statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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differentiable_at.smul (hc : differentiable_at 𝕜 c x) (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λ y, c y • f y) x | (hc.has_fderiv_at.smul hf.has_fderiv_at).differentiable_at | lemma | differentiable_at.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.smul (hc : differentiable_on 𝕜 c s) (hf : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λ y, c y • f y) s | λx hx, (hc x hx).smul (hf x hx) | lemma | differentiable_on.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.smul (hc : differentiable 𝕜 c) (hf : differentiable 𝕜 f) :
differentiable 𝕜 (λ y, c y • f y) | λx, (hc x).smul (hf x) | lemma | differentiable.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_smul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hf : differentiable_within_at 𝕜 f s x) :
fderiv_within 𝕜 (λ y, c y • f y) s x =
c x • fderiv_within 𝕜 f s x + (fderiv_within 𝕜 c s x).smul_right (f x) | (hc.has_fderiv_within_at.smul hf.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_smul (hc : differentiable_at 𝕜 c x) (hf : differentiable_at 𝕜 f x) :
fderiv 𝕜 (λ y, c y • f y) x =
c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smul_right (f x) | (hc.has_fderiv_at.smul hf.has_fderiv_at).fderiv | lemma | fderiv_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.smul_const (hc : has_strict_fderiv_at c c' x) (f : F) :
has_strict_fderiv_at (λ y, c y • f) (c'.smul_right f) x | by simpa only [smul_zero, zero_add] using hc.smul (has_strict_fderiv_at_const f x) | theorem | has_strict_fderiv_at.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.smul_const (hc : has_fderiv_within_at c c' s x) (f : F) :
has_fderiv_within_at (λ y, c y • f) (c'.smul_right f) s x | by simpa only [smul_zero, zero_add] using hc.smul (has_fderiv_within_at_const f x s) | theorem | has_fderiv_within_at.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at",
"has_fderiv_within_at_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.smul_const (hc : has_fderiv_at c c' x) (f : F) :
has_fderiv_at (λ y, c y • f) (c'.smul_right f) x | by simpa only [smul_zero, zero_add] using hc.smul (has_fderiv_at_const f x) | theorem | has_fderiv_at.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at",
"has_fderiv_at_const",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.smul_const
(hc : differentiable_within_at 𝕜 c s x) (f : F) :
differentiable_within_at 𝕜 (λ y, c y • f) s x | (hc.has_fderiv_within_at.smul_const f).differentiable_within_at | lemma | differentiable_within_at.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.smul_const (hc : differentiable_at 𝕜 c x) (f : F) :
differentiable_at 𝕜 (λ y, c y • f) x | (hc.has_fderiv_at.smul_const f).differentiable_at | lemma | differentiable_at.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.smul_const (hc : differentiable_on 𝕜 c s) (f : F) :
differentiable_on 𝕜 (λ y, c y • f) s | λx hx, (hc x hx).smul_const f | lemma | differentiable_on.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.smul_const (hc : differentiable 𝕜 c) (f : F) :
differentiable 𝕜 (λ y, c y • f) | λx, (hc x).smul_const f | lemma | differentiable.smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_smul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (f : F) :
fderiv_within 𝕜 (λ y, c y • f) s x =
(fderiv_within 𝕜 c s x).smul_right f | (hc.has_fderiv_within_at.smul_const f).fderiv_within hxs | lemma | fderiv_within_smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_smul_const (hc : differentiable_at 𝕜 c x) (f : F) :
fderiv 𝕜 (λ y, c y • f) x = (fderiv 𝕜 c x).smul_right f | (hc.has_fderiv_at.smul_const f).fderiv | lemma | fderiv_smul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.mul' {x : E} (ha : has_strict_fderiv_at a a' x)
(hb : has_strict_fderiv_at b b' x) :
