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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|---|---|---|---|---|---|---|---|---|---|---|
differentiable_on_fst {s : set (E × F)} : differentiable_on 𝕜 prod.fst s | differentiable_fst.differentiable_on | lemma | differentiable_on_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.fst (h : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λ x, (f₂ x).1) s | differentiable_fst.comp_differentiable_on h | lemma | differentiable_on.fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_fst : fderiv 𝕜 prod.fst p = fst 𝕜 E F | has_fderiv_at_fst.fderiv | lemma | fderiv_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv.fst (h : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λ x, (f₂ x).1) x = (fst 𝕜 F G).comp (fderiv 𝕜 f₂ x) | h.has_fderiv_at.fst.fderiv | lemma | fderiv.fst | 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 | |
fderiv_within_fst {s : set (E × F)} (hs : unique_diff_within_at 𝕜 s p) :
fderiv_within 𝕜 prod.fst s p = fst 𝕜 E F | has_fderiv_within_at_fst.fderiv_within hs | lemma | fderiv_within_fst | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within.fst (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_within_at 𝕜 f₂ s x) :
fderiv_within 𝕜 (λ x, (f₂ x).1) s x = (fst 𝕜 F G).comp (fderiv_within 𝕜 f₂ s x) | h.has_fderiv_within_at.fst.fderiv_within hs | lemma | fderiv_within.fst | 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_snd : has_strict_fderiv_at (@prod.snd E F) (snd 𝕜 E F) p | (snd 𝕜 E F).has_strict_fderiv_at | lemma | has_strict_fderiv_at_snd | 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.snd (h : has_strict_fderiv_at f₂ f₂' x) :
has_strict_fderiv_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x | has_strict_fderiv_at_snd.comp x h | lemma | has_strict_fderiv_at.snd | 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_snd {L : filter (E × F)} :
has_fderiv_at_filter (@prod.snd E F) (snd 𝕜 E F) p L | (snd 𝕜 E F).has_fderiv_at_filter | lemma | has_fderiv_at_filter_snd | 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.snd (h : has_fderiv_at_filter f₂ f₂' x L) :
has_fderiv_at_filter (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x L | has_fderiv_at_filter_snd.comp x h tendsto_map | lemma | has_fderiv_at_filter.snd | 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_snd : has_fderiv_at (@prod.snd E F) (snd 𝕜 E F) p | has_fderiv_at_filter_snd | lemma | has_fderiv_at_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.snd (h : has_fderiv_at f₂ f₂' x) :
has_fderiv_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') x | h.snd | lemma | has_fderiv_at.snd | 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_snd {s : set (E × F)} :
has_fderiv_within_at (@prod.snd E F) (snd 𝕜 E F) s p | has_fderiv_at_filter_snd | lemma | has_fderiv_within_at_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter_snd",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.snd (h : has_fderiv_within_at f₂ f₂' s x) :
has_fderiv_within_at (λ x, (f₂ x).2) ((snd 𝕜 F G).comp f₂') s x | h.snd | lemma | has_fderiv_within_at.snd | 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_snd : differentiable_at 𝕜 prod.snd p | has_fderiv_at_snd.differentiable_at | lemma | differentiable_at_snd | 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.snd (h : differentiable_at 𝕜 f₂ x) :
differentiable_at 𝕜 (λ x, (f₂ x).2) x | differentiable_at_snd.comp x h | lemma | differentiable_at.snd | 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_snd : differentiable 𝕜 (prod.snd : E × F → F) | λ x, differentiable_at_snd | lemma | differentiable_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.snd (h : differentiable 𝕜 f₂) :
differentiable 𝕜 (λ x, (f₂ x).2) | differentiable_snd.comp h | lemma | differentiable.snd | 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_snd {s : set (E × F)} : differentiable_within_at 𝕜 prod.snd s p | differentiable_at_snd.differentiable_within_at | lemma | differentiable_within_at_snd | 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.snd (h : differentiable_within_at 𝕜 f₂ s x) :
differentiable_within_at 𝕜 (λ x, (f₂ x).2) s x | differentiable_at_snd.comp_differentiable_within_at x h | lemma | differentiable_within_at.snd | 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_on_snd {s : set (E × F)} : differentiable_on 𝕜 prod.snd s | differentiable_snd.differentiable_on | lemma | differentiable_on_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.snd (h : differentiable_on 𝕜 f₂ s) :
