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deriv_cosh (hc : differentiable_at ℝ f x) :
deriv (λx, real.cosh (f x)) x = real.sinh (f x) * (deriv f x) | hc.has_deriv_at.cosh.deriv | lemma | deriv_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"differentiable_at",
"real.cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.sinh (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.sinh (f x)) (real.cosh (f x) * f') x | (real.has_strict_deriv_at_sinh (f x)).comp x hf | lemma | has_strict_deriv_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_deriv_at",
"real.cosh",
"real.has_strict_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.sinh (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.sinh (f x)) (real.cosh (f x) * f') x | (real.has_deriv_at_sinh (f x)).comp x hf | lemma | has_deriv_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_at",
"real.cosh",
"real.has_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.sinh (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.sinh (f x)) (real.cosh (f x) * f') s x | (real.has_deriv_at_sinh (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_within_at",
"real.cosh",
"real.has_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_sinh (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.sinh (f x)) s x = real.cosh (f x) * (deriv_within f s x) | hf.has_deriv_within_at.sinh.deriv_within hxs | lemma | deriv_within_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv_within",
"differentiable_within_at",
"real.cosh",
"real.sinh",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sinh (hc : differentiable_at ℝ f x) :
deriv (λx, real.sinh (f x)) x = real.cosh (f x) * (deriv f x) | hc.has_deriv_at.sinh.deriv | lemma | deriv_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"differentiable_at",
"real.cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.cos (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.cos (f x)) (- real.sin (f x) • f') x | (real.has_strict_deriv_at_cos (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_fderiv_at",
"real.cos",
"real.has_strict_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.cos (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.cos (f x)) (- real.sin (f x) • f') x | (real.has_deriv_at_cos (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_at",
"real.cos",
"real.has_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.cos (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.cos (f x)) (- real.sin (f x) • f') s x | (real.has_deriv_at_cos (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_within_at",
"real.cos",
"real.has_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.cos (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.cos (f x)) s x | hf.has_fderiv_within_at.cos.differentiable_within_at | lemma | differentiable_within_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.cos (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.cos (f x)) x | hc.has_fderiv_at.cos.differentiable_at | lemma | differentiable_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.cos (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.cos (f x)) s | λx h, (hc x h).cos | lemma | differentiable_on.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_on",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.cos (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.cos (f x)) | λx, (hc x).cos | lemma | differentiable.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_cos (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.cos (f x)) s x = - real.sin (f x) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.cos.fderiv_within hxs | lemma | fderiv_within_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"real.cos",
"real.sin",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_cos (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.cos (f x)) x = - real.sin (f x) • (fderiv ℝ f x) | hc.has_fderiv_at.cos.fderiv | lemma | fderiv_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"fderiv",
"real.cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cos {n} (h : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.cos (f x)) | real.cont_diff_cos.comp h | lemma | cont_diff.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.cos {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.cos (f x)) x | real.cont_diff_cos.cont_diff_at.comp x hf | lemma | cont_diff_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_at",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.cos {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.cos (f x)) s | real.cont_diff_cos.comp_cont_diff_on hf | lemma | cont_diff_on.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_on",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cos {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.cos (f x)) s x | real.cont_diff_cos.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_within_at",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sin (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.sin (f x)) (real.cos (f x) • f') x | (real.has_strict_deriv_at_sin (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_fderiv_at",
