statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
has_fderiv_at : has_fderiv_at iso (iso : E →L[𝕜] F) x | iso.to_continuous_linear_map.has_fderiv_at_filter | lemma | continuous_linear_equiv.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at : differentiable_at 𝕜 iso x | iso.has_fderiv_at.differentiable_at | lemma | continuous_linear_equiv.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at :
differentiable_within_at 𝕜 iso s x | iso.differentiable_at.differentiable_within_at | lemma | continuous_linear_equiv.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv : fderiv 𝕜 iso x = iso | iso.has_fderiv_at.fderiv | lemma | continuous_linear_equiv.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 iso s x = iso | iso.to_continuous_linear_map.fderiv_within hxs | lemma | continuous_linear_equiv.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"iso",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable : differentiable 𝕜 iso | λx, iso.differentiable_at | lemma | continuous_linear_equiv.differentiable | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on : differentiable_on 𝕜 iso s | iso.differentiable.differentiable_on | lemma | continuous_linear_equiv.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_within_at_iff {f : G → E} {s : set G} {x : G} :
differentiable_within_at 𝕜 (iso ∘ f) s x ↔ differentiable_within_at 𝕜 f s x | begin
refine ⟨λ H, _, λ H, iso.differentiable.differentiable_at.comp_differentiable_within_at x H⟩,
have : differentiable_within_at 𝕜 (iso.symm ∘ (iso ∘ f)) s x :=
iso.symm.differentiable.differentiable_at.comp_differentiable_within_at x H,
rwa [← function.comp.assoc iso.symm iso f, iso.symm_comp_self] at th... | lemma | continuous_linear_equiv.comp_differentiable_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_at_iff {f : G → E} {x : G} :
differentiable_at 𝕜 (iso ∘ f) x ↔ differentiable_at 𝕜 f x | by rw [← differentiable_within_at_univ, ← differentiable_within_at_univ,
iso.comp_differentiable_within_at_iff] | lemma | continuous_linear_equiv.comp_differentiable_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"differentiable_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_on_iff {f : G → E} {s : set G} :
differentiable_on 𝕜 (iso ∘ f) s ↔ differentiable_on 𝕜 f s | begin
rw [differentiable_on, differentiable_on],
simp only [iso.comp_differentiable_within_at_iff],
end | lemma | continuous_linear_equiv.comp_differentiable_on_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_iff {f : G → E} :
differentiable 𝕜 (iso ∘ f) ↔ differentiable 𝕜 f | begin
rw [← differentiable_on_univ, ← differentiable_on_univ],
exact iso.comp_differentiable_on_iff
end | lemma | continuous_linear_equiv.comp_differentiable_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_on_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_within_at_iff
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_within_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ has_fderiv_within_at f f' s x | begin
refine ⟨λ H, _, λ H, iso.has_fderiv_at.comp_has_fderiv_within_at x H⟩,
have A : f = iso.symm ∘ (iso ∘ f), by { rw [← function.comp.assoc, iso.symm_comp_self], refl },
have B : f' = (iso.symm : F →L[𝕜] E).comp ((iso : E →L[𝕜] F).comp f'),
by rw [← continuous_linear_map.comp_assoc, iso.coe_symm_comp_coe... | lemma | continuous_linear_equiv.comp_has_fderiv_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.comp_assoc",
"continuous_linear_map.id_comp",
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_strict_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_strict_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_strict_fderiv_at f f' x | begin
refine ⟨λ H, _, λ H, iso.has_strict_fderiv_at.comp x H⟩,
convert iso.symm.has_strict_fderiv_at.comp x H; ext z; apply (iso.symm_apply_apply _).symm
end | lemma | continuous_linear_equiv.comp_has_strict_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_fderiv_at f f' x | by simp_rw [← has_fderiv_within_at_univ, iso.comp_has_fderiv_within_at_iff] | lemma | continuous_linear_equiv.comp_has_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_within_at_iff'
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_within_at (iso ∘ f) f' s x ↔
