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Mirrors > Home > MPE Home > Th. List > dvn0 | Structured version Visualization version GIF version |
Description: Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvn0 | ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ V ↦ (𝑆 D 𝑥)) = (𝑥 ∈ V ↦ (𝑆 D 𝑥)) | |
2 | 1 | dvnfval 24513 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))) |
3 | 2 | fveq1d 6666 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘0)) |
4 | 0z 11986 | . . 3 ⊢ 0 ∈ ℤ | |
5 | simpr 487 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
6 | 0nn0 11906 | . . . 4 ⊢ 0 ∈ ℕ0 | |
7 | fvconst2g 6958 | . . . 4 ⊢ ((𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 0 ∈ ℕ0) → ((ℕ0 × {𝐹})‘0) = 𝐹) | |
8 | 5, 6, 7 | sylancl 588 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((ℕ0 × {𝐹})‘0) = 𝐹) |
9 | 4, 8 | seq1i 13377 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘0) = 𝐹) |
10 | 3, 9 | eqtrd 2856 | 1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 {csn 4560 ↦ cmpt 5138 × cxp 5547 ∘ ccom 5553 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 ↑pm cpm 8401 ℂcc 10529 0cc0 10531 ℕ0cn0 11891 seqcseq 13363 D cdv 24455 D𝑛 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 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 cpncn 24527 dvnfre 24543 c1lip2 24589 dvnply2 24870 tayl0 24944 dvntaylp 24953 taylthlem1 24955 dvnmptdivc 42216 dvnxpaek 42220 dvnmul 42221 dvnprodlem3 42226 etransclem46 42559 |
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