differentiable_on 𝕜 (λ x, (f₂ x).2) s | differentiable_snd.comp_differentiable_on h | lemma | differentiable_on.snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_snd : fderiv 𝕜 prod.snd p = snd 𝕜 E F | has_fderiv_at_snd.fderiv | lemma | fderiv_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv.snd (h : differentiable_at 𝕜 f₂ x) :
fderiv 𝕜 (λ x, (f₂ x).2) x = (snd 𝕜 F G).comp (fderiv 𝕜 f₂ x) | h.has_fderiv_at.snd.fderiv | lemma | fderiv.snd | 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 | |
fderiv_within_snd {s : set (E × F)} (hs : unique_diff_within_at 𝕜 s p) :
fderiv_within 𝕜 prod.snd s p = snd 𝕜 E F | has_fderiv_within_at_snd.fderiv_within hs | lemma | fderiv_within_snd | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within.snd (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_within_at 𝕜 f₂ s x) :
fderiv_within 𝕜 (λ x, (f₂ x).2) s x = (snd 𝕜 F G).comp (fderiv_within 𝕜 f₂ s x) | h.has_fderiv_within_at.snd.fderiv_within hs | lemma | fderiv_within.snd | 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.prod_map (hf : has_strict_fderiv_at f f' p.1)
(hf₂ : has_strict_fderiv_at f₂ f₂' p.2) :
has_strict_fderiv_at (prod.map f f₂) (f'.prod_map f₂') p | (hf.comp p has_strict_fderiv_at_fst).prod (hf₂.comp p has_strict_fderiv_at_snd) | theorem | has_strict_fderiv_at.prod_map | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at_fst",
"has_strict_fderiv_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.prod_map (hf : has_fderiv_at f f' p.1)
(hf₂ : has_fderiv_at f₂ f₂' p.2) :
has_fderiv_at (prod.map f f₂) (f'.prod_map f₂') p | (hf.comp p has_fderiv_at_fst).prod (hf₂.comp p has_fderiv_at_snd) | theorem | has_fderiv_at.prod_map | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_fst",
"has_fderiv_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.prod_map (hf : differentiable_at 𝕜 f p.1)
(hf₂ : differentiable_at 𝕜 f₂ p.2) :
differentiable_at 𝕜 (λ p : E × G, (f p.1, f₂ p.2)) p | (hf.comp p differentiable_at_fst).prod (hf₂.comp p differentiable_at_snd) | theorem | differentiable_at.prod_map | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"differentiable_at_fst",
"differentiable_at_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_pi' :
has_strict_fderiv_at Φ Φ' x ↔
∀ i, has_strict_fderiv_at (λ x, Φ x i) ((proj i).comp Φ') x | begin
simp only [has_strict_fderiv_at, continuous_linear_map.coe_pi],
exact is_o_pi
end | lemma | has_strict_fderiv_at_pi' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.coe_pi",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_pi :
has_strict_fderiv_at (λ x i, φ i x) (continuous_linear_map.pi φ') x ↔
∀ i, has_strict_fderiv_at (φ i) (φ' i) x | has_strict_fderiv_at_pi' | lemma | has_strict_fderiv_at_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.pi",
"has_strict_fderiv_at",
"has_strict_fderiv_at_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_pi' :
has_fderiv_at_filter Φ Φ' x L ↔
∀ i, has_fderiv_at_filter (λ x, Φ x i) ((proj i).comp Φ') x L | begin
simp only [has_fderiv_at_filter, continuous_linear_map.coe_pi],
exact is_o_pi
end | lemma | has_fderiv_at_filter_pi' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.coe_pi",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_pi :
has_fderiv_at_filter (λ x i, φ i x) (continuous_linear_map.pi φ') x L ↔
∀ i, has_fderiv_at_filter (φ i) (φ' i) x L | has_fderiv_at_filter_pi' | lemma | has_fderiv_at_filter_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.pi",
"has_fderiv_at_filter",
"has_fderiv_at_filter_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_pi' :
has_fderiv_at Φ Φ' x ↔
∀ i, has_fderiv_at (λ x, Φ x i) ((proj i).comp Φ') x | has_fderiv_at_filter_pi' | lemma | has_fderiv_at_pi' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_pi :
has_fderiv_at (λ x i, φ i x) (continuous_linear_map.pi φ') x ↔
∀ i, has_fderiv_at (φ i) (φ' i) x | has_fderiv_at_filter_pi | lemma | has_fderiv_at_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.pi",
"has_fderiv_at",
"has_fderiv_at_filter_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_pi' :
has_fderiv_within_at Φ Φ' s x ↔
∀ i, has_fderiv_within_at (λ x, Φ x i) ((proj i).comp Φ') s x | has_fderiv_at_filter_pi' | lemma | has_fderiv_within_at_pi' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter_pi'",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_pi :
has_fderiv_within_at (λ x i, φ i x) (continuous_linear_map.pi φ') s x ↔
∀ i, has_fderiv_within_at (φ i) (φ' i) s x | has_fderiv_at_filter_pi | lemma | has_fderiv_within_at_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.pi",
"has_fderiv_at_filter_pi",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_pi :
differentiable_within_at 𝕜 Φ s x ↔
∀ i, differentiable_within_at 𝕜 (λ x, Φ x i) s x | ⟨λ h i, (has_fderiv_within_at_pi'.1 h.has_fderiv_within_at i).differentiable_within_at,
