of the real part function, from `ℂ` to `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_im : continuous im | im_clm.continuous | lemma | complex.continuous_im | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_clm_coe : (coe (im_clm) : ℂ →ₗ[ℝ] ℝ) = im_lm | rfl | lemma | complex.im_clm_coe | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_clm_apply (z : ℂ) : (im_clm : ℂ → ℝ) z = z.im | rfl | lemma | complex.im_clm_apply | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_one_smul_right' (x : E) :
continuous_linear_map.restrict_scalars ℝ ((1 : ℂ →L[ℂ] ℂ).smul_right x : ℂ →L[ℂ] E) =
re_clm.smul_right x + I • im_clm.smul_right x | by { ext ⟨a, b⟩, simp [mk_eq_add_mul_I, add_smul, mul_smul, smul_comm I] } | lemma | complex.restrict_scalars_one_smul_right' | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"add_smul",
"continuous_linear_map.restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_one_smul_right (x : ℂ) :
continuous_linear_map.restrict_scalars ℝ ((1 : ℂ →L[ℂ] ℂ).smul_right x : ℂ →L[ℂ] ℂ) = x • 1 | by { ext1 z, dsimp, apply mul_comm } | lemma | complex.restrict_scalars_one_smul_right | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous_linear_map.restrict_scalars",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_lie : ℂ ≃ₗᵢ[ℝ] ℂ | ⟨conj_ae.to_linear_equiv, abs_conj⟩ | def | complex.conj_lie | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | The complex-conjugation function from `ℂ` to itself is an isometric linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_lie_apply (z : ℂ) : conj_lie z = conj z | rfl | lemma | complex.conj_lie_apply | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_lie_symm : conj_lie.symm = conj_lie | rfl | lemma | complex.conj_lie_symm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_conj : isometry (conj : ℂ → ℂ) | conj_lie.isometry | lemma | complex.isometry_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_conj_conj (z w : ℂ) : dist (conj z) (conj w) = dist z w | isometry_conj.dist_eq z w | lemma | complex.dist_conj_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_conj_conj (z w : ℂ) : nndist (conj z) (conj w) = nndist z w | isometry_conj.nndist_eq z w | lemma | complex.nndist_conj_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_conj_comm (z w : ℂ) : dist (conj z) w = dist z (conj w) | by rw [← dist_conj_conj, conj_conj] | lemma | complex.dist_conj_comm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_conj_comm (z w : ℂ) : nndist (conj z) w = nndist z (conj w) | subtype.ext $ dist_conj_comm _ _ | lemma | complex.nndist_conj_comm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_conj : continuous (conj : ℂ → ℂ) | continuous_star | lemma | complex.continuous_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_eq_id_or_conj_of_continuous {f : ℂ →+* ℂ} (hf : continuous f) :
f = ring_hom.id ℂ ∨ f = conj | begin
refine (real_alg_hom_eq_id_or_conj $ alg_hom.mk' f $ map_real_smul f hf).imp (λ h, _) (λ h, _),
all_goals { convert congr_arg alg_hom.to_ring_hom h, ext1, refl, },
end | lemma | complex.ring_hom_eq_id_or_conj_of_continuous | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"alg_hom.mk'",
"continuous",
"map_real_smul",
"ring_hom.id"
] | The only continuous ring homomorphisms from `ℂ` to `ℂ` are the identity and the complex
conjugation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_cle : ℂ ≃L[ℝ] ℂ | conj_lie | def | complex.conj_cle | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | Continuous linear equiv version of the conj function, from `ℂ` to `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_cle_coe : conj_cle.to_linear_equiv = conj_ae.to_linear_equiv | rfl | lemma | complex.conj_cle_coe | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_cle_apply (z : ℂ) : conj_cle z = conj z | rfl | lemma | complex.conj_cle_apply | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_li : ℝ →ₗᵢ[ℝ] ℂ | ⟨of_real_am.to_linear_map, norm_real⟩ | def | complex.of_real_li | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | Linear isometry version of the canonical embedding of `ℝ` in `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_of_real : isometry (coe : ℝ → ℂ) | of_real_li.isometry | lemma | complex.isometry_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_real : continuous (coe : ℝ → ℂ) | of_real_li.continuous | lemma | complex.continuous_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_eq_of_real_of_continuous {f : ℝ →+* ℂ} (h : continuous f) :
f = complex.of_real | begin
convert congr_arg alg_hom.to_ring_hom
(subsingleton.elim (alg_hom.mk' f $ map_real_smul f h) $ algebra.of_id ℝ ℂ),
ext1, refl,
end | lemma | complex.ring_hom_eq_of_real_of_continuous | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"alg_hom.mk'",
