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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 6879 | . . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0)) | |
2 | 1 | eleq1d 2818 | . . . . 5 ⊢ (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆))) |
3 | 2 | imbi2d 340 | . . . 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 6879 | . . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛)) | |
5 | 4 | eleq1d 2818 | . . . . 5 ⊢ (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆))) |
6 | 5 | imbi2d 340 | . . . 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 6879 | . . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1))) | |
8 | 7 | eleq1d 2818 | . . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))) |
9 | 8 | imbi2d 340 | . . . 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 6879 | . . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁)) | |
11 | 10 | eleq1d 2818 | . . . . 5 ⊢ (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
12 | 11 | imbi2d 340 | . . . 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 4001 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
14 | cnex 11175 | . . . . . . 7 ⊢ ℂ ∈ V | |
15 | plyf 25643 | . . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
16 | 15 | adantl 482 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ) |
17 | fpmg 8847 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
18 | 14, 14, 16, 17 | mp3an12i 1465 | . . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
19 | dvn0 25372 | . . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) | |
20 | 13, 18, 19 | sylancr 587 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹) |
21 | simpr 485 | . . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) | |
22 | 20, 21 | eqeltrd 2833 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘0) ∈ (Poly‘𝑆)) |
23 | dvply2g 25729 | . . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)) | |
24 | 23 | ex 413 | . . . . . . . 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 25373 | . . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) | |
27 | 13, 26 | mp3an1 1448 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
28 | 18, 27 | sylan 580 | . . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛))) |
29 | 28 | eleq1d 2818 | . . . . . . 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 258 | . . . . . 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 414 | . . . . 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 12641 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))) |
34 | 33 | impcom 408 | . 2 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)) |
35 | 34 | 3impa 1110 | 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3945 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 ↑pm cpm 8806 ℂcc 11092 0cc0 11094 1c1 11095 + caddc 11097 ℕ0cn0 12456 SubRingcsubrg 20310 ℂfldccnfld 20880 D cdv 25311 D𝑛 cdvn 25312 Polycply 25629 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-inf2 9620 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 ax-addf 11173 ax-mulf 11174 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-2o 8451 df-er 8688 df-map 8807 df-pm 8808 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-fi 9390 df-sup 9421 df-inf 9422 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-div 11856 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-q 12917 df-rp 12959 df-xneg 13076 df-xadd 13077 df-xmul 13078 df-icc 13315 df-fz 13469 df-fzo 13612 df-fl 13741 df-seq 13951 df-exp 14012 df-hash 14275 df-cj 15030 df-re 15031 df-im 15032 df-sqrt 15166 df-abs 15167 df-clim 15416 df-rlim 15417 df-sum 15617 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17352 df-topn 17353 df-0g 17371 df-gsum 17372 df-topgen 17373 df-pt 17374 df-prds 17377 df-xrs 17432 df-qtop 17437 df-imas 17438 df-xps 17440 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-submnd 18650 df-grp 18799 df-minusg 18800 df-mulg 18925 df-subg 18977 df-cntz 19149 df-cmn 19616 df-mgp 19949 df-ur 19966 df-ring 20018 df-cring 20019 df-subrg 20312 df-psmet 20872 df-xmet 20873 df-met 20874 df-bl 20875 df-mopn 20876 df-fbas 20877 df-fg 20878 df-cnfld 20881 df-top 22327 df-topon 22344 df-topsp 22366 df-bases 22380 df-cld 22454 df-ntr 22455 df-cls 22456 df-nei 22533 df-lp 22571 df-perf 22572 df-cn 22662 df-cnp 22663 df-haus 22750 df-tx 22997 df-hmeo 23190 df-fil 23281 df-fm 23373 df-flim 23374 df-flf 23375 df-xms 23757 df-ms 23758 df-tms 23759 df-cncf 24325 df-0p 25118 df-limc 25314 df-dv 25315 df-dvn 25316 df-ply 25633 df-coe 25635 df-dgr 25636 |
This theorem is referenced by: dvnply 25732 taylthlem2 25817 |
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