λ h, (has_fderiv_within_at_pi.2 (λ i, (h i).has_fderiv_within_at)).differentiable_within_at⟩ | lemma | differentiable_within_at_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_pi :
differentiable_at 𝕜 Φ x ↔ ∀ i, differentiable_at 𝕜 (λ x, Φ x i) x | ⟨λ h i, (has_fderiv_at_pi'.1 h.has_fderiv_at i).differentiable_at,
λ h, (has_fderiv_at_pi.2 (λ i, (h i).has_fderiv_at)).differentiable_at⟩ | lemma | differentiable_at_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_pi :
differentiable_on 𝕜 Φ s ↔ ∀ i, differentiable_on 𝕜 (λ x, Φ x i) s | ⟨λ h i x hx, differentiable_within_at_pi.1 (h x hx) i,
λ h x hx, differentiable_within_at_pi.2 (λ i, h i x hx)⟩ | lemma | differentiable_on_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_pi :
differentiable 𝕜 Φ ↔ ∀ i, differentiable 𝕜 (λ x, Φ x i) | ⟨λ h i x, differentiable_at_pi.1 (h x) i, λ h x, differentiable_at_pi.2 (λ i, h i x)⟩ | lemma | differentiable_pi | 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 | |
fderiv_within_pi (h : ∀ i, differentiable_within_at 𝕜 (φ i) s x)
(hs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ x i, φ i x) s x = pi (λ i, fderiv_within 𝕜 (φ i) s x) | (has_fderiv_within_at_pi.2 (λ i, (h i).has_fderiv_within_at)).fderiv_within hs | lemma | fderiv_within_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_pi (h : ∀ i, differentiable_at 𝕜 (φ i) x) :
fderiv 𝕜 (λ x i, φ i x) x = pi (λ i, fderiv 𝕜 (φ i) x) | (has_fderiv_at_pi.2 (λ i, (h i).has_fderiv_at)).fderiv | lemma | fderiv_pi | analysis.calculus.fderiv | src/analysis/calculus/fderiv/prod.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.restrict_scalars (h : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at f (f'.restrict_scalars 𝕜) x | h | lemma | has_strict_fderiv_at.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.restrict_scalars {L} (h : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter f (f'.restrict_scalars 𝕜) x L | h | lemma | has_fderiv_at_filter.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.restrict_scalars (h : has_fderiv_at f f' x) :
has_fderiv_at f (f'.restrict_scalars 𝕜) x | h | lemma | has_fderiv_at.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.restrict_scalars (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at f (f'.restrict_scalars 𝕜) s x | h | lemma | has_fderiv_within_at.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.restrict_scalars (h : differentiable_at 𝕜' f x) :
differentiable_at 𝕜 f x | (h.has_fderiv_at.restrict_scalars 𝕜).differentiable_at | lemma | differentiable_at.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.restrict_scalars (h : differentiable_within_at 𝕜' f s x) :
differentiable_within_at 𝕜 f s x | (h.has_fderiv_within_at.restrict_scalars 𝕜).differentiable_within_at | lemma | differentiable_within_at.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.restrict_scalars (h : differentiable_on 𝕜' f s) :
differentiable_on 𝕜 f s | λx hx, (h x hx).restrict_scalars 𝕜 | lemma | differentiable_on.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_on",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.restrict_scalars (h : differentiable 𝕜' f) :
differentiable 𝕜 f | λx, (h x).restrict_scalars 𝕜 | lemma | differentiable.restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_of_restrict_scalars
{g' : E →L[𝕜] F} (h : has_fderiv_within_at f g' s x)
(H : f'.restrict_scalars 𝕜 = g') : has_fderiv_within_at f f' s x | by { rw ← H at h, exact h } | lemma | has_fderiv_within_at_of_restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_of_restrict_scalars {g' : E →L[𝕜] F} (h : has_fderiv_at f g' x)
(H : f'.restrict_scalars 𝕜 = g') : has_fderiv_at f f' x | by { rw ← H at h, exact h } | lemma | has_fderiv_at_of_restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.fderiv_restrict_scalars (h : differentiable_at 𝕜' f x) :
fderiv 𝕜 f x = (fderiv 𝕜' f x).restrict_scalars 𝕜 | (h.has_fderiv_at.restrict_scalars 𝕜).fderiv | lemma | differentiable_at.fderiv_restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"fderiv",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_iff_restrict_scalars
(hf : differentiable_within_at 𝕜 f s x) (hs : unique_diff_within_at 𝕜 s x) :
differentiable_within_at 𝕜' f s x ↔
∃ (g' : E →L[𝕜'] F), g'.restrict_scalars 𝕜 = fderiv_within 𝕜 f s x | begin
split,
{ rintros ⟨g', hg'⟩,
exact ⟨g', hs.eq (hg'.restrict_scalars 𝕜) hf.has_fderiv_within_at⟩, },
{ rintros ⟨f', hf'⟩,
exact ⟨f', has_fderiv_within_at_of_restrict_scalars 𝕜 hf.has_fderiv_within_at hf'⟩, },
end | lemma | differentiable_within_at_iff_restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at_of_restrict_scalars",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_iff_restrict_scalars (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜' f x ↔ ∃ (g' : E →L[𝕜'] F), g'.restrict_scalars 𝕜 = fderiv 𝕜 f x | begin
