dvnply2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvnply2		Structured version Visualization version GIF version

Theorem dvnply2 26232

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 6831	. . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	eleq1d 2818	. . . . 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 6831	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
5	4	eleq1d 2818	. . . . 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 6831	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
8	7	eleq1d 2818	. . . . 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 6831	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
11	10	eleq1d 2818	. . . . 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 3954	. . . . . 6 ⊢ ℂ ⊆ ℂ
14		cnex 11097	. . . . . . 7 ⊢ ℂ ∈ V
15		plyf 26140	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
17		fpmg 8801	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ) → 𝐹 ∈ (ℂ ↑_pm ℂ))
18	14, 14, 16, 17	mp3an12i 1467	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
19		dvn0 25863	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
20	13, 18, 19	sylancr 587	. . . . 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 2833	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆))
23		dvply2g 26229	. . . . . . . . 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 726	. . . . . . 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 25864	. . . . . . . . . 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 1450	. . . . . . . . 9 ⊢ ((𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
28	18, 27	sylan 580	. . . . . . . 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 2818	. . . . . . 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 12578	. . 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 1109	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 1086 = wceq 1541 ∈ wcel 2113 Vcvv 3438 ⊆ wss 3899 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_pm cpm 8760 ℂcc 11014 0cc0 11016 1c1 11017 + caddc 11019 ℕ₀cn0 12391 SubRingcsubrg 20494 ℂ_fldccnfld 21301 D cdv 25801 D^𝑛 cdvn 25802 Polycply 26126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9541 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-pre-sup 11094 ax-addf 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8831 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-fsupp 9256 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9406 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-div 11785 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-7 12203 df-8 12204 df-9 12205 df-n0 12392 df-z 12479 df-dec 12599 df-uz 12743 df-q 12857 df-rp 12901 df-xneg 13021 df-xadd 13022 df-xmul 13023 df-icc 13262 df-fz 13418 df-fzo 13565 df-fl 13706 df-seq 13919 df-exp 13979 df-hash 14248 df-cj 15016 df-re 15017 df-im 15018 df-sqrt 15152 df-abs 15153 df-clim 15405 df-rlim 15406 df-sum 15604 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-starv 17186 df-sca 17187 df-vsca 17188 df-ip 17189 df-tset 17190 df-ple 17191 df-ds 17193 df-unif 17194 df-hom 17195 df-cco 17196 df-rest 17336 df-topn 17337 df-0g 17355 df-gsum 17356 df-topgen 17357 df-pt 17358 df-prds 17361 df-xrs 17416 df-qtop 17421 df-imas 17422 df-xps 17424 df-mre 17498 df-mrc 17499 df-acs 17501 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-submnd 18702 df-grp 18859 df-minusg 18860 df-mulg 18991 df-subg 19046 df-cntz 19239 df-cmn 19704 df-abl 19705 df-mgp 20069 df-rng 20081 df-ur 20110 df-ring 20163 df-cring 20164 df-subrng 20471 df-subrg 20495 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-lp 23061 df-perf 23062 df-cn 23152 df-cnp 23153 df-haus 23240 df-tx 23487 df-hmeo 23680 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24245 df-ms 24246 df-tms 24247 df-cncf 24808 df-0p 25608 df-limc 25804 df-dv 25805 df-dvn 25806 df-ply 26130 df-coe 26132 df-dgr 26133

This theorem is referenced by: dvnply 26233 taylthlem2 26319 taylthlem2OLD 26320

W3C validator