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 26215

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 6817	. . . . . 6 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	eleq1d 2814	. . . . 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 6817	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
5	4	eleq1d 2814	. . . . 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 6817	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
8	7	eleq1d 2814	. . . . 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 6817	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
11	10	eleq1d 2814	. . . . 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 3955	. . . . . 6 ⊢ ℂ ⊆ ℂ
14		cnex 11079	. . . . . . 7 ⊢ ℂ ∈ V
15		plyf 26123	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
17		fpmg 8787	. . . . . . 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 25846	. . . . . 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 2829	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ D^𝑛 𝐹)‘0) ∈ (Poly‘𝑆))
23		dvply2g 26212	. . . . . . . . 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 25847	. . . . . . . . . 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 2814	. . . . . . 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 12560	. . 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 2110 Vcvv 3434 ⊆ wss 3900 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 ↑_pm cpm 8746 ℂcc 10996 0cc0 10998 1c1 10999 + caddc 11001 ℕ₀cn0 12373 SubRingcsubrg 20477 ℂ_fldccnfld 21284 D cdv 25784 D^𝑛 cdvn 25785 Polycply 26109

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 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

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 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-seq 13901 df-exp 13961 df-hash 14230 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-clim 15387 df-rlim 15388 df-sum 15586 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-grp 18841 df-minusg 18842 df-mulg 18973 df-subg 19028 df-cntz 19222 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-subrng 20454 df-subrg 20478 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-0p 25591 df-limc 25787 df-dv 25788 df-dvn 25789 df-ply 26113 df-coe 26115 df-dgr 26116

This theorem is referenced by: dvnply 26216 taylthlem2 26302 taylthlem2OLD 26303

W3C validator