"real.cos",
"real.has_strict_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sin (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.sin (f x)) (real.cos (f x) • f') x | (real.has_deriv_at_sin (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_at",
"real.cos",
"real.has_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sin (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.sin (f x)) (real.cos (f x) • f') s x | (real.has_deriv_at_sin (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_within_at",
"real.cos",
"real.has_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sin (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.sin (f x)) s x | hf.has_fderiv_within_at.sin.differentiable_within_at | lemma | differentiable_within_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sin (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.sin (f x)) x | hc.has_fderiv_at.sin.differentiable_at | lemma | differentiable_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sin (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.sin (f x)) s | λx h, (hc x h).sin | lemma | differentiable_on.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_on",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sin (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.sin (f x)) | λx, (hc x).sin | lemma | differentiable.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sin (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.sin (f x)) s x = real.cos (f x) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.sin.fderiv_within hxs | lemma | fderiv_within_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"real.cos",
"real.sin",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sin (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.sin (f x)) x = real.cos (f x) • (fderiv ℝ f x) | hc.has_fderiv_at.sin.fderiv | lemma | fderiv_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"fderiv",
"real.cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.sin {n} (h : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.sin (f x)) | real.cont_diff_sin.comp h | lemma | cont_diff.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.sin {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.sin (f x)) x | real.cont_diff_sin.cont_diff_at.comp x hf | lemma | cont_diff_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_at",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.sin {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.sin (f x)) s | real.cont_diff_sin.comp_cont_diff_on hf | lemma | cont_diff_on.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_on",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.sin {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.sin (f x)) s x | real.cont_diff_sin.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_within_at",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.cosh (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') x | (real.has_strict_deriv_at_cosh (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_fderiv_at",
"real.cosh",
"real.has_strict_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.cosh (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') x | (real.has_deriv_at_cosh (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_at",
"real.cosh",
"real.has_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.cosh (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.cosh (f x)) (real.sinh (f x) • f') s x | (real.has_deriv_at_cosh (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_within_at",
"real.cosh",
"real.has_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.cosh (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.cosh (f x)) s x | hf.has_fderiv_within_at.cosh.differentiable_within_at | lemma | differentiable_within_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.cosh (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.cosh (f x)) x | hc.has_fderiv_at.cosh.differentiable_at | lemma | differentiable_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.cosh (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.cosh (f x)) s | λx h, (hc x h).cosh | lemma | differentiable_on.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_on",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.cosh (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.cosh (f x)) | λx, (hc x).cosh | lemma | differentiable.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_cosh (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.cosh (f x)) s x = real.sinh (f x) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.cosh.fderiv_within hxs | lemma | fderiv_within_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"real.cosh",