has_fderiv_within_at f ((iso.symm : F →L[𝕜] E).comp f') s x | by rw [← iso.comp_has_fderiv_within_at_iff, ← continuous_linear_map.comp_assoc,
iso.coe_comp_coe_symm, continuous_linear_map.id_comp] | lemma | continuous_linear_equiv.comp_has_fderiv_within_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.comp_assoc",
"continuous_linear_map.id_comp",
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_at_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_at (iso ∘ f) f' x ↔ has_fderiv_at f ((iso.symm : F →L[𝕜] E).comp f') x | by simp_rw [← has_fderiv_within_at_univ, iso.comp_has_fderiv_within_at_iff'] | lemma | continuous_linear_equiv.comp_has_fderiv_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_fderiv_within {f : G → E} {s : set G} {x : G}
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderiv_within 𝕜 f s x) | begin
by_cases h : differentiable_within_at 𝕜 f s x,
{ rw [fderiv.comp_fderiv_within x iso.differentiable_at h hxs, iso.fderiv] },
{ have : ¬differentiable_within_at 𝕜 (iso ∘ f) s x,
from mt iso.comp_differentiable_within_at_iff.1 h,
rw [fderiv_within_zero_of_not_differentiable_within_at h,
fd... | lemma | continuous_linear_equiv.comp_fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.comp_zero",
"differentiable_within_at",
"fderiv.comp_fderiv_within",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"iso",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_fderiv {f : G → E} {x : G} :
fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) | begin
rw [← fderiv_within_univ, ← fderiv_within_univ],
exact iso.comp_fderiv_within unique_diff_within_at_univ,
end | lemma | continuous_linear_equiv.comp_fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_within_univ",
"iso",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_differentiable_within_at_iff {f : F → G} {s : set F} {x : E} :
differentiable_within_at 𝕜 (f ∘ iso) (iso ⁻¹' s) x ↔ differentiable_within_at 𝕜 f s (iso x) | begin
refine ⟨λ H, _, λ H, H.comp x iso.differentiable_within_at (maps_to_preimage _ s)⟩,
have : differentiable_within_at 𝕜 ((f ∘ iso) ∘ iso.symm) s (iso x),
{ rw ← iso.symm_apply_apply x at H,
apply H.comp (iso x) iso.symm.differentiable_within_at,
assume y hy,
simpa only [mem_preimage, apply_symm_a... | lemma | continuous_linear_equiv.comp_right_differentiable_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_differentiable_at_iff {f : F → G} {x : E} :
differentiable_at 𝕜 (f ∘ iso) x ↔ differentiable_at 𝕜 f (iso x) | by simp only [← differentiable_within_at_univ, ← iso.comp_right_differentiable_within_at_iff,
preimage_univ] | lemma | continuous_linear_equiv.comp_right_differentiable_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"differentiable_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_differentiable_on_iff {f : F → G} {s : set F} :
differentiable_on 𝕜 (f ∘ iso) (iso ⁻¹' s) ↔ differentiable_on 𝕜 f s | begin
refine ⟨λ H y hy, _, λ H y hy, iso.comp_right_differentiable_within_at_iff.2 (H _ hy)⟩,
rw [← iso.apply_symm_apply y, ← comp_right_differentiable_within_at_iff],
apply H,
simpa only [mem_preimage, apply_symm_apply] using hy,
end | lemma | continuous_linear_equiv.comp_right_differentiable_on_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_differentiable_iff {f : F → G} :
differentiable 𝕜 (f ∘ iso) ↔ differentiable 𝕜 f | by simp only [← differentiable_on_univ, ← iso.comp_right_differentiable_on_iff, preimage_univ] | lemma | continuous_linear_equiv.comp_right_differentiable_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_on_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_has_fderiv_within_at_iff
{f : F → G} {s : set F} {x : E} {f' : F →L[𝕜] G} :
has_fderiv_within_at (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) (iso ⁻¹' s) x ↔
has_fderiv_within_at f f' s (iso x) | begin
refine ⟨λ H, _, λ H, H.comp x iso.has_fderiv_within_at (maps_to_preimage _ s)⟩,
rw [← iso.symm_apply_apply x] at H,
have A : f = (f ∘ iso) ∘ iso.symm, by { rw [function.comp.assoc, iso.self_comp_symm], refl },
have B : f' = (f'.comp (iso : E →L[𝕜] F)).comp (iso.symm : F →L[𝕜] E),
by rw [continuous_... | lemma | continuous_linear_equiv.comp_right_has_fderiv_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.comp_assoc",
