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Mirrors > Home > MPE Home > Th. List > dvnply2 | Structured version Visualization version GIF version |
Description: Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
dvnply2 | ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0)) | |
2 | 1 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆))) |
3 | 2 | imbi2d 343 | . . . 4 ⊢ (𝑥 = 0 → (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆)))) |
4 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛)) | |
5 | 4 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆))) |
6 | 5 | imbi2d 343 | . . . 4 ⊢ (𝑥 = 𝑛 → (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)))) |
7 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1))) | |
8 | 7 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))) |
9 | 8 | imbi2d 343 | . . . 4 ⊢ (𝑥 = (𝑛 + 1) → (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))) |
10 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁)) | |
11 | 10 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
12 | 11 | imbi2d 343 | . . . 4 ⊢ (𝑥 = 𝑁 → (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)))) |
13 | ssid 3988 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
14 | cnex 10612 | . . . . . . 7 ⊢ ℂ ∈ V | |
15 | plyf 24782 | . . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
16 | 15 | adantl 484 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ) |
17 | fpmg 8426 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
18 | 14, 14, 16, 17 | mp3an12i 1461 | . . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
19 | dvn0 24515 | . . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) | |
20 | 13, 18, 19 | sylancr 589 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) |
21 | simpr 487 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) | |
22 | 20, 21 | eqeltrd 2913 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆)) |
23 | dvply2g 24868 | . . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)) | |
24 | 23 | ex 415 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))) |
25 | 24 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))) |
26 | dvnp1 24516 | . . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) | |
27 | 13, 26 | mp3an1 1444 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
28 | 18, 27 | sylan 582 | . . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
29 | 28 | eleq1d 2897 | . . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆) ↔ (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))) |
30 | 25, 29 | sylibrd 261 | . . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))) |
31 | 30 | expcom 416 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))) |
32 | 31 | a2d 29 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))) |
33 | 3, 6, 9, 12, 22, 32 | nn0ind 12071 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
34 | 33 | impcom 410 | . 2 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)) |
35 | 34 | 3impa 1106 | 1 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑pm cpm 8401 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 ℕ0cn0 11891 SubRingcsubrg 19525 ℂfldccnfld 20539 D cdv 24455 D𝑛 cdvn 24456 Polycply 24768 |
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 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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-rmo 3146 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-int 4869 df-iun 4913 df-iin 4914 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-se 5509 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-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-0p 24265 df-limc 24458 df-dv 24459 df-dvn 24460 df-ply 24772 df-coe 24774 df-dgr 24775 |
This theorem is referenced by: dvnply 24871 taylthlem2 24956 |
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