has_strict_fderiv_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) x | ((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_strict_fderiv_at (a x, b x)).comp x
(ha.prod hb) | theorem | has_strict_fderiv_at.mul' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_strict_fderiv_at",
"is_bounded_bilinear_map.has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.mul
(hc : has_strict_fderiv_at c c' x) (hd : has_strict_fderiv_at d d' x) :
has_strict_fderiv_at (λ y, c y * d y) (c x • d' + d x • c') x | by { convert hc.mul' hd, ext z, apply mul_comm } | theorem | has_strict_fderiv_at.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mul'
(ha : has_fderiv_within_at a a' s x) (hb : has_fderiv_within_at b b' s x) :
has_fderiv_within_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) s x | ((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_fderiv_at
(a x, b x)).comp_has_fderiv_within_at x (ha.prod hb) | theorem | has_fderiv_within_at.mul' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_within_at",
"is_bounded_bilinear_map.has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mul
(hc : has_fderiv_within_at c c' s x) (hd : has_fderiv_within_at d d' s x) :
has_fderiv_within_at (λ y, c y * d y) (c x • d' + d x • c') s x | by { convert hc.mul' hd, ext z, apply mul_comm } | theorem | has_fderiv_within_at.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.mul'
(ha : has_fderiv_at a a' x) (hb : has_fderiv_at b b' x) :
has_fderiv_at (λ y, a y * b y) (a x • b' + a'.smul_right (b x)) x | ((continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.has_fderiv_at (a x, b x)).comp x
(ha.prod hb) | theorem | has_fderiv_at.mul' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_at",
"is_bounded_bilinear_map.has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.mul (hc : has_fderiv_at c c' x) (hd : has_fderiv_at d d' x) :
has_fderiv_at (λ y, c y * d y) (c x • d' + d x • c') x | by { convert hc.mul' hd, ext z, apply mul_comm } | theorem | has_fderiv_at.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.mul
(ha : differentiable_within_at 𝕜 a s x) (hb : differentiable_within_at 𝕜 b s x) :
differentiable_within_at 𝕜 (λ y, a y * b y) s x | (ha.has_fderiv_within_at.mul' hb.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.mul (ha : differentiable_at 𝕜 a x) (hb : differentiable_at 𝕜 b x) :
differentiable_at 𝕜 (λ y, a y * b y) x | (ha.has_fderiv_at.mul' hb.has_fderiv_at).differentiable_at | lemma | differentiable_at.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.mul (ha : differentiable_on 𝕜 a s) (hb : differentiable_on 𝕜 b s) :
differentiable_on 𝕜 (λ y, a y * b y) s | λx hx, (ha x hx).mul (hb x hx) | lemma | differentiable_on.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.mul (ha : differentiable 𝕜 a) (hb : differentiable 𝕜 b) :
differentiable 𝕜 (λ y, a y * b y) | λx, (ha x).mul (hb x) | lemma | differentiable.mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.pow (ha : differentiable_within_at 𝕜 a s x) :
∀ n : ℕ, differentiable_within_at 𝕜 (λ x, a x ^ n) s x | | 0 := by simp only [pow_zero, differentiable_within_at_const]
| (n + 1) := by simp only [pow_succ, differentiable_within_at.pow n, ha.mul] | lemma | differentiable_within_at.pow | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"differentiable_within_at_const",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.pow (ha : differentiable_at 𝕜 a x) (n : ℕ) :
differentiable_at 𝕜 (λ x, a x ^ n) x | differentiable_within_at_univ.mp $ ha.differentiable_within_at.pow n | lemma | differentiable_at.pow | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.pow (ha : differentiable_on 𝕜 a s) (n : ℕ) :
differentiable_on 𝕜 (λ x, a x ^ n) s | λ x h, (ha x h).pow n | lemma | differentiable_on.pow | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.pow (ha : differentiable 𝕜 a) (n : ℕ) :