"continuous_linear_map.comp_id",
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_has_fderiv_at_iff {f : F → G} {x : E} {f' : F →L[𝕜] G} :
has_fderiv_at (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) x ↔ has_fderiv_at f f' (iso x) | by simp only [← has_fderiv_within_at_univ, ← comp_right_has_fderiv_within_at_iff, preimage_univ] | lemma | continuous_linear_equiv.comp_right_has_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_has_fderiv_within_at_iff'
{f : F → G} {s : set F} {x : E} {f' : E →L[𝕜] G} :
has_fderiv_within_at (f ∘ iso) f' (iso ⁻¹' s) x ↔
has_fderiv_within_at f (f'.comp (iso.symm : F →L[𝕜] E)) s (iso x) | by rw [← iso.comp_right_has_fderiv_within_at_iff, continuous_linear_map.comp_assoc,
iso.coe_symm_comp_coe, continuous_linear_map.comp_id] | lemma | continuous_linear_equiv.comp_right_has_fderiv_within_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.comp_assoc",
"continuous_linear_map.comp_id",
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_has_fderiv_at_iff' {f : F → G} {x : E} {f' : E →L[𝕜] G} :
has_fderiv_at (f ∘ iso) f' x ↔ has_fderiv_at f (f'.comp (iso.symm : F →L[𝕜] E)) (iso x) | by simp only [← has_fderiv_within_at_univ, ← iso.comp_right_has_fderiv_within_at_iff',
preimage_univ] | lemma | continuous_linear_equiv.comp_right_has_fderiv_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_within_at_univ",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_fderiv_within {f : F → G} {s : set F} {x : E}
(hxs : unique_diff_within_at 𝕜 (iso ⁻¹' s) x) :
fderiv_within 𝕜 (f ∘ iso) (iso ⁻¹'s) x = (fderiv_within 𝕜 f s (iso x)).comp (iso : E →L[𝕜] F) | begin
by_cases h : differentiable_within_at 𝕜 f s (iso x),
{ exact (iso.comp_right_has_fderiv_within_at_iff.2 (h.has_fderiv_within_at)).fderiv_within hxs },
{ have : ¬ differentiable_within_at 𝕜 (f ∘ iso) (iso ⁻¹' s) x,
{ assume h', exact h (iso.comp_right_differentiable_within_at_iff.1 h') },
rw [fderi... | lemma | continuous_linear_equiv.comp_right_fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.zero_comp",
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"iso",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_fderiv {f : F → G} {x : E} :
fderiv 𝕜 (f ∘ iso) x = (fderiv 𝕜 f (iso x)).comp (iso : E →L[𝕜] F) | begin
rw [← fderiv_within_univ, ← fderiv_within_univ, ← iso.comp_right_fderiv_within, preimage_univ],
exact unique_diff_within_at_univ,
end | lemma | continuous_linear_equiv.comp_right_fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_within_univ",
"iso",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at : has_strict_fderiv_at iso (iso : E →L[𝕜] F) x | (iso : E ≃L[𝕜] F).has_strict_fderiv_at | lemma | linear_isometry_equiv.has_strict_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at : has_fderiv_within_at iso (iso : E →L[𝕜] F) s x | (iso : E ≃L[𝕜] F).has_fderiv_within_at | lemma | linear_isometry_equiv.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at : has_fderiv_at iso (iso : E →L[𝕜] F) x | (iso : E ≃L[𝕜] F).has_fderiv_at | lemma | linear_isometry_equiv.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv : fderiv 𝕜 iso x = iso | iso.has_fderiv_at.fderiv | lemma | linear_isometry_equiv.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 iso s x = iso | (iso : E ≃L[𝕜] F).fderiv_within hxs | lemma | linear_isometry_equiv.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"iso",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_within_at_iff {f : G → E} {s : set G} {x : G} :
differentiable_within_at 𝕜 (iso ∘ f) s x ↔ differentiable_within_at 𝕜 f s x | (iso : E ≃L[𝕜] F).comp_differentiable_within_at_iff | lemma | linear_isometry_equiv.comp_differentiable_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_at_iff {f : G → E} {x : G} :
differentiable_at 𝕜 (iso ∘ f) x ↔ differentiable_at 𝕜 f x | (iso : E ≃L[𝕜] F).comp_differentiable_at_iff | lemma | linear_isometry_equiv.comp_differentiable_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_on_iff {f : G → E} {s : set G} :
differentiable_on 𝕜 (iso ∘ f) s ↔ differentiable_on 𝕜 f s | (iso : E ≃L[𝕜] F).comp_differentiable_on_iff | lemma | linear_isometry_equiv.comp_differentiable_on_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_differentiable_iff {f : G → E} :
