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Theorem cpnfval 25906

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 → (𝓑C^𝑛‘𝑆) = (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))

Distinct variable group: 𝑓,𝑛,𝑆

Proof of Theorem cpnfval Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 11221	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5348	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		oveq2 7427	. . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
4		oveq1 7426	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D^𝑛 𝑓) = (𝑆 D^𝑛 𝑓))
5	4	fveq1d 6898	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D^𝑛 𝑓)‘𝑛) = ((𝑆 D^𝑛 𝑓)‘𝑛))
6	5	eleq1d 2810	. . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)))
7	3, 6	rabeqbidv 3436	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} = {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})
8	7	mpteq2dv 5251	. . 3 ⊢ (𝑠 = 𝑆 → (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) = (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
9		df-cpn 25842	. . 3 ⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
10		nn0ex 12511	. . . 4 ⊢ ℕ₀ ∈ V
11	10	mptex 7235	. . 3 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V
12	8, 9, 11	fvmpt 7004	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ → (𝓑C^𝑛‘𝑆) = (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
13	2, 12	sylbir 234	1 ⊢ (𝑆 ⊆ ℂ → (𝓑C^𝑛‘𝑆) = (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3418 ⊆ wss 3944 𝒫 cpw 4604 ↦ cmpt 5232 dom cdm 5678 ‘cfv 6549 (class class class)co 7419 ↑_pm cpm 8846 ℂcc 11138 ℕ₀cn0 12505 –cn→ccncf 24840 D^𝑛 cdvn 25837 𝓑C^𝑛ccpn 25838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-addcl 11200

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12246 df-n0 12506 df-cpn 25842

This theorem is referenced by: fncpn 25907 elcpn 25908

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