"algebra.of_id",
"complex.of_real",
"continuous",
"map_real_smul"
] | The only continuous ring homomorphism from `ℝ` to `ℂ` is the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_real_clm : ℝ →L[ℝ] ℂ | of_real_li.to_continuous_linear_map | def | complex.of_real_clm | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | Continuous linear map version of the canonical embedding of `ℝ` in `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_real_clm_coe : (of_real_clm : ℝ →ₗ[ℝ] ℂ) = of_real_am.to_linear_map | rfl | lemma | complex.of_real_clm_coe | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_clm_apply (x : ℝ) : of_real_clm x = x | rfl | lemma | complex.of_real_clm_apply | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_R_or_C.re_eq_complex_re : ⇑(is_R_or_C.re : ℂ →+ ℝ) = complex.re | rfl | lemma | is_R_or_C.re_eq_complex_re | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_R_or_C.im_eq_complex_im : ⇑(is_R_or_C.im : ℂ →+ ℝ) = complex.im | rfl | lemma | is_R_or_C.im_eq_complex_im | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_coe_norm_of_nonneg {z : ℂ} (hz : 0 ≤ z) : z = ↑‖z‖ | by rw [eq_re_of_real_le hz, is_R_or_C.norm_of_real, _root_.abs_of_nonneg (complex.le_def.2 hz).1] | lemma | complex.eq_coe_norm_of_nonneg | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.norm_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_to_complex {x : ℂ} : reC x = x.re | rfl | lemma | is_R_or_C.re_to_complex | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_to_complex {x : ℂ} : imC x = x.im | rfl | lemma | is_R_or_C.im_to_complex | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
I_to_complex : IC = complex.I | rfl | lemma | is_R_or_C.I_to_complex | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"complex.I"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_to_complex {x : ℂ} : norm_sqC x = complex.norm_sq x | rfl | lemma | is_R_or_C.norm_sq_to_complex | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_conj {f : α → 𝕜} {x : 𝕜} :
has_sum (λ x, conj (f x)) x ↔ has_sum f (conj x) | conj_cle.has_sum | lemma | is_R_or_C.has_sum_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_conj' {f : α → 𝕜} {x : 𝕜} : has_sum (λ x, conj (f x)) (conj x) ↔ has_sum f x | conj_cle.has_sum' | lemma | is_R_or_C.has_sum_conj' | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_conj {f : α → 𝕜} : summable (λ x, conj (f x)) ↔ summable f | summable_star_iff | lemma | is_R_or_C.summable_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"summable",
"summable_star_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_tsum (f : α → 𝕜) : conj (∑' a, f a) = ∑' a, conj (f a) | tsum_star | lemma | is_R_or_C.conj_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"tsum_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_of_real {f : α → ℝ} {x : ℝ} :
has_sum (λ x, (f x : 𝕜)) x ↔ has_sum f x | ⟨λ h, by simpa only [is_R_or_C.re_clm_apply, is_R_or_C.of_real_re] using re_clm.has_sum h,
of_real_clm.has_sum⟩ | lemma | is_R_or_C.has_sum_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.of_real_re",
"is_R_or_C.re_clm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_real {f : α → ℝ} : summable (λ x, (f x : 𝕜)) ↔ summable f | ⟨λ h, by simpa only [is_R_or_C.re_clm_apply, is_R_or_C.of_real_re] using re_clm.summable h,
of_real_clm.summable⟩ | lemma | is_R_or_C.summable_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.of_real_re",
"is_R_or_C.re_clm_apply",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_tsum (f : α → ℝ) : (↑(∑' a, f a) : 𝕜) = ∑' a, f a | begin
by_cases h : summable f,
{ exact continuous_linear_map.map_tsum of_real_clm h },
{ rw [tsum_eq_zero_of_not_summable h,
tsum_eq_zero_of_not_summable ((summable_of_real _).not.mpr h), of_real_zero] }
end | lemma | is_R_or_C.of_real_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"continuous_linear_map.map_tsum",
"summable",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_re {f : α → 𝕜} {x : 𝕜} (h : has_sum f x) : has_sum (λ x, re (f x)) (re x) | re_clm.has_sum h | lemma | is_R_or_C.has_sum_re | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_im {f : α → 𝕜} {x : 𝕜} (h : has_sum f x) : has_sum (λ x, im (f x)) (im x) | im_clm.has_sum h | lemma | is_R_or_C.has_sum_im | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_tsum {f : α → 𝕜} (h : summable f) : re (∑' a, f a) = ∑' a, re (f a) | re_clm.map_tsum h | lemma | is_R_or_C.re_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_tsum {f : α → 𝕜} (h : summable f) : im (∑' a, f a) = ∑' a, im (f a) | im_clm.map_tsum h | lemma | is_R_or_C.im_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff (f : α → 𝕜) (c : 𝕜) :
has_sum f c ↔ has_sum (λ x, re (f x)) (re c) ∧ has_sum (λ x, im (f x)) (im c) | begin