"real.sinh",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_cosh (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.cosh (f x)) x = real.sinh (f x) • (fderiv ℝ f x) | hc.has_fderiv_at.cosh.fderiv | lemma | fderiv_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"fderiv",
"real.cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cosh {n} (h : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.cosh (f x)) | real.cont_diff_cosh.comp h | lemma | cont_diff.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.cosh {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.cosh (f x)) x | real.cont_diff_cosh.cont_diff_at.comp x hf | lemma | cont_diff_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_at",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.cosh {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.cosh (f x)) s | real.cont_diff_cosh.comp_cont_diff_on hf | lemma | cont_diff_on.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_on",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cosh {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.cosh (f x)) s x | real.cont_diff_cosh.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_within_at",
"real.cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sinh (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') x | (real.has_strict_deriv_at_sinh (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_fderiv_at",
"real.cosh",
"real.has_strict_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sinh (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') x | (real.has_deriv_at_sinh (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_at",
"real.cosh",
"real.has_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sinh (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.sinh (f x)) (real.cosh (f x) • f') s x | (real.has_deriv_at_sinh (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_fderiv_within_at",
"real.cosh",
"real.has_deriv_at_sinh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sinh (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.sinh (f x)) s x | hf.has_fderiv_within_at.sinh.differentiable_within_at | lemma | differentiable_within_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sinh (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.sinh (f x)) x | hc.has_fderiv_at.sinh.differentiable_at | lemma | differentiable_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sinh (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.sinh (f x)) s | λx h, (hc x h).sinh | lemma | differentiable_on.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_on",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sinh (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.sinh (f x)) | λx, (hc x).sinh | lemma | differentiable.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sinh (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.sinh (f x)) s x = real.cosh (f x) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.sinh.fderiv_within hxs | lemma | fderiv_within_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"real.cosh",
"real.sinh",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sinh (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.sinh (f x)) x = real.cosh (f x) • (fderiv ℝ f x) | hc.has_fderiv_at.sinh.fderiv | lemma | fderiv_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at",
"fderiv",
"real.cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.sinh {n} (h : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.sinh (f x)) | real.cont_diff_sinh.comp h | lemma | cont_diff.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.sinh {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.sinh (f x)) x | real.cont_diff_sinh.cont_diff_at.comp x hf | lemma | cont_diff_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_at",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.sinh {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.sinh (f x)) s | real.cont_diff_sinh.comp_cont_diff_on hf | lemma | cont_diff_on.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_on",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.sinh {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.sinh (f x)) s x | real.cont_diff_sinh.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff_within_at",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antideriv_cos_comp_const_mul (hz : z ≠ 0) (x : ℝ) :
has_deriv_at (λ y:ℝ, complex.sin (2 * z * y) / (2 * z)) (complex.cos (2 * z * x)) x | begin
have a : has_deriv_at _ _ ↑x := has_deriv_at_mul_const _,
have b : has_deriv_at (λ (y : ℂ), complex.sin (y * (2 * z))) _ ↑x :=
has_deriv_at.comp x (complex.has_deriv_at_sin (x * (2 * z))) a,