differentiable 𝕜 (iso ∘ f) ↔ differentiable 𝕜 f | (iso : E ≃L[𝕜] F).comp_differentiable_iff | lemma | linear_isometry_equiv.comp_differentiable_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_within_at_iff
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_within_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ has_fderiv_within_at f f' s x | (iso : E ≃L[𝕜] F).comp_has_fderiv_within_at_iff | lemma | linear_isometry_equiv.comp_has_fderiv_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_strict_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_strict_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_strict_fderiv_at f f' x | (iso : E ≃L[𝕜] F).comp_has_strict_fderiv_at_iff | lemma | linear_isometry_equiv.comp_has_strict_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_at_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
has_fderiv_at (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ has_fderiv_at f f' x | (iso : E ≃L[𝕜] F).comp_has_fderiv_at_iff | lemma | linear_isometry_equiv.comp_has_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_within_at_iff'
{f : G → E} {s : set G} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_within_at (iso ∘ f) f' s x ↔
has_fderiv_within_at f ((iso.symm : F →L[𝕜] E).comp f') s x | (iso : E ≃L[𝕜] F).comp_has_fderiv_within_at_iff' | lemma | linear_isometry_equiv.comp_has_fderiv_within_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_has_fderiv_at_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
has_fderiv_at (iso ∘ f) f' x ↔ has_fderiv_at f ((iso.symm : F →L[𝕜] E).comp f') x | (iso : E ≃L[𝕜] F).comp_has_fderiv_at_iff' | lemma | linear_isometry_equiv.comp_has_fderiv_at_iff' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_fderiv_within {f : G → E} {s : set G} {x : G}
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderiv_within 𝕜 f s x) | (iso : E ≃L[𝕜] F).comp_fderiv_within hxs | lemma | linear_isometry_equiv.comp_fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"iso",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_fderiv {f : G → E} {x : G} :
fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) | (iso : E ≃L[𝕜] F).comp_fderiv | lemma | linear_isometry_equiv.comp_fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : continuous_at g a) (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_strict_fderiv_at g (f'.symm : F →L[𝕜] E) a | begin
replace hg := hg.prod_map' hg,
replace hfg := hfg.prod_mk_nhds hfg,
have : (λ p : F × F, g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)]
(λ p : F × F, f' (g p.1 - g p.2) - (p.1 - p.2)),
{ refine ((f'.symm : F →L[𝕜] E).is_O_comp _ _).congr (λ x, _) (λ _, rfl),
simp },
refine this.trans_is_o _... | theorem | has_strict_fderiv_at.of_local_left_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_at",
"has_strict_fderiv_at"
] | If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a` in the strict sense, then `g` has the derivative `f'⁻¹` at `a`
in the strict sense.
This is one of the easy parts of the inverse function theorem: it assumes that we already have an
invers... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : continuous_at g a) (hf : has_fderiv_at f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) :
has_fderiv_at g (f'.symm : F →L[𝕜] E) a | begin
have : (λ x : F, g x - g a - f'.symm (x - a)) =O[𝓝 a] (λ x : F, f' (g x - g a) - (x - a)),
{ refine ((f'.symm : F →L[𝕜] E).is_O_comp _ _).congr (λ x, _) (λ _, rfl),
simp },
refine this.trans_is_o _, clear this,
refine ((hf.comp_tendsto hg).symm.congr' (hfg.mono _)
(eventually_of_forall $ λ _, rf... | theorem | has_fderiv_at.of_local_left_inverse | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_at",
"has_fderiv_at"
] | If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
invertible derivative `f'` at `g a`, then `g` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.has_strict_fderiv_at_symm (f : local_homeomorph E F) {f' : E ≃L[𝕜] F} {a : F}
(ha : a ∈ f.target) (htff' : has_strict_fderiv_at f (f' : E →L[𝕜] F) (f.symm a)) :
has_strict_fderiv_at f.symm (f'.symm : F →L[𝕜] E) a | htff'.of_local_left_inverse (f.symm.continuous_at ha) (f.eventually_right_inverse ha) | lemma | local_homeomorph.has_strict_fderiv_at_symm | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"local_homeomorph"