differentiable 𝕜 (λ x, a x ^ n) | λx, (ha x).pow n | lemma | differentiable.pow | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_mul' (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (hb : differentiable_within_at 𝕜 b s x) :
fderiv_within 𝕜 (λ y, a y * b y) s x =
a x • fderiv_within 𝕜 b s x + (fderiv_within 𝕜 a s x).smul_right (b x) | (ha.has_fderiv_within_at.mul' hb.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_mul' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_mul (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
fderiv_within 𝕜 (λ y, c y * d y) s x =
c x • fderiv_within 𝕜 d s x + d x • fderiv_within 𝕜 c s x | (hc.has_fderiv_within_at.mul hd.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_mul' (ha : differentiable_at 𝕜 a x) (hb : differentiable_at 𝕜 b x) :
fderiv 𝕜 (λ y, a y * b y) x =
a x • fderiv 𝕜 b x + (fderiv 𝕜 a x).smul_right (b x) | (ha.has_fderiv_at.mul' hb.has_fderiv_at).fderiv | lemma | fderiv_mul' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_mul (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
fderiv 𝕜 (λ y, c y * d y) x =
c x • fderiv 𝕜 d x + d x • fderiv 𝕜 c x | (hc.has_fderiv_at.mul hd.has_fderiv_at).fderiv | lemma | fderiv_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.mul_const' (ha : has_strict_fderiv_at a a' x) (b : 𝔸) :
has_strict_fderiv_at (λ y, a y * b) (a'.smul_right b) x | (((continuous_linear_map.mul 𝕜 𝔸).flip b).has_strict_fderiv_at).comp x ha | theorem | has_strict_fderiv_at.mul_const' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.mul_const (hc : has_strict_fderiv_at c c' x) (d : 𝔸') :
has_strict_fderiv_at (λ y, c y * d) (d • c') x | by { convert hc.mul_const' d, ext z, apply mul_comm } | theorem | has_strict_fderiv_at.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mul_const' (ha : has_fderiv_within_at a a' s x) (b : 𝔸) :
has_fderiv_within_at (λ y, a y * b) (a'.smul_right b) s x | (((continuous_linear_map.mul 𝕜 𝔸).flip b).has_fderiv_at).comp_has_fderiv_within_at x ha | theorem | has_fderiv_within_at.mul_const' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mul_const (hc : has_fderiv_within_at c c' s x) (d : 𝔸') :
has_fderiv_within_at (λ y, c y * d) (d • c') s x | by { convert hc.mul_const' d, ext z, apply mul_comm } | theorem | has_fderiv_within_at.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.mul_const' (ha : has_fderiv_at a a' x) (b : 𝔸) :
has_fderiv_at (λ y, a y * b) (a'.smul_right b) x | (((continuous_linear_map.mul 𝕜 𝔸).flip b).has_fderiv_at).comp x ha | theorem | has_fderiv_at.mul_const' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.mul_const (hc : has_fderiv_at c c' x) (d : 𝔸') :
has_fderiv_at (λ y, c y * d) (d • c') x | by { convert hc.mul_const' d, ext z, apply mul_comm } | theorem | has_fderiv_at.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.mul_const
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
differentiable_within_at 𝕜 (λ y, a y * b) s x | (ha.has_fderiv_within_at.mul_const' b).differentiable_within_at | lemma | differentiable_within_at.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.mul_const (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
differentiable_at 𝕜 (λ y, a y * b) x | (ha.has_fderiv_at.mul_const' b).differentiable_at | lemma | differentiable_at.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.mul_const (ha : differentiable_on 𝕜 a s) (b : 𝔸) :
differentiable_on 𝕜 (λ y, a y * b) s | λx hx, (ha x hx).mul_const b | lemma | differentiable_on.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.mul_const (ha : differentiable 𝕜 a) (b : 𝔸) :
differentiable 𝕜 (λ y, a y * b) | λx, (ha x).mul_const b | lemma | differentiable.mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_mul_const' (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
fderiv_within 𝕜 (λ y, a y * b) s x = (fderiv_within 𝕜 a s x).smul_right b | (ha.has_fderiv_within_at.mul_const' b).fderiv_within hxs | lemma | fderiv_within_mul_const' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_mul_const (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (d : 𝔸') :
