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Theorem elcpn 24634
 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 24632 . . . . 5 (𝑆 ⊆ ℂ → (𝓑C𝑛𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
21fveq1d 6661 . . . 4 (𝑆 ⊆ ℂ → ((𝓑C𝑛𝑆)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)})‘𝑁))
3 fveq2 6659 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑁))
43eleq1d 2837 . . . . . 6 (𝑛 = 𝑁 → (((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)))
54rabbidv 3393 . . . . 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 2759 . . . . 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 7184 . . . . . 6 (ℂ ↑pm 𝑆) ∈ V
87rabex 5203 . . . . 5 {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)} ∈ V
95, 6, 8fvmpt 6760 . . . 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 2814 . . 3 ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝓑C𝑛𝑆)‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)})
1110eleq2d 2838 . 2 ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁) ↔ 𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ)}))
12 oveq2 7159 . . . . 5 (𝑓 = 𝐹 → (𝑆 D𝑛 𝑓) = (𝑆 D𝑛 𝐹))
1312fveq1d 6661 . . . 4 (𝑓 = 𝐹 → ((𝑆 D𝑛 𝑓)‘𝑁) = ((𝑆 D𝑛 𝐹)‘𝑁))
14 dmeq 5744 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
1514oveq1d 7166 . . . 4 (𝑓 = 𝐹 → (dom 𝑓cn→ℂ) = (dom 𝐹cn→ℂ))
1613, 15eleq12d 2847 . . 3 (𝑓 = 𝐹 → (((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓cn→ℂ) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ)))
1716elrab 3603 . 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 290 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 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075   ⊆ wss 3859   ↦ cmpt 5113  dom cdm 5525  ‘cfv 6336  (class class class)co 7151   ↑pm cpm 8418  ℂcc 10574  ℕ0cn0 11935  –cn→ccncf 23578   D𝑛 cdvn 24564  𝓑C𝑛ccpn 24565 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-1cn 10634  ax-addcl 10636 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-nn 11676  df-n0 11936  df-cpn 24569 This theorem is referenced by:  cpnord  24635  cpncn  24636  cpnres  24637  c1lip2  24698  plycpn  24985
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