convert (b.comp_of_real).div_const (2 * z),
{ ext1 x, rw mul_comm _ (2 * z) },
{ field_simp, rw mul_comm _ (2... | lemma | euler_sine.antideriv_cos_comp_const_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.cos",
"complex.has_deriv_at_sin",
"complex.sin",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at_mul_const",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antideriv_sin_comp_const_mul (hz : z ≠ 0) (x : ℝ) :
has_deriv_at (λ y:ℝ, -complex.cos (2 * z * y) / (2 * z)) (complex.sin (2 * z * x)) x | begin
have a : has_deriv_at _ _ ↑x := has_deriv_at_mul_const _,
have b : has_deriv_at (λ (y : ℂ), complex.cos (y * (2 * z))) _ ↑x :=
has_deriv_at.comp x (complex.has_deriv_at_cos (x * (2 * z))) a,
convert ((b.comp_of_real).div_const (2 * z)).neg,
{ ext1 x, rw mul_comm _ (2 * z), field_simp },
{ field_simp... | lemma | euler_sine.antideriv_sin_comp_const_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.cos",
"complex.has_deriv_at_cos",
"complex.sin",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at_mul_const",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_mul_cos_pow_aux (hn : 2 ≤ n) (hz : z ≠ 0):
(∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ n) =
n / (2 * z) * ∫ x:ℝ in 0..π/2, complex.sin (2 * z * x) * sin x * cos x ^ (n - 1) | begin
have der1 : ∀ (x : ℝ), (x ∈ uIcc 0 (π/2)) → has_deriv_at (λ y, (↑(cos y)) ^ n : ℝ → ℂ)
(-n * sin x * cos x ^ (n - 1)) x,
{ intros x hx,
have b : has_deriv_at (λ y, ↑(cos y) : ℝ → ℂ) (-sin x) x,
by simpa using (has_deriv_at_cos x).of_real_comp,
convert has_deriv_at.comp x (has_deriv_at_pow _ ... | lemma | euler_sine.integral_cos_mul_cos_pow_aux | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.continuous_of_real",
"complex.cos",
"complex.of_real_zero",
"complex.sin",
"complex.sin_zero",
"continuous.interval_integrable",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at_pow",
"mul_comm",
"mul_zero",
"ring",
"zero_div",
"zero_mul",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_mul_sin_mul_cos_pow_eq (hn : 2 ≤ n) (hz : z ≠ 0) :
∫ x:ℝ in 0..π/2, complex.sin (2 * z * x) * sin x * cos x ^ (n - 1) =
n / (2 * z) * (∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ n) -
(n - 1) / (2 * z) * (∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ (n - 2)) | begin
have der1 : ∀ (x : ℝ), (x ∈ uIcc 0 (π/2)) →
has_deriv_at (λ y, (sin y) * (cos y) ^ (n - 1) : ℝ → ℂ)
(cos x ^ n - (n - 1) * sin x ^ 2 * cos x ^ (n - 2)) x,
{ intros x hx,
have c := has_deriv_at.comp (x:ℂ) (has_deriv_at_pow (n - 1) _) (complex.has_deriv_at_cos x),
convert ((complex.has_deriv_at_... | lemma | euler_sine.integral_sin_mul_sin_mul_cos_pow_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.continuous_of_real",
"complex.cos",
"complex.has_deriv_at_cos",
"complex.has_deriv_at_sin",
"complex.of_real_cos",
"complex.of_real_sin",
"complex.of_real_zero",
"complex.sin",
"complex.sin_sq",
"continuous.interval_integrable",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at_pow... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_mul_cos_pow (hn : 2 ≤ n) (hz : z ≠ 0) :
(1 - 4 * z ^ 2 / n ^ 2) * (∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ n) =
(n - 1 : ℂ) / n * ∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ (n - 2) | begin
have nne : (n : ℂ) ≠ 0,
{ contrapose! hn, rw nat.cast_eq_zero at hn, rw hn, exact zero_lt_two },
have := integral_cos_mul_cos_pow_aux hn hz,
rw [integral_sin_mul_sin_mul_cos_pow_eq hn hz, sub_eq_neg_add, mul_add, ←sub_eq_iff_eq_add]
at this,
convert congr_arg (λ u:ℂ, -u * (2 * z) ^ 2 / n ^ 2) this u... | lemma | euler_sine.integral_cos_mul_cos_pow | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.cos",
"nat.cast_eq_zero",
"ring",
"zero_lt_two"
] | Note this also holds for `z = 0`, but we do not need this case for `sin_pi_mul_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_cos_mul_cos_pow_even (n : ℕ) (hz : z ≠ 0) :
(1 - z ^ 2 / (n + 1) ^ 2) * (∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ (2 * n + 2)) =
(2 * n + 1 : ℂ) / (2 * n + 2) * ∫ x:ℝ in 0..π/2, complex.cos (2 * z * x) * cos x ^ (2 * n) | begin
convert integral_cos_mul_cos_pow (by linarith : 2 ≤ 2 * n + 2) hz using 3,
{ simp only [nat.cast_add, nat.cast_mul, nat.cast_two],
nth_rewrite_rhs 2 ←mul_one (2:ℂ),
rw [←mul_add, mul_pow, ←div_div],
ring },
{ push_cast, ring },
{ push_cast, ring },
end | lemma | euler_sine.integral_cos_mul_cos_pow_even | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.cos",
"mul_pow",
"nat.cast_add",
"nat.cast_mul",
"nat.cast_two",
"ring"
] | Note this also holds for `z = 0`, but we do not need this case for `sin_pi_mul_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_cos_pow_eq (n : ℕ) :
(∫ (x:ℝ) in 0..π/2, cos x ^ n) = 1 / 2 * (∫ (x:ℝ) in 0..π, (sin x) ^ n) | begin
rw [mul_comm (1/2 : ℝ), ←div_eq_iff (one_div_ne_zero (two_ne_zero' ℝ)), ←div_mul, div_one,
mul_two],
