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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 6645 | . . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0)) | |
2 | 1 | eleq1d 2874 | . . . . 5 ⊢ (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆))) |
3 | 2 | imbi2d 344 | . . . 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 6645 | . . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛)) | |
5 | 4 | eleq1d 2874 | . . . . 5 ⊢ (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆))) |
6 | 5 | imbi2d 344 | . . . 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 6645 | . . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1))) | |
8 | 7 | eleq1d 2874 | . . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))) |
9 | 8 | imbi2d 344 | . . . 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 6645 | . . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁)) | |
11 | 10 | eleq1d 2874 | . . . . 5 ⊢ (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
12 | 11 | imbi2d 344 | . . . 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 3937 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
14 | cnex 10607 | . . . . . . 7 ⊢ ℂ ∈ V | |
15 | plyf 24795 | . . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
16 | 15 | adantl 485 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ) |
17 | fpmg 8415 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
18 | 14, 14, 16, 17 | mp3an12i 1462 | . . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
19 | dvn0 24527 | . . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) | |
20 | 13, 18, 19 | sylancr 590 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) |
21 | simpr 488 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) | |
22 | 20, 21 | eqeltrd 2890 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆)) |
23 | dvply2g 24881 | . . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)) | |
24 | 23 | ex 416 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))) |
25 | 24 | ad2antrr 725 | . . . . . . 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 24528 | . . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) | |
27 | 13, 26 | mp3an1 1445 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
28 | 18, 27 | sylan 583 | . . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
29 | 28 | eleq1d 2874 | . . . . . . 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 262 | . . . . . 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 417 | . . . . 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 12065 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
34 | 33 | impcom 411 | . 2 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)) |
35 | 34 | 3impa 1107 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 ℕ0cn0 11885 SubRingcsubrg 19524 ℂfldccnfld 20091 D cdv 24466 D𝑛 cdvn 24467 Polycply 24781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-0p 24274 df-limc 24469 df-dv 24470 df-dvn 24471 df-ply 24785 df-coe 24787 df-dgr 24788 |
This theorem is referenced by: dvnply 24884 taylthlem2 24969 |
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