refine ⟨λ h, ⟨has_sum_re _ h, has_sum_im _ h⟩, _⟩,
rintro ⟨h₁, h₂⟩,
rw ←is_R_or_C.re_add_im c,
convert ((has_sum_of_real 𝕜).mpr h₁).add (((has_sum_of_real 𝕜).mpr h₂).mul_right I),
simp_rw is_R_or_C.re_add_im,
end | lemma | is_R_or_C.has_sum_iff | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.re_add_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_conj {f : α → ℂ} {x : ℂ} :
has_sum (λ x, conj (f x)) x ↔ has_sum f (conj x) | is_R_or_C.has_sum_conj _ | lemma | complex.has_sum_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_conj' {f : α → ℂ} {x : ℂ} : has_sum (λ x, conj (f x)) (conj x) ↔ has_sum f x | is_R_or_C.has_sum_conj' _ | lemma | complex.has_sum_conj' | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_conj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_conj {f : α → ℂ} : summable (λ x, conj (f x)) ↔ summable f | is_R_or_C.summable_conj _ | lemma | complex.summable_conj | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.summable_conj",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_tsum (f : α → ℂ) : conj (∑' a, f a) = ∑' a, conj (f a) | is_R_or_C.conj_tsum _ | lemma | complex.conj_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.conj_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_of_real {f : α → ℝ} {x : ℝ} :
has_sum (λ x, (f x : ℂ)) x ↔ has_sum f x | is_R_or_C.has_sum_of_real _ | lemma | complex.has_sum_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_real {f : α → ℝ} : summable (λ x, (f x : ℂ)) ↔ summable f | is_R_or_C.summable_of_real _ | lemma | complex.summable_of_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.summable_of_real",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_tsum (f : α → ℝ) : (↑(∑' a, f a) : ℂ) = ∑' a, f a | is_R_or_C.of_real_tsum _ _ | lemma | complex.of_real_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.of_real_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_re {f : α → ℂ} {x : ℂ} (h : has_sum f x) : has_sum (λ x, (f x).re) x.re | is_R_or_C.has_sum_re _ h | lemma | complex.has_sum_re | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_re"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_im {f : α → ℂ} {x : ℂ} (h : has_sum f x) : has_sum (λ x, (f x).im) x.im | is_R_or_C.has_sum_im _ h | lemma | complex.has_sum_im | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_tsum {f : α → ℂ} (h : summable f) : (∑' a, f a).re = ∑' a, (f a).re | is_R_or_C.re_tsum _ h | lemma | complex.re_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.re_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_tsum {f : α → ℂ} (h : summable f) : (∑' a, f a).im = ∑' a, (f a).im | is_R_or_C.im_tsum _ h | lemma | complex.im_tsum | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"is_R_or_C.im_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_iff (f : α → ℂ) (c : ℂ) :
has_sum f c ↔ has_sum (λ x, (f x).re) c.re ∧ has_sum (λ x, (f x).im) c.im | is_R_or_C.has_sum_iff _ _ | lemma | complex.has_sum_iff | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"has_sum",
"is_R_or_C.has_sum_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_boundary_rect_of_has_fderiv_at_real_off_countable (f : ℂ → E)
(f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (s : set ℂ) (hs : s.countable)
(Hc : continuous_on f ([z.re, w.re] ×ℂ [z.im, w.im]))
(Hd : ∀ x ∈ (Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im)) \ s,
has_fderiv_at f (f' x) x)
(... | begin
set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equiv_real_prod_clm.symm,
have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y), from λ x y, (mk_eq_add_mul_I x y).symm,
have he₁ : e (1, 0) = 1 := rfl, have he₂ : e (0, 1) = I := rfl,
simp only [he] at *,
set F : (ℝ × ℝ) → E := f ∘ e,
set F' : (ℝ × ℝ) → (ℝ × ℝ) →L[ℝ] E := λ p, (f' (e... | lemma | complex.integral_boundary_rect_of_has_fderiv_at_real_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_map.comp_apply",
"continuous_linear_map.neg_apply",
"continuous_linear_map.smul_apply",
"continuous_on",
"has_fderiv_at",
"interval_integral.integral_neg",
"interval_integral.integral_smul",
"interval_integral.integral_symm",
"measurable_equiv.... | Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_boundary_rect_of_continuous_on_of_has_fderiv_at_real (f : ℂ → E)
(f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ)
(Hc : continuous_on f ([z.re, w.re] ×ℂ [z.im, w.im]))
(Hd : ∀ x ∈ (Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im)),
has_fderiv_at f (f' x) x)
(Hi : integrable_on (λ z, I • f' ... | integral_boundary_rect_of_has_fderiv_at_real_off_countable f f' z w ∅ countable_empty Hc
(λ x hx, Hd x hx.1) Hi | lemma | complex.integral_boundary_rect_of_continuous_on_of_has_fderiv_at_real | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_on",