rw [← differentiable_within_at_univ, ← fderiv_within_univ],
exact differentiable_within_at_iff_restrict_scalars 𝕜
hf.differentiable_within_at unique_diff_within_at_univ,
end | lemma | differentiable_at_iff_restrict_scalars | analysis.calculus.fderiv | src/analysis/calculus/fderiv/restrict_scalars.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"differentiable_within_at_iff_restrict_scalars",
"differentiable_within_at_univ",
"fderiv",
"fderiv_within_univ",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.star (h : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x | (starL' 𝕜 : F ≃L[𝕜] F).to_continuous_linear_map.has_strict_fderiv_at.comp x h | theorem | has_strict_fderiv_at.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"has_strict_fderiv_at",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.star (h : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter (λ x, star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x L | (starL' 𝕜 : F ≃L[𝕜] F).to_continuous_linear_map.has_fderiv_at_filter.comp x h filter.tendsto_map | theorem | has_fderiv_at_filter.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"filter.tendsto_map",
"has_fderiv_at_filter",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.star (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') s x | h.star | theorem | has_fderiv_within_at.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"has_fderiv_within_at",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.star (h : has_fderiv_at f f' x) :
has_fderiv_at (λ x, star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x | h.star | theorem | has_fderiv_at.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"has_fderiv_at",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.star (h : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λ y, star (f y)) s x | h.has_fderiv_within_at.star.differentiable_within_at | lemma | differentiable_within_at.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_star_iff :
differentiable_within_at 𝕜 (λ y, star (f y)) s x ↔ differentiable_within_at 𝕜 f s x | (starL' 𝕜 : F ≃L[𝕜] F).comp_differentiable_within_at_iff | lemma | differentiable_within_at_star_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_within_at",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.star (h : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λ y, star (f y)) x | h.has_fderiv_at.star.differentiable_at | lemma | differentiable_at.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_star_iff :
differentiable_at 𝕜 (λ y, star (f y)) x ↔ differentiable_at 𝕜 f x | (starL' 𝕜 : F ≃L[𝕜] F).comp_differentiable_at_iff | lemma | differentiable_at_star_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_at",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.star (h : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λ y, star (f y)) s | λ x hx, (h x hx).star | lemma | differentiable_on.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_star_iff :
differentiable_on 𝕜 (λ y, star (f y)) s ↔ differentiable_on 𝕜 f s | (starL' 𝕜 : F ≃L[𝕜] F).comp_differentiable_on_iff | lemma | differentiable_on_star_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable_on",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.star (h : differentiable 𝕜 f) :
differentiable 𝕜 (λ y, star (f y)) | λx, (h x).star | lemma | differentiable.star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_star_iff : differentiable 𝕜 (λ y, star (f y)) ↔ differentiable 𝕜 f | (starL' 𝕜 : F ≃L[𝕜] F).comp_differentiable_iff | lemma | differentiable_star_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"differentiable",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_star (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ y, star (f y)) s x =
((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L fderiv_within 𝕜 f s x | (starL' 𝕜 : F ≃L[𝕜] F).comp_fderiv_within hxs | lemma | fderiv_within_star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"fderiv_within",
"starL'",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_star :
fderiv 𝕜 (λ y, star (f y)) x = ((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L fderiv 𝕜 f x | (starL' 𝕜 : F ≃L[𝕜] F).comp_fderiv | lemma | fderiv_star | analysis.calculus.fderiv | src/analysis/calculus/fderiv/star.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.equiv",
"analysis.normed_space.star.basic"
] | [
"fderiv",
"starL'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_max_aux₁ [complete_space F] {f : ℂ → F} {z w : ℂ}
(hd : diff_cont_on_cl ℂ f (ball z (dist w z)))
(hz : is_max_on (norm ∘ f) (closed_ball z (dist w z)) z) :