fderiv_within 𝕜 (λ y, c y * d) s x = d • fderiv_within 𝕜 c s x | (hc.has_fderiv_within_at.mul_const d).fderiv_within hxs | lemma | fderiv_within_mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_mul_const' (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
fderiv 𝕜 (λ y, a y * b) x = (fderiv 𝕜 a x).smul_right b | (ha.has_fderiv_at.mul_const' b).fderiv | lemma | fderiv_mul_const' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_mul_const (hc : differentiable_at 𝕜 c x) (d : 𝔸') :
fderiv 𝕜 (λ y, c y * d) x = d • fderiv 𝕜 c x | (hc.has_fderiv_at.mul_const d).fderiv | lemma | fderiv_mul_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_mul (ha : has_strict_fderiv_at a a' x) (b : 𝔸) :
has_strict_fderiv_at (λ y, b * a y) (b • a') x | (((continuous_linear_map.mul 𝕜 𝔸) b).has_strict_fderiv_at).comp x ha | theorem | has_strict_fderiv_at.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_mul
(ha : has_fderiv_within_at a a' s x) (b : 𝔸) :
has_fderiv_within_at (λ y, b * a y) (b • a') s x | (((continuous_linear_map.mul 𝕜 𝔸) b).has_fderiv_at).comp_has_fderiv_within_at x ha | theorem | has_fderiv_within_at.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_mul (ha : has_fderiv_at a a' x) (b : 𝔸) :
has_fderiv_at (λ y, b * a y) (b • a') x | (((continuous_linear_map.mul 𝕜 𝔸) b).has_fderiv_at).comp x ha | theorem | has_fderiv_at.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"continuous_linear_map.mul",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.const_mul
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
differentiable_within_at 𝕜 (λ y, b * a y) s x | (ha.has_fderiv_within_at.const_mul b).differentiable_within_at | lemma | differentiable_within_at.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.const_mul (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
differentiable_at 𝕜 (λ y, b * a y) x | (ha.has_fderiv_at.const_mul b).differentiable_at | lemma | differentiable_at.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.const_mul (ha : differentiable_on 𝕜 a s) (b : 𝔸) :
differentiable_on 𝕜 (λ y, b * a y) s | λx hx, (ha x hx).const_mul b | lemma | differentiable_on.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.const_mul (ha : differentiable 𝕜 a) (b : 𝔸) :
differentiable 𝕜 (λ y, b * a y) | λx, (ha x).const_mul b | lemma | differentiable.const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_const_mul (hxs : unique_diff_within_at 𝕜 s x)
(ha : differentiable_within_at 𝕜 a s x) (b : 𝔸) :
fderiv_within 𝕜 (λ y, b * a y) s x = b • fderiv_within 𝕜 a s x | (ha.has_fderiv_within_at.const_mul b).fderiv_within hxs | lemma | fderiv_within_const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const_mul (ha : differentiable_at 𝕜 a x) (b : 𝔸) :
fderiv 𝕜 (λ y, b * a y) x = b • fderiv 𝕜 a x | (ha.has_fderiv_at.const_mul b).fderiv | lemma | fderiv_const_mul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_ring_inverse (x : Rˣ) :
has_fderiv_at ring.inverse (-mul_left_right 𝕜 R ↑x⁻¹ ↑x⁻¹) x | begin
have h_is_o : (λ (t : R), inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] (λ (t : R), t),
{ refine (inverse_add_norm_diff_second_order x).trans_is_o ((is_o_norm_norm).mp _),
simp only [norm_pow, norm_norm],
have h12 : 1 < 2 := by norm_num,
convert (asymptotics.is_o_pow_pow h12).comp_tendsto te... | lemma | has_fderiv_at_ring_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"asymptotics.is_o_pow_pow",
"has_fderiv_at",
"has_fderiv_at_filter",
"norm_norm",
"norm_pow",
"one_mul",
"ring.inverse",
"units.inv_mul"
] | At an invertible element `x` of a normed algebra `R`, the Fréchet derivative of the inversion
operation is the linear map `λ t, - x⁻¹ * t * x⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at_inverse {x : R} (hx : is_unit x) :