] | If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has an
invertible derivative `f'` in the sense of strict differentiability at `f.symm a`, then `f.symm` has
the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.has_fderiv_at_symm (f : local_homeomorph E F) {f' : E ≃L[𝕜] F} {a : F}
(ha : a ∈ f.target) (htff' : has_fderiv_at f (f' : E →L[𝕜] F) (f.symm a)) :
has_fderiv_at f.symm (f'.symm : F →L[𝕜] E) a | htff'.of_local_left_inverse (f.symm.continuous_at ha) (f.eventually_right_inverse ha) | lemma | local_homeomorph.has_fderiv_at_symm | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"local_homeomorph"
] | If `f` is a local homeomorphism defined on a neighbourhood of `f.symm a`, and `f` has an
invertible derivative `f'` at `f.symm a`, then `f.symm` has the derivative `f'⁻¹` at `a`.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at.eventually_ne (h : has_fderiv_within_at f f' s x)
(hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) :
∀ᶠ z in 𝓝[s \ {x}] x, f z ≠ f x | begin
rw [nhds_within, diff_eq, ← inf_principal, ← inf_assoc, eventually_inf_principal],
have A : (λ z, z - x) =O[𝓝[s] x] (λ z, f' (z - x)) :=
(is_O_iff.2 $ hf'.imp $ λ C hC, eventually_of_forall $ λ z, hC _),
have : (λ z, f z - f x) ~[𝓝[s] x] (λ z, f' (z - x)) := h.trans_is_O A,
simpa [not_imp_not, sub_e... | lemma | has_fderiv_within_at.eventually_ne | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"inf_assoc",
"nhds_within",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.eventually_ne (h : has_fderiv_at f f' x) (hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) :
∀ᶠ z in 𝓝[≠] x, f z ≠ f x | by simpa only [compl_eq_univ_diff] using (has_fderiv_within_at_univ.2 h).eventually_ne hf' | lemma | has_fderiv_at.eventually_ne | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_real_equiv {L : filter E} :
tendsto (λ x' : E, ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔
tendsto (λ x' : E, ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) | begin
symmetry,
rw [tendsto_iff_norm_tendsto_zero], refine tendsto_congr (λ x', _),
have : ‖x' - x‖⁻¹ ≥ 0, from inv_nonneg.mpr (norm_nonneg _),
simp [norm_smul, abs_of_nonneg this]
end | theorem | has_fderiv_at_filter_real_equiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"abs_of_nonneg",
"filter",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.lim_real (hf : has_fderiv_at f f' x) (v : E) :
tendsto (λ (c:ℝ), c • (f (x + c⁻¹ • v) - f x)) at_top (𝓝 (f' v)) | begin
apply hf.lim v,
rw tendsto_at_top_at_top,
exact λ b, ⟨b, λ a ha, le_trans ha (le_abs_self _)⟩
end | lemma | has_fderiv_at.lim_real | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.maps_to_tangent_cone {x : E} (h : has_fderiv_within_at f f' s x) :
maps_to f' (tangent_cone_at 𝕜 s x) (tangent_cone_at 𝕜 (f '' s) (f x)) | begin
rintros v ⟨c, d, dtop, clim, cdlim⟩,
refine ⟨c, (λn, f (x + d n) - f x), mem_of_superset dtop _, clim,
h.lim at_top dtop clim cdlim⟩,
simp [-mem_image, mem_image_of_mem] {contextual := tt}
end | lemma | has_fderiv_within_at.maps_to_tangent_cone | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"tangent_cone_at"
] | The image of a tangent cone under the differential of a map is included in the tangent cone to
the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at.unique_diff_within_at {x : E} (h : has_fderiv_within_at f f' s x)
(hs : unique_diff_within_at 𝕜 s x) (h' : dense_range f') :
unique_diff_within_at 𝕜 (f '' s) (f x) | begin
refine ⟨h'.dense_of_maps_to f'.continuous hs.1 _,
h.continuous_within_at.mem_closure_image hs.2⟩,
show submodule.span 𝕜 (tangent_cone_at 𝕜 s x) ≤
(submodule.span 𝕜 (tangent_cone_at 𝕜 (f '' s) (f x))).comap f',
rw [submodule.span_le],
exact h.maps_to_tangent_cone.mono (subset.refl _) submodule.... | lemma | has_fderiv_within_at.unique_diff_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"dense_range",
"has_fderiv_within_at",
"submodule.span",
"submodule.span_le",
"submodule.subset_span",
"tangent_cone_at",
"unique_diff_within_at"
] | If a set has the unique differentiability property at a point x, then the image of this set