‖f w‖ = ‖f z‖ | begin
/- Consider a circle of radius `r = dist w z`. -/
set r : ℝ := dist w z,
have hw : w ∈ closed_ball z r, from mem_closed_ball.2 le_rfl,
/- Assume the converse. Since `‖f w‖ ≤ ‖f z‖`, we have `‖f w‖ < ‖f z‖`. -/
refine (is_max_on_iff.1 hz _ hw).antisymm (not_lt.1 _),
rintro hw_lt : ‖f w‖ < ‖f z‖,
have... | lemma | complex.norm_max_aux₁ | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"circle_integral.norm_integral_lt_of_norm_le_const_of_lt",
"complete_space",
"continuous_on",
"continuous_on_const",
"diff_cont_on_cl",
"div_eq_inv_mul",
"div_lt_div_right",
"is_max_on",
"le_div_iff",
"le_rfl",
"mul_assoc",
"mul_comm",
"mul_div_cancel'",
"mul_inv_cancel_left₀",
"ne_of_ap... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_max_aux₂ {f : ℂ → F} {z w : ℂ} (hd : diff_cont_on_cl ℂ f (ball z (dist w z)))
(hz : is_max_on (norm ∘ f) (closed_ball z (dist w z)) z) :
‖f w‖ = ‖f z‖ | begin
set e : F →L[ℂ] F̂ := uniform_space.completion.to_complL,
have he : ∀ x, ‖e x‖ = ‖x‖, from uniform_space.completion.norm_coe,
replace hz : is_max_on (norm ∘ (e ∘ f)) (closed_ball z (dist w z)) z,
by simpa only [is_max_on, (∘), he] using hz,
simpa only [he] using norm_max_aux₁ (e.differentiable.comp_di... | lemma | complex.norm_max_aux₂ | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"diff_cont_on_cl",
"is_max_on",
"uniform_space.completion.norm_coe",
"uniform_space.completion.to_complL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_max_aux₃ {f : ℂ → F} {z w : ℂ} {r : ℝ} (hr : dist w z = r)
(hd : diff_cont_on_cl ℂ f (ball z r)) (hz : is_max_on (norm ∘ f) (ball z r) z) :
‖f w‖ = ‖f z‖ | begin
subst r,
rcases eq_or_ne w z with rfl|hne, { refl },
rw ← dist_ne_zero at hne,
exact norm_max_aux₂ hd (closure_ball z hne ▸ hz.closure hd.continuous_on.norm)
end | lemma | complex.norm_max_aux₃ | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"closure_ball",
"diff_cont_on_cl",
"dist_ne_zero",
"eq_or_ne",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_on_closed_ball_of_is_max_on {f : E → F} {z : E} {r : ℝ}
(hd : diff_cont_on_cl ℂ f (ball z r)) (hz : is_max_on (norm ∘ f) (ball z r) z) :
eq_on (norm ∘ f) (const E ‖f z‖) (closed_ball z r) | begin
intros w hw,
rw [mem_closed_ball, dist_comm] at hw,
rcases eq_or_ne z w with rfl|hne, { refl },
set e : ℂ → E := line_map z w,
have hde : differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z,
suffices : ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖, by simpa [e],
have hr : dist (1 : ℂ) 0... | lemma | complex.norm_eq_on_closed_ball_of_is_max_on | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"coe_nndist",
"diff_cont_on_cl",
"differentiable",
"dist_comm",
"eq_or_ne",
"is_max_on",
"lipschitz_with_line_map",
"mul_one"
] | **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_norm_of_is_max_on_of_ball_subset {f : E → F} {s : set E} {z w : E}
(hd : diff_cont_on_cl ℂ f s) (hz : is_max_on (norm ∘ f) s z) (hsub : ball z (dist w z) ⊆ s) :
‖f w‖ = ‖f z‖ | norm_eq_on_closed_ball_of_is_max_on (hd.mono hsub) (hz.on_subset hsub) (mem_closed_ball.2 le_rfl) | lemma | complex.norm_eq_norm_of_is_max_on_of_ball_subset | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"diff_cont_on_cl",
"is_max_on",
"le_rfl"
] | **Maximum modulus principle**: if `f : E → F` is complex differentiable on a set `s`, the norm
of `f` takes it maximum on `s` at `z`, and `w` is a point such that the closed ball with center `z`
and radius `dist w z` is included in `s`, then `‖f w‖ = ‖f z‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eventually_eq_of_is_local_max {f : E → F} {c : E}
(hd : ∀ᶠ z in 𝓝 c, differentiable_at ℂ f z) (hc : is_local_max (norm ∘ f) c) :
∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ | begin
rcases nhds_basis_closed_ball.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩,
exact nhds_basis_closed_ball.eventually_iff.2 ⟨r, hr₀, norm_eq_on_closed_ball_of_is_max_on
(differentiable_on.diff_cont_on_cl $
λ x hx, (hr $ closure_ball_subset_closed_ball hx).1.differentiable_within_at)
(λ x hx, (hr... | lemma | complex.norm_eventually_eq_of_is_local_max | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_at",
"differentiable_on.diff_cont_on_cl",
"differentiable_within_at",
"is_local_max"
] | **Maximum modulus principle**: if `f : E → F` is complex differentiable in a neighborhood of `c`
and the norm `‖f z‖` has a local maximum at `c`, then `‖f z‖` is locally constant in a neighborhood
of `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_set_of_mem_nhds_and_is_max_on_norm {f : E → F} {s : set E}
(hd : differentiable_on ℂ f s) :
is_open {z | s ∈ 𝓝 z ∧ is_max_on (norm ∘ f) s z} | begin