differentiable_at 𝕜 (@ring.inverse R _) x | let ⟨u, hu⟩ := hx in hu ▸ (has_fderiv_at_ring_inverse u).differentiable_at | lemma | differentiable_at_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"has_fderiv_at_ring_inverse",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_inverse {x : R} (hx : is_unit x) (s : set R):
differentiable_within_at 𝕜 (@ring.inverse R _) s x | (differentiable_at_inverse hx).differentiable_within_at | lemma | differentiable_within_at_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at_inverse",
"differentiable_within_at",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_inverse : differentiable_on 𝕜 (@ring.inverse R _) {x | is_unit x} | λ x hx, differentiable_within_at_inverse hx _ | lemma | differentiable_on_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on",
"differentiable_within_at_inverse",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_inverse (x : Rˣ) :
fderiv 𝕜 (@ring.inverse R _) x = - mul_left_right 𝕜 R ↑x⁻¹ ↑x⁻¹ | (has_fderiv_at_ring_inverse x).fderiv | lemma | fderiv_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"fderiv",
"has_fderiv_at_ring_inverse",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.inverse (hf : differentiable_within_at 𝕜 h S z)
(hz : is_unit (h z)) :
differentiable_within_at 𝕜 (λ x, ring.inverse (h x)) S z | (differentiable_at_inverse hz).comp_differentiable_within_at z hf | lemma | differentiable_within_at.inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at_inverse",
"differentiable_within_at",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.inverse (hf : differentiable_at 𝕜 h z) (hz : is_unit (h z)) :
differentiable_at 𝕜 (λ x, ring.inverse (h x)) z | (differentiable_at_inverse hz).comp z hf | lemma | differentiable_at.inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"differentiable_at_inverse",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.inverse (hf : differentiable_on 𝕜 h S) (hz : ∀ x ∈ S, is_unit (h x)) :
differentiable_on 𝕜 (λ x, ring.inverse (h x)) S | λ x h, (hf x h).inverse (hz x h) | lemma | differentiable_on.inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.inverse (hf : differentiable 𝕜 h) (hz : ∀ x, is_unit (h x)) :
differentiable 𝕜 (λ x, ring.inverse (h x)) | λ x, (hf x).inverse (hz x) | lemma | differentiable.inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable",
"is_unit",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_inv' {x : R} (hx : x ≠ 0) :
has_fderiv_at has_inv.inv (-mul_left_right 𝕜 R x⁻¹ x⁻¹) x | by simpa using has_fderiv_at_ring_inverse (units.mk0 _ hx) | lemma | has_fderiv_at_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at",
"has_fderiv_at_ring_inverse",
"units.mk0"
] | At an invertible element `x` of a normed division algebra `R`, the Fréchet derivative of the
inversion operation is the linear map `λ t, - x⁻¹ * t * x⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at_inv' {x : R} (hx : x ≠ 0) : differentiable_at 𝕜 has_inv.inv x | (has_fderiv_at_inv' hx).differentiable_at | lemma | differentiable_at_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"has_fderiv_at_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_inv' {x : R} (hx : x ≠ 0) (s : set R):
differentiable_within_at 𝕜 (λx, x⁻¹) s x | (differentiable_at_inv' hx).differentiable_within_at | lemma | differentiable_within_at_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at_inv'",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_inv' : differentiable_on 𝕜 (λ x : R, x⁻¹) {x | x ≠ 0} | λ x hx, differentiable_within_at_inv' hx _ | lemma | differentiable_on_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on",
"differentiable_within_at_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_inv' {x : R} (hx : x ≠ 0) :
fderiv 𝕜 has_inv.inv x = - mul_left_right 𝕜 R x⁻¹ x⁻¹ | (has_fderiv_at_inv' hx).fderiv | lemma | fderiv_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"fderiv",
"has_fderiv_at_inv'"