have L : interval_integrable _ volume 0 (π / 2) := (continuous_sin.pow n).interval_integrable _ _,
have R : interval_integrable _ volume (π / 2) π := (continuous_sin.pow n).interval_integrable _ _,
rw ... | lemma | euler_sine.integral_cos_pow_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"div_one",
"interval_integrable",
"mul_comm",
"mul_two",
"one_div_ne_zero",
"ring",
"two_ne_zero'"
] | Relate the integral `cos x ^ n` over `[0, π/2]` to the integral of `sin x ^ n` over `[0, π]`,
which is studied in `data.real.pi.wallis` and other places. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_cos_pow_pos (n : ℕ) : 0 < (∫ (x:ℝ) in 0..π/2, cos x ^ n) | (integral_cos_pow_eq n).symm ▸ (mul_pos one_half_pos (integral_sin_pow_pos _)) | lemma | euler_sine.integral_cos_pow_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"integral_sin_pow_pos",
"one_half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_mul_eq (z : ℂ) (n : ℕ) :
complex.sin (π * z) = π * z * (∏ j in finset.range n, (1 - z ^ 2 / (j + 1) ^ 2)) *
(∫ x in 0..π/2, complex.cos (2 * z * x) * cos x ^ (2 * n)) / ↑∫ x in 0..π/2, cos x ^ (2 * n) | begin
rcases eq_or_ne z 0 with rfl | hz,
{ simp },
induction n with n hn,
{ simp_rw [mul_zero, pow_zero, mul_one, finset.prod_range_zero, mul_one, integral_one, sub_zero],
rw [integral_cos_mul_complex (mul_ne_zero two_ne_zero hz), complex.of_real_zero, mul_zero,
complex.sin_zero, zero_div, sub_zero,
... | lemma | euler_sine.sin_pi_mul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.cos",
"complex.of_real_div",
"complex.of_real_mul",
"complex.of_real_zero",
"complex.sin",
"complex.sin_zero",
"eq_or_ne",
"finset.prod_range_succ",
"finset.prod_range_zero",
"finset.range",
"integral_cos_mul_complex",
"integral_one",
"integral_sin_pow",
"mul_comm",
"mul_ne_zero... | Finite form of Euler's sine product, with remainder term expressed as a ratio of cosine
integrals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_cos_pow_mul_div {f : ℝ → ℂ} (hf : continuous_on f (Icc 0 (π/2))) :
tendsto (λ (n : ℕ), (∫ x:ℝ in 0..π/2, ↑(cos x) ^ n * f x) / ↑(∫ x:ℝ in 0..π/2, (cos x) ^ n))
at_top (𝓝 $ f 0) | begin
simp_rw [div_eq_inv_mul _ (coe _), ←complex.of_real_inv, integral_of_le (pi_div_two_pos.le),
←measure_theory.integral_Icc_eq_integral_Ioc, ←complex.of_real_pow, ←complex.real_smul],
have c_lt : ∀ (y : ℝ), y ∈ Icc 0 (π / 2) → y ≠ 0 → cos y < cos 0, from λ y hy hy',
cos_lt_cos_of_nonneg_of_le_pi_div_two... | lemma | euler_sine.tendsto_integral_cos_pow_mul_div | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"closure",
"closure_Ioo",
"continuous_on",
"div_eq_inv_mul",
"interior",
"interior_Icc",
"tendsto_set_integral_pow_smul_of_unique_maximum_of_is_compact_of_continuous_on",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.complex.tendsto_euler_sin_prod (z : ℂ) :
tendsto (λ n:ℕ, ↑π * z * (∏ j in finset.range n, (1 - z ^ 2 / (j + 1) ^ 2)))
at_top (𝓝 $ complex.sin (π * z)) | begin
have A : tendsto (λ n:ℕ, ↑π * z * (∏ j in finset.range n, (1 - z ^ 2 / (j + 1) ^ 2)) *
(∫ x in 0..π / 2, complex.cos (2 * z * x) * cos x ^ (2 * n)) /
↑∫ x in 0..π / 2, cos x ^ (2 * n))
at_top (𝓝 $ _) := tendsto.congr (λ n, (sin_pi_mul_eq z n)) tendsto_const_nhds,
have : 𝓝 (complex.sin (π * z)) =... | lemma | complex.tendsto_euler_sin_prod | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.continuous_of_real",
"complex.cos",
"complex.cos_zero",
"complex.of_real_zero",
"complex.sin",
"continuous_on",
"finset.range",
"mul_comm",
"mul_div_assoc",
"mul_one",
"mul_zero",
"one_ne_zero",
"tendsto_const_nhds",
"zero_lt_two"
] | Euler's infinite product formula for the complex sine function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.real.tendsto_euler_sin_prod (x : ℝ) :
tendsto (λ n:ℕ, π * x * (∏ j in finset.range n, (1 - x ^ 2 / (j + 1) ^ 2)))
at_top (𝓝 $ sin (π * x)) | begin
convert (complex.continuous_re.tendsto _).comp (complex.tendsto_euler_sin_prod x),
{ ext1 n,
rw [function.comp_app, ←complex.of_real_mul, complex.of_real_mul_re],
suffices : ∏ (j : ℕ) in finset.range n, (1 - (x:ℂ) ^ 2 / (↑j + 1) ^ 2) =
↑∏ (j : ℕ) in finset.range n, (1 - x ^ 2 / (↑j + 1) ^ 2), by... | lemma | real.tendsto_euler_sin_prod | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/euler_sine_prod.lean | [
"analysis.special_functions.integrals",
"measure_theory.integral.peak_function"
] | [
"complex.of_real_mul_re",
"complex.of_real_prod",
"complex.of_real_re",
"complex.tendsto_euler_sin_prod",
"finset.prod_congr",
"finset.range"
] | Euler's infinite product formula for the real sine function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arcsin : ℝ → ℝ | coe ∘ Icc_extend (neg_le_self zero_le_one) sin_order_iso.symm | def | real.arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"zero_le_one"
] | Inverse of the `sin` function, returns values in the range `-π / 2 ≤ arcsin x ≤ π / 2`.