under a map with onto derivative has also the unique differentiability property at the image point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on.image {f' : E → E →L[𝕜] F} (hs : unique_diff_on 𝕜 s)
(hf' : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hd : ∀ x ∈ s, dense_range (f' x)) :
unique_diff_on 𝕜 (f '' s) | ball_image_iff.2 $ λ x hx, (hf' x hx).unique_diff_within_at (hs x hx) (hd x hx) | lemma | unique_diff_on.image | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"dense_range",
"has_fderiv_within_at",
"unique_diff_on",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.unique_diff_within_at_of_continuous_linear_equiv
{x : E} (e' : E ≃L[𝕜] F) (h : has_fderiv_within_at f (e' : E →L[𝕜] F) s x)
(hs : unique_diff_within_at 𝕜 s x) :
unique_diff_within_at 𝕜 (f '' s) (f x) | h.unique_diff_within_at hs e'.surjective.dense_range | lemma | has_fderiv_within_at.unique_diff_within_at_of_continuous_linear_equiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.unique_diff_on_image (e : E ≃L[𝕜] F) (h : unique_diff_on 𝕜 s) :
unique_diff_on 𝕜 (e '' s) | h.image (λ x _, e.has_fderiv_within_at) (λ x hx, e.surjective.dense_range) | lemma | continuous_linear_equiv.unique_diff_on_image | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.unique_diff_on_image_iff (e : E ≃L[𝕜] F) :
unique_diff_on 𝕜 (e '' s) ↔ unique_diff_on 𝕜 s | ⟨λ h, e.symm_image_image s ▸ e.symm.unique_diff_on_image h, e.unique_diff_on_image⟩ | lemma | continuous_linear_equiv.unique_diff_on_image_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.unique_diff_on_preimage_iff (e : F ≃L[𝕜] E) :
unique_diff_on 𝕜 (e ⁻¹' s) ↔ unique_diff_on 𝕜 s | by rw [← e.image_symm_eq_preimage, e.symm.unique_diff_on_image_iff] | lemma | continuous_linear_equiv.unique_diff_on_preimage_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_strict_fderiv_at {x : E} :
has_strict_fderiv_at e e x | (is_o_zero _ _).congr_left $ λ x, by simp only [e.map_sub, sub_self] | theorem | continuous_linear_map.has_strict_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_fderiv_at_filter :
has_fderiv_at_filter e e x L | (is_o_zero _ _).congr_left $ λ x, by simp only [e.map_sub, sub_self] | lemma | continuous_linear_map.has_fderiv_at_filter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_fderiv_within_at : has_fderiv_within_at e e s x | e.has_fderiv_at_filter | lemma | continuous_linear_map.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_fderiv_at : has_fderiv_at e e x | e.has_fderiv_at_filter | lemma | continuous_linear_map.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.differentiable_at : differentiable_at 𝕜 e x | e.has_fderiv_at.differentiable_at | lemma | continuous_linear_map.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.differentiable_within_at : differentiable_within_at 𝕜 e s x | e.differentiable_at.differentiable_within_at | lemma | continuous_linear_map.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.fderiv : fderiv 𝕜 e x = e | e.has_fderiv_at.fderiv | lemma | continuous_linear_map.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 e s x = e | begin
rw differentiable_at.fderiv_within e.differentiable_at hxs,
exact e.fderiv
end | lemma | continuous_linear_map.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at.fderiv_within",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.differentiable : differentiable 𝕜 e | λx, e.differentiable_at | lemma | continuous_linear_map.differentiable | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.differentiable_on : differentiable_on 𝕜 e s | e.differentiable.differentiable_on | lemma | continuous_linear_map.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.has_fderiv_at_filter (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_at_filter f h.to_continuous_linear_map x L | h.to_continuous_linear_map.has_fderiv_at_filter | lemma | is_bounded_linear_map.has_fderiv_at_filter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at_filter",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.has_fderiv_within_at (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_within_at f h.to_continuous_linear_map s x | h.has_fderiv_at_filter | lemma | is_bounded_linear_map.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.has_fderiv_at (h : is_bounded_linear_map 𝕜 f) :