] | Non-commutative version of `fderiv_inv` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_within_inv' {s : set R} {x : R} (hx : x ≠ 0) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ x, x⁻¹) s x = - mul_left_right 𝕜 R x⁻¹ x⁻¹ | begin
rw differentiable_at.fderiv_within (differentiable_at_inv' hx) hxs,
exact fderiv_inv' hx
end | lemma | fderiv_within_inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at.fderiv_within",
"differentiable_at_inv'",
"fderiv_inv'",
"fderiv_within",
"unique_diff_within_at"
] | Non-commutative version of `fderiv_within_inv` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_within_at.inv' (hf : differentiable_within_at 𝕜 h S z) (hz : h z ≠ 0) :
differentiable_within_at 𝕜 (λ x, (h x)⁻¹) S z | (differentiable_at_inv' hz).comp_differentiable_within_at z hf | lemma | differentiable_within_at.inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at_inv'",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.inv' (hf : differentiable_at 𝕜 h z) (hz : h z ≠ 0) :
differentiable_at 𝕜 (λ x, (h x)⁻¹) z | (differentiable_at_inv' hz).comp z hf | lemma | differentiable_at.inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at",
"differentiable_at_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.inv' (hf : differentiable_on 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
differentiable_on 𝕜 (λ x, (h x)⁻¹) S | λ x h, (hf x h).inv' (hz x h) | lemma | differentiable_on.inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.inv' (hf : differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
differentiable 𝕜 (λ x, (h x)⁻¹) | λ x, (hf x).inv' (hz x) | lemma | differentiable.inv' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.prod
(hf₁ : has_strict_fderiv_at f₁ f₁' x) (hf₂ : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x | hf₁.prod_left hf₂ | lemma | has_strict_fderiv_at.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.prod
(hf₁ : has_fderiv_at_filter f₁ f₁' x L) (hf₂ : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x L | hf₁.prod_left hf₂ | lemma | has_fderiv_at_filter.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.prod
(hf₁ : has_fderiv_within_at f₁ f₁' s x) (hf₂ : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') s x | hf₁.prod hf₂ | lemma | has_fderiv_within_at.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.prod (hf₁ : has_fderiv_at f₁ f₁' x) (hf₂ : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λx, (f₁ x, f₂ x)) (f₁'.prod f₂') x | hf₁.prod hf₂ | lemma | has_fderiv_at.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_prod_mk_left (e₀ : E) (f₀ : F) :
has_fderiv_at (λ e : E, (e, f₀)) (inl 𝕜 E F) e₀ | (has_fderiv_at_id e₀).prod (has_fderiv_at_const f₀ e₀) | lemma | has_fderiv_at_prod_mk_left | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_const",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_prod_mk_right (e₀ : E) (f₀ : F) :
has_fderiv_at (λ f : F, (e₀, f)) (inr 𝕜 E F) f₀ | (has_fderiv_at_const e₀ f₀).prod (has_fderiv_at_id f₀) | lemma | has_fderiv_at_prod_mk_right | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_const",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.prod
(hf₁ : differentiable_within_at 𝕜 f₁ s x) (hf₂ : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λx:E, (f₁ x, f₂ x)) s x | (hf₁.has_fderiv_within_at.prod hf₂.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.prod (hf₁ : differentiable_at 𝕜 f₁ x) (hf₂ : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λx:E, (f₁ x, f₂ x)) x | (hf₁.has_fderiv_at.prod hf₂.has_fderiv_at).differentiable_at | lemma | differentiable_at.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.prod (hf₁ : differentiable_on 𝕜 f₁ s) (hf₂ : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λx:E, (f₁ x, f₂ x)) s | λx hx, differentiable_within_at.prod (hf₁ x hx) (hf₂ x hx) | lemma | differentiable_on.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"differentiable_within_at.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.prod (hf₁ : differentiable 𝕜 f₁) (hf₂ : differentiable 𝕜 f₂) :