refine is_open_iff_mem_nhds.2 (λ z hz, (eventually_eventually_nhds.2 hz.1).and _),
replace hd : ∀ᶠ w in 𝓝 z, differentiable_at ℂ f w, from hd.eventually_differentiable_at hz.1,
exact (norm_eventually_eq_of_is_local_max hd $ (hz.2.is_local_max hz.1)).mono
(λ x hx y hy, le_trans (hz.2 hy) hx.ge)
end | lemma | complex.is_open_set_of_mem_nhds_and_is_max_on_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_at",
"differentiable_on",
"is_max_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_on_of_is_preconnected_of_is_max_on {f : E → F} {U : set E} {c : E}
(hc : is_preconnected U) (ho : is_open U) (hd : differentiable_on ℂ f U) (hcU : c ∈ U)
(hm : is_max_on (norm ∘ f) U c) :
eq_on (norm ∘ f) (const E ‖f c‖) U | begin
set V := U ∩ {z | is_max_on (norm ∘ f) U z},
have hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖, from λ x hx, le_antisymm (hm hx.1) (hx.2 hcU),
suffices : U ⊆ V, from λ x hx, hV x (this hx),
have hVo : is_open V,
{ simpa only [ho.mem_nhds_iff, set_of_and, set_of_mem_eq]
using is_open_set_of_mem_nhds_and_is_max_on_n... | lemma | complex.norm_eq_on_of_is_preconnected_of_is_max_on | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_on",
"disjoint",
"eq_or_ne",
"is_max_on",
"is_open",
"is_open_ne",
"is_preconnected"
] | **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_on_closure_of_is_preconnected_of_is_max_on {f : E → F} {U : set E} {c : E}
(hc : is_preconnected U) (ho : is_open U) (hd : diff_cont_on_cl ℂ f U) (hcU : c ∈ U)
(hm : is_max_on (norm ∘ f) U c) :
eq_on (norm ∘ f) (const E ‖f c‖) (closure U) | (norm_eq_on_of_is_preconnected_of_is_max_on hc ho hd.differentiable_on hcU hm).of_subset_closure
hd.continuous_on.norm continuous_on_const subset_closure subset.rfl | lemma | complex.norm_eq_on_closure_of_is_preconnected_of_is_max_on | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"closure",
"continuous_on_const",
"diff_cont_on_cl",
"is_max_on",
"is_open",
"is_preconnected",
"subset_closure"
] | **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U` and is
continuous on its closure. Suppose that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then
`‖f x‖ = ‖f c‖` for all `x ∈ closu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_of_is_preconnected_of_is_max_on_norm {f : E → F} {U : set E} {c : E}
(hc : is_preconnected U) (ho : is_open U) (hd : differentiable_on ℂ f U) (hcU : c ∈ U)
(hm : is_max_on (norm ∘ f) U c) :
eq_on f (const E (f c)) U | λ x hx,
have H₁ : ‖f x‖ = ‖f c‖, from norm_eq_on_of_is_preconnected_of_is_max_on hc ho hd hcU hm hx,
have H₂ : ‖f x + f c‖ = ‖f c + f c‖,
from norm_eq_on_of_is_preconnected_of_is_max_on hc ho (hd.add_const _) hcU hm.norm_add_self hx,
eq_of_norm_eq_of_norm_add_eq H₁ $ by simp only [H₂, same_ray.rfl.norm_add, H₁] | lemma | complex.eq_on_of_is_preconnected_of_is_max_on_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_on",
"eq_of_norm_eq_of_norm_add_eq",
"is_max_on",
"is_open",
"is_preconnected"
] | **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `f x = f c` for all `x ∈ U`.
TODO: change assumption from `is_max_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_closure_of_is_preconnected_of_is_max_on_norm {f : E → F} {U : set E} {c : E}
(hc : is_preconnected U) (ho : is_open U) (hd : diff_cont_on_cl ℂ f U) (hcU : c ∈ U)
(hm : is_max_on (norm ∘ f) U c) :
eq_on f (const E (f c)) (closure U) | (eq_on_of_is_preconnected_of_is_max_on_norm hc ho hd.differentiable_on hcU hm).of_subset_closure
hd.continuous_on continuous_on_const subset_closure subset.rfl | lemma | complex.eq_on_closure_of_is_preconnected_of_is_max_on_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"closure",
"continuous_on_const",
"diff_cont_on_cl",
"is_max_on",
"is_open",
"is_preconnected",
"subset_closure"
] | **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U` and is
continuous on its closure. Suppose that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then
`f x = f c` for all `x ∈ closure U... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_is_max_on_of_ball_subset {f : E → F} {s : set E} {z w : E} (hd : diff_cont_on_cl ℂ f s)
(hz : is_max_on (norm ∘ f) s z) (hsub : ball z (dist w z) ⊆ s) :
f w = f z | have H₁ : ‖f w‖ = ‖f z‖, from norm_eq_norm_of_is_max_on_of_ball_subset hd hz hsub,
have H₂ : ‖f w + f z‖ = ‖f z + f z‖,
from norm_eq_norm_of_is_max_on_of_ball_subset (hd.add_const _) hz.norm_add_self hsub,
eq_of_norm_eq_of_norm_add_eq H₁ $ by simp only [H₂, same_ray.rfl.norm_add, H₁] | lemma | complex.eq_of_is_max_on_of_ball_subset | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"diff_cont_on_cl",
"eq_of_norm_eq_of_norm_add_eq",
"is_max_on"
] | **Maximum modulus principle**. Let `f : E → F` be a function between complex normed spaces.
