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Theorem dvnp1 25803

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 1138	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
2		nn0uz 12811	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
4		seqp1 13957	. . . 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 6853	. . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁) ∈ V
7		fvex 6853	. . . 4 ⊢ ((ℕ₀ × {𝐹})‘(𝑁 + 1)) ∈ V
8	6, 7	opco1i 8081	. . 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 2780	. 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 2729	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
11	10	dvnfval 25800	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹})))
12	11	3adant3 1132	. . 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 6842	. 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 6853	. . . 4 ⊢ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V
15		oveq2 7377	. . . . 5 ⊢ (𝑥 = ((𝑆 D^𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
16		ovex 7402	. . . . 5 ⊢ (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) ∈ V
17	15, 10, 16	fvmpt 6950	. . . 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 6842	. . . 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 6844	. . 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 2778	. 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 2774	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 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3444 ⊆ wss 3911 {csn 4585 ↦ cmpt 5183 × cxp 5629 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 ↑_pm cpm 8777 ℂcc 11042 0cc0 11044 1c1 11045 + caddc 11047 ℕ₀cn0 12418 ℤ_≥cuz 12769 seqcseq 13942 D cdv 25740 D^𝑛 cdvn 25741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-seq 13943 df-dvn 25745

This theorem is referenced by: dvn1 25804 dvnadd 25807 dvnres 25809 cpnord 25813 dvnfre 25832 c1lip2 25879 dvnply2 26171 dvntaylp 26255 taylthlem1 26257 taylthlem2 26258 taylthlem2OLD 26259 dvnmptdivc 45909 dvnmptconst 45912 dvnxpaek 45913 dvnmul 45914 etransclem2 46207

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