differentiable 𝕜 (λx:E, (f₁ x, f₂ x)) | λ x, differentiable_at.prod (hf₁ x) (hf₂ x) | lemma | differentiable.prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_at.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.fderiv_prod
(hf₁ : differentiable_at 𝕜 f₁ x) (hf₂ : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λx:E, (f₁ x, f₂ x)) x = (fderiv 𝕜 f₁ x).prod (fderiv 𝕜 f₂ x) | (hf₁.has_fderiv_at.prod hf₂.has_fderiv_at).fderiv | lemma | differentiable_at.fderiv_prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.fderiv_within_prod
(hf₁ : differentiable_within_at 𝕜 f₁ s x) (hf₂ : differentiable_within_at 𝕜 f₂ s x)
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λx:E, (f₁ x, f₂ x)) s x =
(fderiv_within 𝕜 f₁ s x).prod (fderiv_within 𝕜 f₂ s x) | (hf₁.has_fderiv_within_at.prod hf₂.has_fderiv_within_at).fderiv_within hxs | lemma | differentiable_within_at.fderiv_within_prod | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_fst : has_strict_fderiv_at (@prod.fst E F) (fst 𝕜 E F) p | (fst 𝕜 E F).has_strict_fderiv_at | lemma | has_strict_fderiv_at_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.fst (h : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x | has_strict_fderiv_at_fst.comp x h | lemma | has_strict_fderiv_at.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_fst {L : filter (E × F)} :
has_fderiv_at_filter (@prod.fst E F) (fst 𝕜 E F) p L | (fst 𝕜 E F).has_fderiv_at_filter | lemma | has_fderiv_at_filter_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"filter",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.fst (h : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x L | has_fderiv_at_filter_fst.comp x h tendsto_map | lemma | has_fderiv_at_filter.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_fst : has_fderiv_at (@prod.fst E F) (fst 𝕜 E F) p | has_fderiv_at_filter_fst | lemma | has_fderiv_at_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.fst (h : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') x | h.fst | lemma | has_fderiv_at.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_fst {s : set (E × F)} :
has_fderiv_within_at (@prod.fst E F) (fst 𝕜 E F) s p | has_fderiv_at_filter_fst | lemma | has_fderiv_within_at_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter_fst",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.fst (h : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λ x, (f₂ x).1) ((fst 𝕜 F G).comp f₂') s x | h.fst | lemma | has_fderiv_within_at.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_fst : differentiable_at 𝕜 prod.fst p | has_fderiv_at_fst.differentiable_at | lemma | differentiable_at_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.fst (h : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λ x, (f₂ x).1) x | differentiable_at_fst.comp x h | lemma | differentiable_at.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_fst : differentiable 𝕜 (prod.fst : E × F → E) | λ x, differentiable_at_fst | lemma | differentiable_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_at_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.fst (h : differentiable 𝕜 f₂) :
differentiable 𝕜 (λ x, (f₂ x).1) | differentiable_fst.comp h | lemma | differentiable.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_fst {s : set (E × F)} : differentiable_within_at 𝕜 prod.fst s p | differentiable_at_fst.differentiable_within_at | lemma | differentiable_within_at_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.fst (h : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λ x, (f₂ x).1) s x | differentiable_at_fst.comp_differentiable_within_at x h | lemma | differentiable_within_at.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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