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Theorem dvnp1 24516

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 1134	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
2		nn0uz 12274	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2923	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
4		seqp1 13378	. . . 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 6678	. . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁) ∈ V
7		fvex 6678	. . . 4 ⊢ ((ℕ₀ × {𝐹})‘(𝑁 + 1)) ∈ V
8	6, 7	algrflem 7813	. . 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	syl6eq 2872	. 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 2821	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
11	10	dvnfval 24513	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹})))
12	11	3adant3 1128	. . 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 6667	. 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 6678	. . . 4 ⊢ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V
15		oveq2 7158	. . . . 5 ⊢ (𝑥 = ((𝑆 D^𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
16		ovex 7183	. . . . 5 ⊢ (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) ∈ V
17	15, 10, 16	fvmpt 6763	. . . 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 6667	. . . 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 6669	. . 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	syl5eqr 2870	. 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 2866	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 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 {csn 4561 ↦ cmpt 5139 × cxp 5548 ∘ ccom 5554 ‘cfv 6350 (class class class)co 7150 1^st c1st 7681 ↑_pm cpm 8401 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 ℕ₀cn0 11891 ℤ_≥cuz 12237 seqcseq 13363 D cdv 24455 D^𝑛 cdvn 24456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-dvn 24460

This theorem is referenced by: dvn1 24517 dvnadd 24520 dvnres 24522 cpnord 24526 dvnfre 24543 c1lip2 24589 dvnply2 24870 dvntaylp 24953 taylthlem1 24955 taylthlem2 24956 dvnmptdivc 42215 dvnmptconst 42218 dvnxpaek 42219 dvnmul 42220 etransclem2 42514

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