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Theorem dvnp1 25873

Description: Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
dvnp1	⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))

Proof of Theorem dvnp1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1135	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
2		nn0uz 12900	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
4		seqp1 14019	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘(𝑁 + 1)) = ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st )((ℕ₀ × {𝐹})‘(𝑁 + 1))))
5	3, 4	syl 17	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘(𝑁 + 1)) = ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st )((ℕ₀ × {𝐹})‘(𝑁 + 1))))
6		fvex 6913	. . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁) ∈ V
7		fvex 6913	. . . 4 ⊢ ((ℕ₀ × {𝐹})‘(𝑁 + 1)) ∈ V
8	6, 7	opco1i 8134	. . 3 ⊢ ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st )((ℕ₀ × {𝐹})‘(𝑁 + 1))) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁))
9	5, 8	eqtrdi 2783	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘(𝑁 + 1)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)))
10		eqid 2727	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
11	10	dvnfval 25870	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹})))
12	11	3adant3 1129	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (𝑆 D^𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹})))
13	12	fveq1d 6902	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑁 + 1)) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘(𝑁 + 1)))
14		fvex 6913	. . . 4 ⊢ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V
15		oveq2 7432	. . . . 5 ⊢ (𝑥 = ((𝑆 D^𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
16		ovex 7457	. . . . 5 ⊢ (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) ∈ V
17	15, 10, 16	fvmpt 7008	. . . 4 ⊢ (((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V → ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D^𝑛 𝐹)‘𝑁)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
18	14, 17	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D^𝑛 𝐹)‘𝑁)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁))
19	12	fveq1d 6902	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁))
20	19	fveq2d 6904	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D^𝑛 𝐹)‘𝑁)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)))
21	18, 20	eqtr3id 2781	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁)))
22	9, 13, 21	3eqtr4d 2777	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 {csn 4630 ↦ cmpt 5233 × cxp 5678 ∘ ccom 5684 ‘cfv 6551 (class class class)co 7424 1^st c1st 7995 ↑_pm cpm 8850 ℂcc 11142 0cc0 11144 1c1 11145 + caddc 11147 ℕ₀cn0 12508 ℤ_≥cuz 12858 seqcseq 14004 D cdv 25810 D^𝑛 cdvn 25811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-seq 14005 df-dvn 25815

This theorem is referenced by: dvn1 25874 dvnadd 25877 dvnres 25879 cpnord 25883 dvnfre 25902 c1lip2 25949 dvnply2 26240 dvntaylp 26324 taylthlem1 26326 taylthlem2 26327 taylthlem2OLD 26328 dvnmptdivc 45328 dvnmptconst 45331 dvnxpaek 45332 dvnmul 45333 etransclem2 45626

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