Theorem dvnfval 24991

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 24937	. . 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 767	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑠 = 𝑆)
4	3	oveq1d 7270	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑠 D 𝑥) = (𝑆 D 𝑥))
5	4	mpteq2dv 5172	. . . . . 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 2797	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = 𝐺)
8	7	coeq1d 5759	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → ((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ) = (𝐺 ∘ 1st ))
9	8	seqeq2d 13656	. . 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 769	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10	sneqd 4570	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → {𝑓} = {𝐹})
12	11	xpeq2d 5610	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (ℕ0 × {𝑓}) = (ℕ0 × {𝐹}))
13	12	seqeq3d 13657	. . 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 2778	. 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 484	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
16	15	oveq2d 7271	. 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
17		simpl 482	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ⊆ ℂ)
18		cnex 10883	. . . 4 ⊢ ℂ ∈ V
19	18	elpw2 5264	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
20	17, 19	sylibr 233	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ∈ 𝒫 ℂ)
21		simpr 484	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
22		seqex 13651	. . 3 ⊢ seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V
23	22	a1i 11	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})) ∈ V)
24	2, 14, 16, 20, 21, 23	ovmpodx 7402	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558   ↦ cmpt 5153   × cxp 5578   ∘ ccom 5584  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802   ↑_pm cpm 8574  ℂcc 10800  0cc0 10802  ℕ₀cn0 12163  seqcseq 13649   D cdv 24932   D^𝑛 cdvn 24933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seq 13650  df-dvn 24937

This theorem is referenced by:  dvnff  24992  dvn0  24993  dvnp1  24994