Theorem elcpn 25914

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 25912	. . . . 5 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
2	1	fveq1d 6837	. . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝓑C𝑛‘𝑆)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})‘𝑁))
3		fveq2 6835	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 𝑓)‘𝑛) = ((𝑆 D𝑛 𝑓)‘𝑁))
4	3	eleq1d 2822	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)))
5	4	rabbidv 3397	. . . . 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 2737	. . . . 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 7394	. . . . . 6 ⊢ (ℂ ↑pm 𝑆) ∈ V
8	7	rabex 5277	. . . . 5 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)} ∈ V
9	5, 6, 8	fvmpt 6942	. . . 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 2792	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝓑C𝑛‘𝑆)‘𝑁) = {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)})
11	10	eleq2d 2823	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ 𝐹 ∈ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ)}))
12		oveq2 7369	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑆 D𝑛 𝑓) = (𝑆 D𝑛 𝐹))
13	12	fveq1d 6837	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑆 D𝑛 𝑓)‘𝑁) = ((𝑆 D𝑛 𝐹)‘𝑁))
14		dmeq 5853	. . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
15	14	oveq1d 7376	. . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓–cn→ℂ) = (dom 𝐹–cn→ℂ))
16	13, 15	eleq12d 2831	. . 3 ⊢ (𝑓 = 𝐹 → (((𝑆 D𝑛 𝑓)‘𝑁) ∈ (dom 𝑓–cn→ℂ) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
17	16	elrab 3635	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390   ⊆ wss 3890   ↦ cmpt 5167  dom cdm 5625  ‘cfv 6493  (class class class)co 7361   ↑_pm cpm 8768  ℂcc 11030  ℕ₀cn0 12431  –cn→ccncf 24856   D^𝑛 cdvn 25844  𝓑C^𝑛ccpn 25845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-1cn 11090  ax-addcl 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-nn 12169  df-n0 12432  df-cpn 25849

This theorem is referenced by:  cpnord  25915  cpncn  25916  cpnres  25917  c1lip2  25978  plycpn  26269