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Theorem cpnfval 24523

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 10612	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5240	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		oveq2 7158	. . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
4		oveq1 7157	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D^𝑛 𝑓) = (𝑆 D^𝑛 𝑓))
5	4	fveq1d 6666	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D^𝑛 𝑓)‘𝑛) = ((𝑆 D^𝑛 𝑓)‘𝑛))
6	5	eleq1d 2897	. . . . 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 3485	. . . 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 5154	. . 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 24461	. . 3 ⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
10		nn0ex 11897	. . . 4 ⊢ ℕ₀ ∈ V
11	10	mptex 6980	. . 3 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V
12	8, 9, 11	fvmpt 6762	. 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 237	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 2110 {crab 3142 ⊆ wss 3935 𝒫 cpw 4538 ↦ cmpt 5138 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 ↑_pm cpm 8401 ℂcc 10529 ℕ₀cn0 11891 –cn→ccncf 23478 D^𝑛 cdvn 24456 𝓑C^𝑛ccpn 24457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-n0 11892 df-cpn 24461

This theorem is referenced by: fncpn 24524 elcpn 24525

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