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Theorem cpnfval 23446
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
cpnfval (𝑆 ⊆ ℂ → (Cn𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
Distinct variable group:   𝑓,𝑛,𝑆

Proof of Theorem cpnfval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cnex 9874 . . 3 ℂ ∈ V
21elpw2 4750 . 2 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3 oveq2 6535 . . . . 5 (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
4 oveq1 6534 . . . . . . 7 (𝑠 = 𝑆 → (𝑠 D𝑛 𝑓) = (𝑆 D𝑛 𝑓))
54fveq1d 6090 . . . . . 6 (𝑠 = 𝑆 → ((𝑠 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑛))
65eleq1d 2671 . . . . 5 (𝑠 = 𝑆 → (((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)))
73, 6rabeqbidv 3167 . . . 4 (𝑠 = 𝑆 → {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)} = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)})
87mpteq2dv 4667 . . 3 (𝑠 = 𝑆 → (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)}))
9 df-cpn 23384 . . 3 Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
10 nn0ex 11148 . . . 4 0 ∈ V
1110mptex 6368 . . 3 (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}) ∈ V
128, 9, 11fvmpt 6176 . 2 (𝑆 ∈ 𝒫 ℂ → (Cn𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
132, 12sylbir 223 1 (𝑆 ⊆ ℂ → (Cn𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  {crab 2899  wss 3539  𝒫 cpw 4107  cmpt 4637  dom cdm 5028  cfv 5790  (class class class)co 6527  pm cpm 7723  cc 9791  0cn0 11142  cnccncf 22435   D𝑛 cdvn 23379  Cnccpn 23380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-i2m1 9861  ax-1ne0 9862  ax-rrecex 9865  ax-cnre 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-nn 10871  df-n0 11143  df-cpn 23384
This theorem is referenced by:  fncpn  23447  elcpn  23448
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