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Theorem dvnfval 23491
 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 23438 . . 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 790 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → 𝑠 = 𝑆)
43oveq1d 6564 . . . . . . 7 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (𝑠 D 𝑥) = (𝑆 D 𝑥))
54mpteq2dv 4673 . . . . . 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 2662 . . . . 5 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = 𝐺)
87coeq1d 5205 . . . 4 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → ((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ) = (𝐺 ∘ 1st ))
98seqeq2d 12670 . . 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 792 . . . . . 6 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → 𝑓 = 𝐹)
1110sneqd 4137 . . . . 5 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → {𝑓} = {𝐹})
1211xpeq2d 5063 . . . 4 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → (ℕ0 × {𝑓}) = (ℕ0 × {𝐹}))
1312seqeq3d 12671 . . 3 (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆𝑓 = 𝐹)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
149, 13eqtrd 2644 . 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 6565 . 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 9896 . . . 4 ℂ ∈ V
1918elpw2 4755 . . 3 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
2017, 19sylibr 223 . 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 12665 . . 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 6685 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 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057   ↑pm cpm 7745  ℂcc 9813  0cc0 9815  ℕ0cn0 11169  seqcseq 12663   D cdv 23433   D𝑛 cdvn 23434 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-dvn 23438 This theorem is referenced by:  dvnff  23492  dvn0  23493  dvnp1  23494
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