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Mirrors > Home > MPE Home > Th. List > dvnp1 | Structured version Visualization version GIF version |
Description: Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvnp1 | ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1137 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | nn0uz 12630 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | eleqtrdi 2849 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ (ℤ≥‘0)) |
4 | seqp1 13746 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘(𝑁 + 1)) = ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st )((ℕ0 × {𝐹})‘(𝑁 + 1)))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘(𝑁 + 1)) = ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st )((ℕ0 × {𝐹})‘(𝑁 + 1)))) |
6 | fvex 6779 | . . . 4 ⊢ (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁) ∈ V | |
7 | fvex 6779 | . . . 4 ⊢ ((ℕ0 × {𝐹})‘(𝑁 + 1)) ∈ V | |
8 | 6, 7 | opco1i 7953 | . . 3 ⊢ ((seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁)((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st )((ℕ0 × {𝐹})‘(𝑁 + 1))) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁)) |
9 | 5, 8 | eqtrdi 2794 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘(𝑁 + 1)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁))) |
10 | eqid 2738 | . . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥)) | |
11 | 10 | dvnfval 25096 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))) |
12 | 11 | 3adant3 1131 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑆 D𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))) |
13 | 12 | fveq1d 6768 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘(𝑁 + 1))) |
14 | fvex 6779 | . . . 4 ⊢ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ V | |
15 | oveq2 7275 | . . . . 5 ⊢ (𝑥 = ((𝑆 D𝑛 𝐹)‘𝑁) → (𝑆 D 𝑥) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁))) | |
16 | ovex 7300 | . . . . 5 ⊢ (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁)) ∈ V | |
17 | 15, 10, 16 | fvmpt 6867 | . . . 4 ⊢ (((𝑆 D𝑛 𝐹)‘𝑁) ∈ V → ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D𝑛 𝐹)‘𝑁)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁))) |
18 | 14, 17 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D𝑛 𝐹)‘𝑁)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁)) |
19 | 12 | fveq1d 6768 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁)) |
20 | 19 | fveq2d 6770 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘((𝑆 D𝑛 𝐹)‘𝑁)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁))) |
21 | 18, 20 | eqtr3id 2792 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁)) = ((𝑥 ∈ V ↦ (𝑆 D 𝑥))‘(seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘𝑁))) |
22 | 9, 13, 21 | 3eqtr4d 2788 | 1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3429 ⊆ wss 3886 {csn 4561 ↦ cmpt 5156 × cxp 5582 ∘ ccom 5588 ‘cfv 6426 (class class class)co 7267 1st c1st 7818 ↑pm cpm 8603 ℂcc 10879 0cc0 10881 1c1 10882 + caddc 10884 ℕ0cn0 12243 ℤ≥cuz 12592 seqcseq 13731 D cdv 25037 D𝑛 cdvn 25038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-n0 12244 df-z 12330 df-uz 12593 df-seq 13732 df-dvn 25042 |
This theorem is referenced by: dvn1 25100 dvnadd 25103 dvnres 25105 cpnord 25109 dvnfre 25126 c1lip2 25172 dvnply2 25457 dvntaylp 25540 taylthlem1 25542 taylthlem2 25543 dvnmptdivc 43460 dvnmptconst 43463 dvnxpaek 43464 dvnmul 43465 etransclem2 43758 |
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