has_fderiv_at f h.to_continuous_linear_map x | h.has_fderiv_at_filter | lemma | is_bounded_linear_map.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.differentiable_at (h : is_bounded_linear_map 𝕜 f) :
differentiable_at 𝕜 f x | h.has_fderiv_at.differentiable_at | lemma | is_bounded_linear_map.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.differentiable_within_at (h : is_bounded_linear_map 𝕜 f) :
differentiable_within_at 𝕜 f s x | h.differentiable_at.differentiable_within_at | lemma | is_bounded_linear_map.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.fderiv (h : is_bounded_linear_map 𝕜 f) :
fderiv 𝕜 f x = h.to_continuous_linear_map | has_fderiv_at.fderiv (h.has_fderiv_at) | lemma | is_bounded_linear_map.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"fderiv",
"has_fderiv_at.fderiv",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.fderiv_within (h : is_bounded_linear_map 𝕜 f)
(hxs : unique_diff_within_at 𝕜 s x) : fderiv_within 𝕜 f s x = h.to_continuous_linear_map | begin
rw differentiable_at.fderiv_within h.differentiable_at hxs,
exact h.fderiv
end | lemma | is_bounded_linear_map.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at.fderiv_within",
"fderiv_within",
"is_bounded_linear_map",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.differentiable (h : is_bounded_linear_map 𝕜 f) :
differentiable 𝕜 f | λx, h.differentiable_at | lemma | is_bounded_linear_map.differentiable | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map.differentiable_on (h : is_bounded_linear_map 𝕜 f) :
differentiable_on 𝕜 f s | h.differentiable.differentiable_on | lemma | is_bounded_linear_map.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/linear.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_on",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.clm_comp (hc : has_strict_fderiv_at c c' x)
(hd : has_strict_fderiv_at d d' x) : has_strict_fderiv_at (λ y, (c y).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x | (is_bounded_bilinear_map_comp.has_strict_fderiv_at (c x, d x)).comp x $ hc.prod hd | lemma | has_strict_fderiv_at.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.clm_comp (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).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') s x | (is_bounded_bilinear_map_comp.has_fderiv_at (c x, d x)).comp_has_fderiv_within_at x $ hc.prod hd | lemma | has_fderiv_within_at.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.clm_comp (hc : has_fderiv_at c c' x)
(hd : has_fderiv_at d d' x) : has_fderiv_at (λ y, (c y).comp (d y))
((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x | (is_bounded_bilinear_map_comp.has_fderiv_at (c x, d x)).comp x $ hc.prod hd | lemma | has_fderiv_at.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.clm_comp
(hc : differentiable_within_at 𝕜 c s x) (hd : differentiable_within_at 𝕜 d s x) :
differentiable_within_at 𝕜 (λ y, (c y).comp (d y)) s x | (hc.has_fderiv_within_at.clm_comp hd.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.clm_comp | 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.clm_comp (hc : differentiable_at 𝕜 c x)
(hd : differentiable_at 𝕜 d x) : differentiable_at 𝕜 (λ y, (c y).comp (d y)) x | (hc.has_fderiv_at.clm_comp hd.has_fderiv_at).differentiable_at | lemma | differentiable_at.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.clm_comp (hc : differentiable_on 𝕜 c s) (hd : differentiable_on 𝕜 d s) :
differentiable_on 𝕜 (λ y, (c y).comp (d y)) s | λx hx, (hc x hx).clm_comp (hd x hx) | lemma | differentiable_on.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.clm_comp (hc : differentiable 𝕜 c) (hd : differentiable 𝕜 d) :
differentiable 𝕜 (λ y, (c y).comp (d y)) | λx, (hc x).clm_comp (hd x) | lemma | differentiable.clm_comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_clm_comp (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).comp (d y)) s x =
(compL 𝕜 F G H (c x)).comp (fderiv_within 𝕜 d s x) +
((compL 𝕜 F G H).flip (d x)).comp (fderiv_within 𝕜 c s x) | (hc.has_fderiv_within_at.clm_comp hd.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_clm_comp | 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_clm_comp (hc : differentiable_at 𝕜 c x) (hd : differentiable_at 𝕜 d x) :
