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Theorem cpnfval 25096

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 10952	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5269	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		oveq2 7283	. . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
4		oveq1 7282	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D^𝑛 𝑓) = (𝑆 D^𝑛 𝑓))
5	4	fveq1d 6776	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D^𝑛 𝑓)‘𝑛) = ((𝑆 D^𝑛 𝑓)‘𝑛))
6	5	eleq1d 2823	. . . . 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 3420	. . . 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 5176	. . 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 25033	. . 3 ⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
10		nn0ex 12239	. . . 4 ⊢ ℕ₀ ∈ V
11	10	mptex 7099	. . 3 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V
12	8, 9, 11	fvmpt 6875	. 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 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 𝒫 cpw 4533 ↦ cmpt 5157 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 ↑_pm cpm 8616 ℂcc 10869 ℕ₀cn0 12233 –cn→ccncf 24039 D^𝑛 cdvn 25028 𝓑C^𝑛ccpn 25029

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-1cn 10929 ax-addcl 10931

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-nn 11974 df-n0 12234 df-cpn 25033

This theorem is referenced by: fncpn 25097 elcpn 25098

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