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Theorem cpnfval 25832

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 11090	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 5273	. 2 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3		oveq2 7357	. . . . 5 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
4		oveq1 7356	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 D^𝑛 𝑓) = (𝑆 D^𝑛 𝑓))
5	4	fveq1d 6824	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠 D^𝑛 𝑓)‘𝑛) = ((𝑆 D^𝑛 𝑓)‘𝑛))
6	5	eleq1d 2813	. . . . 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 3413	. . . 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 5186	. . 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 25768	. . 3 ⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
10		nn0ex 12390	. . . 4 ⊢ ℕ₀ ∈ V
11	10	mptex 7159	. . 3 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑆) ∣ ((𝑆 D^𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) ∈ V
12	8, 9, 11	fvmpt 6930	. 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 235	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 1540 ∈ wcel 2109 {crab 3394 ⊆ wss 3903 𝒫 cpw 4551 ↦ cmpt 5173 dom cdm 5619 ‘cfv 6482 (class class class)co 7349 ↑_pm cpm 8754 ℂcc 11007 ℕ₀cn0 12384 –cn→ccncf 24767 D^𝑛 cdvn 25763 𝓑C^𝑛ccpn 25764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-1cn 11067 ax-addcl 11069

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-nn 12129 df-n0 12385 df-cpn 25768

This theorem is referenced by: fncpn 25833 elcpn 25834

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