Suppose that the codomain `F` is a strictly convex space, `f` is complex differentiable on a set
`s`, `f` is continuous on the closure of `s`, the norm of `f` takes it maximum on `s` at `z`, and
`w` is a point such that the clos... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_closed_ball_of_is_max_on_norm {f : E → F} {z : E} {r : ℝ}
(hd : diff_cont_on_cl ℂ f (ball z r)) (hz : is_max_on (norm ∘ f) (ball z r) z) :
eq_on f (const E (f z)) (closed_ball z r) | λ x hx, eq_of_is_max_on_of_ball_subset hd hz $ ball_subset_ball hx | lemma | complex.eq_on_closed_ball_of_is_max_on_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"diff_cont_on_cl",
"is_max_on"
] | **Maximum modulus principle** on a closed ball. Suppose that a function `f : E → F` from a
normed complex space to a strictly convex normed complex space has the following properties:
- it is continuous on a closed ball `metric.closed_ball z r`,
- it is complex differentiable on the corresponding open ball;
- the norm... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eq_of_is_local_max_norm {f : E → F} {c : E}
(hd : ∀ᶠ z in 𝓝 c, differentiable_at ℂ f z) (hc : is_local_max (norm ∘ f) c) :
∀ᶠ y in 𝓝 c, f y = f c | begin
rcases nhds_basis_closed_ball.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩,
exact nhds_basis_closed_ball.eventually_iff.2 ⟨r, hr₀, eq_on_closed_ball_of_is_max_on_norm
(differentiable_on.diff_cont_on_cl $
λ x hx, (hr $ closure_ball_subset_closed_ball hx).1.differentiable_within_at)
(λ x hx, (hr... | lemma | complex.eventually_eq_of_is_local_max_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_at",
"differentiable_on.diff_cont_on_cl",
"differentiable_within_at",
"is_local_max"
] | **Maximum modulus principle**: if `f : E → F` is complex differentiable in a neighborhood of `c`
and the norm `‖f z‖` has a local maximum at `c`, then `f` is locally constant in a neighborhood
of `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eq_or_eq_zero_of_is_local_min_norm {f : E → ℂ} {c : E}
(hf : ∀ᶠ z in 𝓝 c, differentiable_at ℂ f z) (hc : is_local_min (norm ∘ f) c) :
(∀ᶠ z in 𝓝 c, f z = f c) ∨ (f c = 0) | begin
refine or_iff_not_imp_right.mpr (λ h, _),
have h1 : ∀ᶠ z in 𝓝 c, f z ≠ 0 := hf.self_of_nhds.continuous_at.eventually_ne h,
have h2 : is_local_max (norm ∘ f)⁻¹ c := hc.inv (h1.mono (λ z, norm_pos_iff.mpr)),
have h3 : is_local_max (norm ∘ f⁻¹) c := by { refine h2.congr (eventually_of_forall _); simp },
h... | lemma | complex.eventually_eq_or_eq_zero_of_is_local_min_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"differentiable_at",
"is_local_max",
"is_local_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_frontier_is_max_on_norm [finite_dimensional ℂ E] {f : E → F} {U : set E}
(hb : bounded U) (hne : U.nonempty) (hd : diff_cont_on_cl ℂ f U) :
∃ z ∈ frontier U, is_max_on (norm ∘ f) (closure U) z | begin
have hc : is_compact (closure U), from hb.is_compact_closure,
obtain ⟨w, hwU, hle⟩ : ∃ w ∈ closure U, is_max_on (norm ∘ f) (closure U) w,
from hc.exists_forall_ge hne.closure hd.continuous_on.norm,
rw [closure_eq_interior_union_frontier, mem_union] at hwU,
cases hwU, rotate, { exact ⟨w, hwU, hle⟩ },
... | lemma | complex.exists_mem_frontier_is_max_on_norm | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"closure",
"closure_eq_interior_union_frontier",
"diff_cont_on_cl",
"dist_comm",
"exists_mem_frontier_inf_dist_compl_eq_dist",
"finite_dimensional",
"frontier",
"frontier_interior_subset",
"interior",
"interior_subset",
"interior_subset_closure",
"is_compact",
"is_max_on",
"ne_top_of_le_ne... | **Maximum modulus principle**: if `f : E → F` is complex differentiable on a nonempty bounded
set `U` and is continuous on its closure, then there exists a point `z ∈ frontier U` such that
`λ z, ‖f z‖` takes it maximum value on `closure U` at `z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_of_forall_mem_frontier_norm_le {f : E → F} {U : set E} (hU : bounded U)
(hd : diff_cont_on_cl ℂ f U) {C : ℝ} (hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C)
{z : E} (hz : z ∈ closure U) :
‖f z‖ ≤ C | begin
rw [closure_eq_self_union_frontier, union_comm, mem_union] at hz,
cases hz, { exact hC z hz },
/- In case of a finite dimensional domain, one can just apply
`complex.exists_mem_frontier_is_max_on_norm`. To make it work in any Banach space, we restrict
the function to a line first. -/
rcases exists_ne ... | lemma | complex.norm_le_of_forall_mem_frontier_norm_le | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"antilipschitz_with",
"antilipschitz_with_line_map",
"closure",
"closure_eq_self_union_frontier",
"diff_cont_on_cl",
"differentiable",
"exists_ne",
"frontier",
"subset_closure"
] | **Maximum modulus principle**: if `f : E → F` is complex differentiable on a bounded set `U` and
