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Theorem dvnfval 23730
Description: Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
dvnfval.1 𝐺 = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
Assertion
Ref Expression
dvnfval ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑆
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem dvnfval
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvn 23677 . . 3 D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
21a1i 11 . 2 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓}))))
3 simprl 809 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → 𝑠 = 𝑆)
43oveq1d 6705 . . . . . . 7 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (𝑠 D 𝑥) = (𝑆 D 𝑥))
54mpteq2dv 4778 . . . . . 6 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥)))
6 dvnfval.1 . . . . . 6 𝐺 = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
75, 6syl6eqr 2703 . . . . 5 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = 𝐺)
87coeq1d 5316 . . . 4 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → ((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ) = (𝐺 ∘ 1st ))
98seqeq2d 12848 . . 3 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝑓})))
10 simprr 811 . . . . . 6 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → 𝑓 = 𝐹)
1110sneqd 4222 . . . . 5 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → {𝑓} = {𝐹})
1211xpeq2d 5173 . . . 4 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (ℕ0 × {𝑓}) = (ℕ0 × {𝐹}))
1312seqeq3d 12849 . . 3 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
149, 13eqtrd 2685 . 2 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
15 simpr 476 . . 3 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
1615oveq2d 6706 . 2 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
17 simpl 472 . . 3 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ⊆ ℂ)
18 cnex 10055 . . . 4 ℂ ∈ V
1918elpw2 4858 . . 3 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
2017, 19sylibr 224 . 2 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ∈ 𝒫 ℂ)
21 simpr 476 . 2 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
22 seqex 12843 . . 3 seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V
2322a1i 11 . 2 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V)
242, 14, 16, 20, 21, 23ovmpt2dx 6829 1 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191  {csn 4210  cmpt 4762   × cxp 5141  ccom 5147  (class class class)co 6690  cmpt2 6692  1st c1st 7208  pm cpm 7900  cc 9972  0cc0 9974  0cn0 11330  seqcseq 12841   D cdv 23672   D𝑛 cdvn 23673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seq 12842  df-dvn 23677
This theorem is referenced by:  dvnff  23731  dvn0  23732  dvnp1  23733
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