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Theorem elcpn 25314
Description: Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
elcpn ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ))))

Proof of Theorem elcpn
Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpnfval 25312 . . . . 5 (𝑆 ⊆ ℂ → (𝓑C𝑛𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
21fveq1d 6849 . . . 4 (𝑆 ⊆ ℂ → ((𝓑C𝑛𝑆)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)})‘𝑁))
3 fveq2 6847 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑁))
43eleq1d 2823 . . . . . 6 (𝑛 = 𝑁 → (((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)))
54rabbidv 3418 . . . . 5 (𝑛 = 𝑁 → {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)} = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)})
6 eqid 2737 . . . . 5 (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)})
7 ovex 7395 . . . . . 6 (ℂ ↑pm 𝑆) ∈ V
87rabex 5294 . . . . 5 {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)} ∈ V
95, 6, 8fvmpt 6953 . . . 4 (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)})‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)})
102, 9sylan9eq 2797 . . 3 ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝓑C𝑛𝑆)‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)})
1110eleq2d 2824 . 2 ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁) ↔ 𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)}))
12 oveq2 7370 . . . . 5 (𝑓 = 𝐹 → (𝑆 D𝑛 𝑓) = (𝑆 D𝑛 𝐹))
1312fveq1d 6849 . . . 4 (𝑓 = 𝐹 → ((𝑆 D𝑛 𝑓)‘𝑁) = ((𝑆 D𝑛 𝐹)‘𝑁))
14 dmeq 5864 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
1514oveq1d 7377 . . . 4 (𝑓 = 𝐹 → (dom 𝑓cn→ℂ) = (dom 𝐹cn→ℂ))
1613, 15eleq12d 2832 . . 3 (𝑓 = 𝐹 → (((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ)))
1716elrab 3650 . 2 (𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)} ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ)))
1811, 17bitrdi 287 1 ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3410  wss 3915  cmpt 5193  dom cdm 5638  cfv 6501  (class class class)co 7362  pm cpm 8773  cc 11056  0cn0 12420  cnccncf 24255   D𝑛 cdvn 25244  𝓑C𝑛ccpn 25245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-1cn 11116  ax-addcl 11118
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-nn 12161  df-n0 12421  df-cpn 25249
This theorem is referenced by:  cpnord  25315  cpncn  25316  cpnres  25317  c1lip2  25378  plycpn  25665
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