`‖f z‖ ≤ C` for any `z ∈ frontier U`, then the same is true for any `z ∈ closure U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_closure_of_eq_on_frontier {f g : E → F} {U : set E} (hU : bounded U)
(hf : diff_cont_on_cl ℂ f U) (hg : diff_cont_on_cl ℂ g U) (hfg : eq_on f g (frontier U)) :
eq_on f g (closure U) | begin
suffices H : ∀ z ∈ closure U, ‖(f - g) z‖ ≤ 0, by simpa [sub_eq_zero] using H,
refine λ z hz, norm_le_of_forall_mem_frontier_norm_le hU (hf.sub hg) (λ w hw, _) hz,
simp [hfg hw]
end | lemma | complex.eq_on_closure_of_eq_on_frontier | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"closure",
"diff_cont_on_cl",
"frontier"
] | If two complex differentiable functions `f g : E → F` are equal on the boundary of a bounded set
`U`, then they are equal on `closure U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_of_eq_on_frontier {f g : E → F} {U : set E} (hU : bounded U)
(hf : diff_cont_on_cl ℂ f U) (hg : diff_cont_on_cl ℂ g U) (hfg : eq_on f g (frontier U)) :
eq_on f g U | (eq_on_closure_of_eq_on_frontier hU hf hg hfg).mono subset_closure | lemma | complex.eq_on_of_eq_on_frontier | analysis.complex | src/analysis/complex/abs_max.lean | [
"analysis.complex.cauchy_integral",
"analysis.normed_space.completion",
"analysis.normed_space.extr",
"topology.algebra.order.extr_closure"
] | [
"diff_cont_on_cl",
"frontier",
"subset_closure"
] | If two complex differentiable functions `f g : E → F` are equal on the boundary of a bounded set
`U`, then they are equal on `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
same_ray_iff : same_ray ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg | begin
rcases eq_or_ne x 0 with rfl | hx,
{ simp },
rcases eq_or_ne y 0 with rfl | hy,
{ simp },
simp only [hx, hy, false_or, same_ray_iff_norm_smul_eq, arg_eq_arg_iff hx hy],
field_simp [hx, hy],
rw [mul_comm, eq_comm]
end | lemma | complex.same_ray_iff | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [
"eq_or_ne",
"mul_comm",
"same_ray",
"same_ray_iff_norm_smul_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_iff_arg_div_eq_zero : same_ray ℝ x y ↔ arg (x / y) = 0 | begin
rw [←real.angle.to_real_zero, ←arg_coe_angle_eq_iff_eq_to_real, same_ray_iff],
by_cases hx : x = 0, { simp [hx] },
by_cases hy : y = 0, { simp [hy] },
simp [hx, hy, arg_div_coe_angle, sub_eq_zero]
end | lemma | complex.same_ray_iff_arg_div_eq_zero | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add_eq_iff : (x + y).abs = x.abs + y.abs ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg | same_ray_iff_norm_add.symm.trans same_ray_iff | lemma | complex.abs_add_eq_iff | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_eq_iff : (x - y).abs = |x.abs - y.abs| ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg | same_ray_iff_norm_sub.symm.trans same_ray_iff | lemma | complex.abs_sub_eq_iff | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
same_ray_of_arg_eq (h : x.arg = y.arg) : same_ray ℝ x y | same_ray_iff.mpr $ or.inr $ or.inr h | lemma | complex.same_ray_of_arg_eq | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [
"same_ray"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add_eq (h : x.arg = y.arg) : (x + y).abs = x.abs + y.abs | (same_ray_of_arg_eq h).norm_add | lemma | complex.abs_add_eq | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_eq (h : x.arg = y.arg) : (x - y).abs = ‖x.abs - y.abs‖ | (same_ray_of_arg_eq h).norm_sub | lemma | complex.abs_sub_eq | analysis.complex | src/analysis/complex/arg.lean | [
"analysis.inner_product_space.basic",
"analysis.special_functions.complex.arg"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_abs (z : ℂ) : ‖z‖ = abs z | rfl | lemma | complex.norm_eq_abs | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_exp_of_real_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 | by simp only [norm_eq_abs, abs_exp_of_real_mul_I] | lemma | complex.norm_exp_of_real_mul_I | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.normed_space.complex_to_real : normed_space ℝ E | normed_space.restrict_scalars ℝ ℂ E | instance | normed_space.complex_to_real | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [
"normed_space",
"normed_space.restrict_scalars"
] | The module structure from `module.complex_to_real` is a normed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq (z w : ℂ) : dist z w = abs (z - w) | rfl | lemma | complex.dist_eq | analysis.complex | src/analysis/complex/basic.lean | [
"data.complex.module",
"data.complex.exponential",
"data.is_R_or_C.basic",
"topology.algebra.infinite_sum.module",
"topology.instances.real_vector_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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