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Theorem cpnfval 26060

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 11181	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5305	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		oveq2 7419	. . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
4		oveq1 7418	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D^𝑛 𝑓) = (𝑆 D^𝑛 𝑓))
5	4	fveq1d 6884	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D^𝑛 𝑓)‘𝑛) = ((𝑆 D^𝑛 𝑓)‘𝑛))
6	5	eleq1d 2854	. . . . 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 3440	. . . 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 5207	. . 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 25997	. . 3 ⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
10		nn0ex 12510	. . . 4 ⊢ ℕ₀ ∈ V
11	10	mptex 7222	. . 3 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V
12	8, 9, 11	fvmpt 6990	. 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 238	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 1567 ∈ wcel 2149 {crab 3422 ⊆ wss 3911 𝒫 cpw 4565 ↦ cmpt 5194 dom cdm 5662 ‘cfv 6537 (class class class)co 7411 ↑_pm cpm 8825 ℂcc 11098 ℕ₀cn0 12504 –cn→ccncf 25004 D^𝑛 cdvn 25992 𝓑C^𝑛ccpn 25993

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-1cn 11158 ax-addcl 11160

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7414 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-nn 12234 df-n0 12505 df-cpn 25997

This theorem is referenced by: fncpn 26061 elcpn 26062

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