dvnply2 - Metamath Proof Explorer

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Theorem dvnply2 26427

Description: Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Assertion**
Ref	Expression
dvnply2	⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))

Proof of Theorem dvnply2 Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆)))
3	2	imbi2d 343	. . . 4 ⊢ (𝑥 = 0 → (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆))))
4		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
5	4	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)))
6	5	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆))))
7		fveq2 6881	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
8	7	eleq1d 2846	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))
9	8	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))))
10		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
11	10	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)))
12	11	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆)) ↔ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))))
13		ssid 3958	. . . . . 6 ⊢ ℂ ⊆ ℂ
14		cnex 11180	. . . . . . 7 ⊢ ℂ ∈ V
15		plyf 26334	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
16	15	adantl 486	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
17		fpmg 8865	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑_pm ℂ))
18	14, 14, 16, 17	mp3an12i 1492	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
19		dvn0 26062	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
20	13, 18, 19	sylancr 598	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
21		simpr 489	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
22	20, 21	eqeltrd 2861	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆))
23		dvply2g 26425	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))
24	23	ex 417	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)))
25	24	ad2antrr 738	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)))
26		dvnp1 26063	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
27	13, 26	mp3an1 1475	. . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
28	18, 27	sylan 591	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
29	28	eleq1d 2846	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆) ↔ (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)))
30	25, 29	sylibrd 262	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))
31	30	expcom 418	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))))
32	31	a2d 30	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆))))
33	3, 6, 9, 12, 22, 32	nn0ind 12690	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)))
34	33	impcom 412	. 2 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑁 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))
35	34	3impa 1125	1 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 Vcvv 3453 ⊆ wss 3904 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_pm cpm 8824 ℂcc 11097 0cc0 11099 1c1 11100 + caddc 11102 ℕ₀cn0 12503 SubRingcsubrg 20653 ℂ_fldccnfld 21501 D cdv 26001 D^𝑛 cdvn 26002 Polycply 26320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-icc 13378 df-fz 13535 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-mulg 19133 df-subg 19188 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-subrng 20630 df-subrg 20654 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-lp 23272 df-perf 23273 df-cn 23363 df-cnp 23364 df-haus 23451 df-tx 23698 df-hmeo 23891 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-tms 24458 df-cncf 25016 df-0p 25808 df-limc 26004 df-dv 26005 df-dvn 26006 df-ply 26324 df-coe 26326 df-dgr 26327

This theorem is referenced by: dvnply 26428 taylthlem2 26513

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