dvnply2 - Metamath Proof Explorer

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Theorem dvnply2 26253

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 6841	. . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆)))
3	2	imbi2d 340	. . . 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 6841	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
5	4	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)))
6	5	imbi2d 340	. . . 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 6841	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
8	7	eleq1d 2822	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ∈ (Poly‘𝑆)))
9	8	imbi2d 340	. . . 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 6841	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
11	10	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ∈ (Poly‘𝑆) ↔ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)))
12	11	imbi2d 340	. . . 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 3945	. . . . . 6 ⊢ ℂ ⊆ ℂ
14		cnex 11119	. . . . . . 7 ⊢ ℂ ∈ V
15		plyf 26163	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
17		fpmg 8816	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑_pm ℂ))
18	14, 14, 16, 17	mp3an12i 1468	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
19		dvn0 25891	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
20	13, 18, 19	sylancr 588	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
21		simpr 484	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
22	20, 21	eqeltrd 2837	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆))
23		dvply2g 26251	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆)) → (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆))
24	23	ex 412	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘𝑆) → (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ∈ (Poly‘𝑆)))
25	24	ad2antrr 727	. . . . . . 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 25892	. . . . . . . . . 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 1451	. . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
28	18, 27	sylan 581	. . . . . . . 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 2822	. . . . . . 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 259	. . . . . 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 413	. . . . 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 29	. . . 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 12624	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆)))
34	33	impcom 407	. 2 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑁 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))
35	34	3impa 1110	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ↑_pm cpm 8774 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 ℕ₀cn0 12437 SubRingcsubrg 20546 ℂ_fldccnfld 21352 D cdv 25830 D^𝑛 cdvn 25831 Polycply 26149

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-mulg 19044 df-subg 19099 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-subrng 20523 df-subrg 20547 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-0p 25637 df-limc 25833 df-dv 25834 df-dvn 25835 df-ply 26153 df-coe 26155 df-dgr 26156

This theorem is referenced by: dvnply 26254 taylthlem2 26339

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