Theorem elcpn 25863

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
elcpn	⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))

Proof of Theorem elcpn

Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpnfval 25861	. . . . 5 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
2	1	fveq1d 6899	. . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝓑C𝑛‘𝑆)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})‘𝑁))
3		fveq2 6897	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑁))
4	3	eleq1d 2814	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)))
5	4	rabbidv 3437	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)})
6		eqid 2728	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})
7		ovex 7453	. . . . . 6 ⊢ (ℂ ↑pm 𝑆) ∈ V
8	7	rabex 5334	. . . . 5 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)} ∈ V
9	5, 6, 8	fvmpt 7005	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)})
10	2, 9	sylan9eq 2788	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝓑C𝑛‘𝑆)‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)})
11	10	eleq2d 2815	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ 𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)}))
12		oveq2 7428	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑆 D𝑛 𝑓) = (𝑆 D𝑛 𝐹))
13	12	fveq1d 6899	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑆 D𝑛 𝑓)‘𝑁) = ((𝑆 D𝑛 𝐹)‘𝑁))
14		dmeq 5906	. . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
15	14	oveq1d 7435	. . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓–cn→ℂ) = (dom 𝐹–cn→ℂ))
16	13, 15	eleq12d 2823	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
17	16	elrab 3682	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)} ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
18	11, 17	bitrdi 287	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429   ⊆ wss 3947   ↦ cmpt 5231  dom cdm 5678  ‘cfv 6548  (class class class)co 7420   ↑_pm cpm 8845  ℂcc 11136  ℕ₀cn0 12502  –cn→ccncf 24795   D^𝑛 cdvn 25792  𝓑C^𝑛ccpn 25793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-nn 12243  df-n0 12503  df-cpn 25797

This theorem is referenced by:  cpnord  25864  cpncn  25865  cpnres  25866  c1lip2  25930  plycpn  26223