Theorem dvnfval 25169

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 25115	. . 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 768	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑠 = 𝑆)
4	3	oveq1d 7332	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑠 D 𝑥) = (𝑆 D 𝑥))
5	4	mpteq2dv 5189	. . . . . 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 2795	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ V ↦ (𝑠 D 𝑥)) = 𝐺)
8	7	coeq1d 5791	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → ((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ) = (𝐺 ∘ 1st ))
9	8	seqeq2d 13808	. . 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 770	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
11	10	sneqd 4583	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → {𝑓} = {𝐹})
12	11	xpeq2d 5638	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑠 = 𝑆 ∧ 𝑓 = 𝐹)) → (ℕ0 × {𝑓}) = (ℕ0 × {𝐹}))
13	12	seqeq3d 13809	. . 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 2777	. 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 485	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
16	15	oveq2d 7333	. 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ 𝑠 = 𝑆) → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆))
17		simpl 483	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ⊆ ℂ)
18		cnex 11032	. . . 4 ⊢ ℂ ∈ V
19	18	elpw2 5284	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
20	17, 19	sylibr 233	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝑆 ∈ 𝒫 ℂ)
21		simpr 485	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
22		seqex 13803	. . 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 7466	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441   ⊆ wss 3897  𝒫 cpw 4545  {csn 4571   ↦ cmpt 5170   × cxp 5606   ∘ ccom 5612  (class class class)co 7317   ∈ cmpo 7319  1^st c1st 7876   ↑_pm cpm 8666  ℂcc 10949  0cc0 10951  ℕ₀cn0 12313  seqcseq 13801   D cdv 25110   D^𝑛 cdvn 25111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-un 7630  ax-inf2 9477  ax-cnex 11007

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-seq 13802  df-dvn 25115

This theorem is referenced by:  dvnff  25170  dvn0  25171  dvnp1  25172