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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 24504 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))) |
3 | 2 | fveq1d 6658 | . 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = (seq0(((𝑥 ∈ V ↦ (𝑆 D 𝑥)) ∘ 1st ), (ℕ0 × {𝐹}))‘0)) |
4 | 0z 11979 | . . 3 ⊢ 0 ∈ ℤ | |
5 | simpr 487 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
6 | 0nn0 11899 | . . . 4 ⊢ 0 ∈ ℕ0 | |
7 | fvconst2g 6950 | . . . 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 13373 | . 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 1537 ∈ wcel 2114 Vcvv 3486 ⊆ wss 3924 {csn 4553 ↦ cmpt 5132 × cxp 5539 ∘ ccom 5545 ‘cfv 6341 (class class class)co 7142 1st c1st 7673 ↑pm cpm 8393 ℂcc 10521 0cc0 10523 ℕ0cn0 11884 seqcseq 13359 D cdv 24446 D𝑛 cdvn 24447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-n0 11885 df-z 11969 df-uz 12231 df-seq 13360 df-dvn 24451 |
This theorem is referenced by: dvn1 24508 dvnadd 24511 dvnres 24513 cpncn 24518 dvnfre 24534 c1lip2 24580 dvnply2 24862 tayl0 24936 dvntaylp 24945 taylthlem1 24947 dvnmptdivc 42313 dvnxpaek 42317 dvnmul 42318 dvnprodlem3 42323 etransclem46 42655 |
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