dvnp1 - Metamath Proof Explorer

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Theorem dvnp1 26135

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 1156	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
2		nn0uz 12916	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2875	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
4		seqp1 14070	. . . 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 18	. . 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 6898	. . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁) ∈ V
7		fvex 6898	. . . 4 ⊢ ((ℕ₀ × {𝐹})‘(𝑁 + 1)) ∈ V
8	6, 7	opco1i 8126	. . 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 2816	. 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 2765	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
11	10	dvnfval 26132	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹})))
12	11	3adant3 1150	. . 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 6887	. 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 6898	. . . 4 ⊢ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V
15		oveq2 7427	. . . . 5 ⊢ (𝑥 = ((𝑆 D^𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
16		ovex 7452	. . . . 5 ⊢ (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) ∈ V
17	15, 10, 16	fvmpt 6993	. . . 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 6887	. . . 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 6889	. . 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 2814	. 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 2810	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

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ⊆ wss 3906 {csn 4591 ↦ cmpt 5194 × cxp 5661 ∘ ccom 5667 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 ↑_pm cpm 8831 ℂcc 11113 0cc0 11115 1c1 11116 + caddc 11118 ℕ₀cn0 12519 ℤ_≥cuz 12878 seqcseq 14055 D cdv 26073 D^𝑛 cdvn 26074

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-n0 12520 df-z 12607 df-uz 12879 df-seq 14056 df-dvn 26078

This theorem is used by: dvn1 26136 dvnadd 26139 dvnres 26141 cpnord 26145 dvnfre 26162 c1lip2 26208 dvnply2 26499 dvntaylp 26585 taylthlem1 26587 taylthlem2 26588 dvnmptdivc 46710 dvnmptconst 46713 dvnxpaek 46714 dvnmul 46715 etransclem2 47008

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