It defaults to `-π / 2` on `(-∞, -1)` and to `π / 2` to `(1, ∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arcsin_mem_Icc (x : ℝ) : arcsin x ∈ Icc (-(π / 2)) (π / 2) | subtype.coe_prop _ | lemma | real.arcsin_mem_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"subtype.coe_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) | by { rw [arcsin, range_comp coe], simp [Icc] } | lemma | real.range_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_le_pi_div_two (x : ℝ) : arcsin x ≤ π / 2 | (arcsin_mem_Icc x).2 | lemma | real.arcsin_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_div_two_le_arcsin (x : ℝ) : -(π / 2) ≤ arcsin x | (arcsin_mem_Icc x).1 | lemma | real.neg_pi_div_two_le_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_proj_Icc (x : ℝ) :
arcsin (proj_Icc (-1) 1 (neg_le_self zero_le_one) x) = arcsin x | by rw [arcsin, function.comp_app, Icc_extend_coe, function.comp_app, Icc_extend] | lemma | real.arcsin_proj_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_arcsin' {x : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) : sin (arcsin x) = x | by simpa [arcsin, Icc_extend_of_mem _ _ hx, -order_iso.apply_symm_apply]
using subtype.ext_iff.1 (sin_order_iso.apply_symm_apply ⟨x, hx⟩) | lemma | real.sin_arcsin' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"order_iso.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arcsin x) = x | sin_arcsin' ⟨hx₁, hx₂⟩ | lemma | real.sin_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_sin' {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin (sin x) = x | inj_on_sin (arcsin_mem_Icc _) hx $ by rw [sin_arcsin (neg_one_le_sin _) (sin_le_one _)] | lemma | real.arcsin_sin' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_sin {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : arcsin (sin x) = x | arcsin_sin' ⟨hx₁, hx₂⟩ | lemma | real.arcsin_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_arcsin : strict_mono_on arcsin (Icc (-1) 1) | (subtype.strict_mono_coe _).comp_strict_mono_on $
sin_order_iso.symm.strict_mono.strict_mono_on_Icc_extend _ | lemma | real.strict_mono_on_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"strict_mono_on",
"subtype.strict_mono_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_arcsin : monotone arcsin | (subtype.mono_coe _).comp $ sin_order_iso.symm.monotone.Icc_extend _ | lemma | real.monotone_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"monotone",
"subtype.mono_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on_arcsin : inj_on arcsin (Icc (-1) 1) | strict_mono_on_arcsin.inj_on | lemma | real.inj_on_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) :
arcsin x = arcsin y ↔ x = y | inj_on_arcsin.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ | lemma | real.arcsin_inj | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_arcsin : continuous arcsin | continuous_subtype_coe.comp sin_order_iso.symm.continuous.Icc_extend' | lemma | real.continuous_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_arcsin {x : ℝ} : continuous_at arcsin x | continuous_arcsin.continuous_at | lemma | real.continuous_at_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_of_sin_eq {x y : ℝ} (h₁ : sin x = y) (h₂ : x ∈ Icc (-(π / 2)) (π / 2)) :
arcsin y = x | begin
subst y,
exact inj_on_sin (arcsin_mem_Icc _) h₂ (sin_arcsin' (sin_mem_Icc x))