"has_fderiv_at"
] | Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable on the corresponding open rectangle, and
$\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the integral of `f` over
the boundary of the rectangle is equal to the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_boundary_rect_of_differentiable_on_real (f : ℂ → E) (z w : ℂ)
(Hd : differentiable_on ℝ f ([z.re, w.re] ×ℂ [z.im, w.im]))
(Hi : integrable_on (λ z, I • fderiv ℝ f z 1 - fderiv ℝ f z I) ([z.re, w.re] ×ℂ [z.im, w.im])) :
(∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) ... | integral_boundary_rect_of_has_fderiv_at_real_off_countable f (fderiv ℝ f) z w ∅ countable_empty
Hd.continuous_on
(λ x hx, Hd.has_fderiv_at $ by simpa only [← mem_interior_iff_mem_nhds,
interior_re_prod_im, uIcc, interior_Icc] using hx.1) Hi | lemma | complex.integral_boundary_rect_of_differentiable_on_real | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"differentiable_on",
"fderiv",
"interior_Icc",
"mem_interior_iff_mem_nhds"
] | Suppose that a function `f : ℂ → E` is *real* differentiable on a closed rectangle with opposite
corners at `z w : ℂ` and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then
the integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_boundary_rect_eq_zero_of_differentiable_on_off_countable (f : ℂ → E)
(z w : ℂ) (s : set ℂ) (hs : s.countable) (Hc : continuous_on f ([z.re, w.re] ×ℂ [z.im, w.im]))
(Hd : ∀ x ∈ (Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im)) \ s,
differentiable_at ℂ f x) :
(∫ x : ℝ in z.re... | by refine (integral_boundary_rect_of_has_fderiv_at_real_off_countable f
(λ z, (fderiv ℂ f z).restrict_scalars ℝ) z w s hs Hc
(λ x hx, (Hd x hx).has_fderiv_at.restrict_scalars ℝ) _).trans _;
simp [← continuous_linear_map.map_smul] | lemma | complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_linear_map.map_smul",
"continuous_on",
"differentiable_at",
"fderiv",
"has_fderiv_at.restrict_scalars",
"restrict_scalars"
] | **Cauchy-Goursat theorem** for a rectangle: the integral of a complex differentiable function
over the boundary of a rectangle equals zero. More precisely, if `f` is continuous on a closed
rectangle and is complex differentiable at all but countably many points of the corresponding open
rectangle, then its integral ove... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_boundary_rect_eq_zero_of_continuous_on_of_differentiable_on (f : ℂ → E) (z w : ℂ)
(Hc : continuous_on f ([z.re, w.re] ×ℂ [z.im, w.im]))
(Hd : differentiable_on ℂ f
(Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im))) :
(∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z... | integral_boundary_rect_eq_zero_of_differentiable_on_off_countable f z w ∅ countable_empty
Hc $ λ x hx, Hd.differentiable_at $ (is_open_Ioo.re_prod_im is_open_Ioo).mem_nhds hx.1 | lemma | complex.integral_boundary_rect_eq_zero_of_continuous_on_of_differentiable_on | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"continuous_on",
"differentiable_on",
"is_open_Ioo"
] | **Cauchy-Goursat theorem for a rectangle**: the integral of a complex differentiable function
over the boundary of a rectangle equals zero. More precisely, if `f` is continuous on a closed
rectangle and is complex differentiable on the corresponding open rectangle, then its integral over
the boundary of the rectangle e... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_boundary_rect_eq_zero_of_differentiable_on (f : ℂ → E) (z w : ℂ)
(H : differentiable_on ℂ f ([z.re, w.re] ×ℂ [z.im, w.im])) :
(∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
(I • ∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
I • ∫ y : ℝ in z.im..w.im, f (re z ... | integral_boundary_rect_eq_zero_of_continuous_on_of_differentiable_on f z w H.continuous_on $
H.mono $
inter_subset_inter (preimage_mono Ioo_subset_Icc_self) (preimage_mono Ioo_subset_Icc_self) | lemma | complex.integral_boundary_rect_eq_zero_of_differentiable_on | analysis.complex | src/analysis/complex/cauchy_integral.lean | [
"measure_theory.measure.lebesgue.complex",
"measure_theory.integral.divergence_theorem",
"measure_theory.integral.circle_integral",
"analysis.calculus.dslope",
"analysis.analytic.basic",
"analysis.complex.re_im_topology",
"analysis.calculus.diff_cont_on_cl",
"data.real.cardinality"
] | [
"differentiable_on"
] | **Cauchy-Goursat theorem** for a rectangle: the integral of a complex differentiable function
over the boundary of a rectangle equals zero. More precisely, if `f` is complex differentiable on a
closed rectangle, then its integral over the boundary of the rectangle equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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