dvnply2 - Metamath Proof Explorer

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Theorem dvnply2 26268

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 6844	. . . . . 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 6844	. . . . . 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 6844	. . . . . 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 6844	. . . . . 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 3958	. . . . . 6 ⊢ ℂ ⊆ ℂ
14		cnex 11121	. . . . . . 7 ⊢ ℂ ∈ V
15		plyf 26176	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
17		fpmg 8820	. . . . . . 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 25899	. . . . . 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 26265	. . . . . . . . 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 25900	. . . . . . . . . 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 12601	. . 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 3442 ⊆ wss 3903 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ↑_pm cpm 8778 ℂcc 11038 0cc0 11040 1c1 11041 + caddc 11043 ℕ₀cn0 12415 SubRingcsubrg 20519 ℂ_fldccnfld 21326 D cdv 25837 D^𝑛 cdvn 25838 Polycply 26162

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-icc 13282 df-fz 13438 df-fzo 13585 df-fl 13726 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-rlim 15426 df-sum 15624 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-rest 17356 df-topn 17357 df-0g 17375 df-gsum 17376 df-topgen 17377 df-pt 17378 df-prds 17381 df-xrs 17437 df-qtop 17442 df-imas 17443 df-xps 17445 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-grp 18883 df-minusg 18884 df-mulg 19015 df-subg 19070 df-cntz 19263 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-subrng 20496 df-subrg 20520 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-nei 23059 df-lp 23097 df-perf 23098 df-cn 23188 df-cnp 23189 df-haus 23276 df-tx 23523 df-hmeo 23716 df-fil 23807 df-fm 23899 df-flim 23900 df-flf 23901 df-xms 24281 df-ms 24282 df-tms 24283 df-cncf 24844 df-0p 25644 df-limc 25840 df-dv 25841 df-dvn 25842 df-ply 26166 df-coe 26168 df-dgr 26169

This theorem is referenced by: dvnply 26269 taylthlem2 26355 taylthlem2OLD 26356

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