end | lemma | real.arcsin_eq_of_sin_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_zero : arcsin 0 = 0 | arcsin_eq_of_sin_eq sin_zero ⟨neg_nonpos.2 pi_div_two_pos.le, pi_div_two_pos.le⟩ | lemma | real.arcsin_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_one : arcsin 1 = π / 2 | arcsin_eq_of_sin_eq sin_pi_div_two $ right_mem_Icc.2 (neg_le_self pi_div_two_pos.le) | lemma | real.arcsin_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_of_one_le {x : ℝ} (hx : 1 ≤ x) : arcsin x = π / 2 | by rw [← arcsin_proj_Icc, proj_Icc_of_right_le _ hx, subtype.coe_mk, arcsin_one] | lemma | real.arcsin_of_one_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_neg_one : arcsin (-1) = -(π / 2) | arcsin_eq_of_sin_eq (by rw [sin_neg, sin_pi_div_two]) $
left_mem_Icc.2 (neg_le_self pi_div_two_pos.le) | lemma | real.arcsin_neg_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) | by rw [← arcsin_proj_Icc, proj_Icc_of_le_left _ hx, subtype.coe_mk, arcsin_neg_one] | lemma | real.arcsin_of_le_neg_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_neg (x : ℝ) : arcsin (-x) = -arcsin x | begin
cases le_total x (-1) with hx₁ hx₁,
{ rw [arcsin_of_le_neg_one hx₁, neg_neg, arcsin_of_one_le (le_neg.2 hx₁)] },
cases le_total 1 x with hx₂ hx₂,
{ rw [arcsin_of_one_le hx₂, arcsin_of_le_neg_one (neg_le_neg hx₂)] },
refine arcsin_eq_of_sin_eq _ _,
{ rw [sin_neg, sin_arcsin hx₁ hx₂] },
{ exact ⟨neg_l... | lemma | real.arcsin_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_le_iff_le_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) :
arcsin x ≤ y ↔ x ≤ sin y | by rw [← arcsin_sin' hy, strict_mono_on_arcsin.le_iff_le hx (sin_mem_Icc _), arcsin_sin' hy] | lemma | real.arcsin_le_iff_le_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_le_iff_le_sin' {x y : ℝ} (hy : y ∈ Ico (-(π / 2)) (π / 2)) :
arcsin x ≤ y ↔ x ≤ sin y | begin
cases le_total x (-1) with hx₁ hx₁,
{ simp [arcsin_of_le_neg_one hx₁, hy.1, hx₁.trans (neg_one_le_sin _)] },
cases lt_or_le 1 x with hx₂ hx₂,
{ simp [arcsin_of_one_le hx₂.le, hy.2.not_le, (sin_le_one y).trans_lt hx₂] },
exact arcsin_le_iff_le_sin ⟨hx₁, hx₂⟩ (mem_Icc_of_Ico hy)
end | lemma | real.arcsin_le_iff_le_sin' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_arcsin_iff_sin_le {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) :
x ≤ arcsin y ↔ sin x ≤ y | by rw [← neg_le_neg_iff, ← arcsin_neg,
arcsin_le_iff_le_sin ⟨neg_le_neg hy.2, neg_le.2 hy.1⟩ ⟨neg_le_neg hx.2, neg_le.2 hx.1⟩,
sin_neg, neg_le_neg_iff] | lemma | real.le_arcsin_iff_sin_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_arcsin_iff_sin_le' {x y : ℝ} (hx : x ∈ Ioc (-(π / 2)) (π / 2)) :
x ≤ arcsin y ↔ sin x ≤ y | by rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin' ⟨neg_le_neg hx.2, neg_lt.2 hx.1⟩,
sin_neg, neg_le_neg_iff] | lemma | real.le_arcsin_iff_sin_le' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_lt_iff_lt_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) :
arcsin x < y ↔ x < sin y | not_le.symm.trans $ (not_congr $ le_arcsin_iff_sin_le hy hx).trans not_le | lemma | real.arcsin_lt_iff_lt_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_lt_iff_lt_sin' {x y : ℝ} (hy : y ∈ Ioc (-(π / 2)) (π / 2)) :
arcsin x < y ↔ x < sin y | not_le.symm.trans $ (not_congr $ le_arcsin_iff_sin_le' hy).trans not_le | lemma | real.arcsin_lt_iff_lt_sin' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_arcsin_iff_sin_lt {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) :
x < arcsin y ↔ sin x < y | not_le.symm.trans $ (not_congr $ arcsin_le_iff_le_sin hy hx).trans not_le | lemma | real.lt_arcsin_iff_sin_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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