dvnp1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvnp1		Structured version Visualization version GIF version

Theorem dvnp1 24525

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 12283	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2926	. . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
4		seqp1 13387	. . . 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 6686	. . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝐹}))‘𝑁) ∈ V
7		fvex 6686	. . . 4 ⊢ ((ℕ₀ × {𝐹})‘(𝑁 + 1)) ∈ V
8	6, 7	algrflem 7822	. . 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 2875	. 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 2824	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥))
11	10	dvnfval 24522	. . . 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 6675	. 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 6686	. . . 4 ⊢ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ V
15		oveq2 7167	. . . . 5 ⊢ (𝑥 = ((𝑆 D^𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)))
16		ovex 7192	. . . . 5 ⊢ (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑁)) ∈ V
17	15, 10, 16	fvmpt 6771	. . . 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 6675	. . . 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 6677	. . 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 2873	. 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 2869	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 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 {csn 4570 ↦ cmpt 5149 × cxp 5556 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 1^st c1st 7690 ↑_pm cpm 8410 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 ℕ₀cn0 11900 ℤ_≥cuz 12246 seqcseq 13372 D cdv 24464 D^𝑛 cdvn 24465

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-dvn 24469

This theorem is referenced by: dvn1 24526 dvnadd 24529 dvnres 24531 cpnord 24535 dvnfre 24552 c1lip2 24598 dvnply2 24879 dvntaylp 24962 taylthlem1 24964 taylthlem2 24965 dvnmptdivc 42229 dvnmptconst 42232 dvnxpaek 42233 dvnmul 42234 etransclem2 42528

W3C validator