fderiv 𝕜 (λ y, (c y).comp (d y)) x =
(compL 𝕜 F G H (c x)).comp (fderiv 𝕜 d x) +
((compL 𝕜 F G H).flip (d x)).comp (fderiv 𝕜 c x) | (hc.has_fderiv_at.clm_comp hd.has_fderiv_at).fderiv | lemma | fderiv_clm_comp | 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.clm_apply (hc : has_strict_fderiv_at c c' x)
(hu : has_strict_fderiv_at u u' x) :
has_strict_fderiv_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) x | (is_bounded_bilinear_map_apply.has_strict_fderiv_at (c x, u x)).comp x (hc.prod hu) | lemma | has_strict_fderiv_at.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.clm_apply (hc : has_fderiv_within_at c c' s x)
(hu : has_fderiv_within_at u u' s x) :
has_fderiv_within_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) s x | (is_bounded_bilinear_map_apply.has_fderiv_at (c x, u x)).comp_has_fderiv_within_at x (hc.prod hu) | lemma | has_fderiv_within_at.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.clm_apply (hc : has_fderiv_at c c' x) (hu : has_fderiv_at u u' x) :
has_fderiv_at (λ y, (c y) (u y)) ((c x).comp u' + c'.flip (u x)) x | (is_bounded_bilinear_map_apply.has_fderiv_at (c x, u x)).comp x (hc.prod hu) | lemma | has_fderiv_at.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.clm_apply
(hc : differentiable_within_at 𝕜 c s x) (hu : differentiable_within_at 𝕜 u s x) :
differentiable_within_at 𝕜 (λ y, (c y) (u y)) s x | (hc.has_fderiv_within_at.clm_apply hu.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.clm_apply | 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.clm_apply (hc : differentiable_at 𝕜 c x)
(hu : differentiable_at 𝕜 u x) : differentiable_at 𝕜 (λ y, (c y) (u y)) x | (hc.has_fderiv_at.clm_apply hu.has_fderiv_at).differentiable_at | lemma | differentiable_at.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.clm_apply (hc : differentiable_on 𝕜 c s) (hu : differentiable_on 𝕜 u s) :
differentiable_on 𝕜 (λ y, (c y) (u y)) s | λx hx, (hc x hx).clm_apply (hu x hx) | lemma | differentiable_on.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.clm_apply (hc : differentiable 𝕜 c) (hu : differentiable 𝕜 u) :
differentiable 𝕜 (λ y, (c y) (u y)) | λx, (hc x).clm_apply (hu x) | lemma | differentiable.clm_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_clm_apply (hxs : unique_diff_within_at 𝕜 s x)
(hc : differentiable_within_at 𝕜 c s x) (hu : differentiable_within_at 𝕜 u s x) :
fderiv_within 𝕜 (λ y, (c y) (u y)) s x =
((c x).comp (fderiv_within 𝕜 u s x) + (fderiv_within 𝕜 c s x).flip (u x)) | (hc.has_fderiv_within_at.clm_apply hu.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_clm_apply | 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_clm_apply (hc : differentiable_at 𝕜 c x) (hu : differentiable_at 𝕜 u x) :
fderiv 𝕜 (λ y, (c y) (u y)) x = ((c x).comp (fderiv 𝕜 u x) + (fderiv 𝕜 c x).flip (u x)) | (hc.has_fderiv_at.clm_apply hu.has_fderiv_at).fderiv | lemma | fderiv_clm_apply | 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 (hc : has_strict_fderiv_at c c' x)
(hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) x | (is_bounded_bilinear_map_smul.has_strict_fderiv_at (c x, f x)).comp x $
hc.prod hf | theorem | has_strict_fderiv_at.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.smul
(hc : has_fderiv_within_at c c' s x) (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) s x | (is_bounded_bilinear_map_smul.has_fderiv_at (c x, f x)).comp_has_fderiv_within_at x $
hc.prod hf | theorem | has_fderiv_within_at.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.smul (hc : has_fderiv_at c c' x) (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ y, c y • f y) (c x • f' + c'.smul_right (f x)) x | (is_bounded_bilinear_map_smul.has_fderiv_at (c x, f x)).comp x $
hc.prod hf | theorem | has_fderiv_at.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.smul
(hc : differentiable_within_at 𝕜 c s x) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λ y, c y • f y) s x | (hc.has_fderiv_within_at.smul hf.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/mul.lean | [
"analysis.calculus.fderiv.bilinear"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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