Theorem dvnfval 26149

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.

Step	Hyp	Ref	Expression
1		df-dvn 26095	. . 3 ⊢ D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
2	1	a1i 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 783	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑠 = 𝑆)
4	3	oveq1d 7431	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑠 D 𝑥) = (𝑆 D 𝑥))
5	4	mpteq2dv 5203	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥)))
6		dvnfval.1	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
7	5, 6	eqtr4di 2815	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = 𝐺)
8	7	coeq1d 5845	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → ((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ) = (𝐺 ∘ 1st ))
9	8	seqeq2d 14072	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝑓})))
10		simprr 785	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10	sneqd 4599	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → {𝑓} = {𝐹})
12	11	xpeq2d 5689	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (ℕ0 × {𝑓}) = (ℕ0 × {𝐹}))
13	12	seqeq3d 14073	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
14	9, 13	eqtrd 2797	. 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
15		simpr 490	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
16	15	oveq2d 7432	. 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
17		cnex 11206	. . . 4 ⊢ ℂ ∈ V
18	17	elpw2 5303	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
19	18	biranri 511	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ∈ 𝒫 ℂ)
20		simpr 490	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
21		seqex 14067	. . 3 ⊢ seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V
22	21	a1i 11	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V)
23	2, 14, 16, 19, 20, 22	ovmpodx 7567	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587   ↦ cmpt 5190   × cxp 5657   ∘ ccom 5663  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7987   ↑_pm cpm 8830  ℂcc 11123  0cc0 11125  ℕ₀cn0 12529  seqcseq 14065   D cdv 26090   D^𝑛 cdvn 26091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11181

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seq 14066  df-dvn 26095

This theorem is used by:  dvnff  26150  dvn0  26151  dvnp1  26152