Theorem dvply2g 24040

Description: The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
dvply2g	|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )

Proof of Theorem dvply2g

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyf 23954	. . . . . 6 |- ( F e. (Poly ` S ) -> F : CC --> CC )
2	1	adantl 482	. . . . 5 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F : CC --> CC )
3	2	feqmptd 6249	. . . 4 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F = ( a e. CC |-> ( F ` a ) ) )
4		simplr 792	. . . . . 6 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ a e. CC ) -> F e. (Poly ` S ) )
5		dgrcl 23989	. . . . . . . . . 10 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
6	5	adantl 482	. . . . . . . . 9 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> (deg ` F ) e. NN0 )
7	6	nn0zd 11480	. . . . . . . 8 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> (deg ` F ) e. ZZ )
8	7	adantr 481	. . . . . . 7 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ a e. CC ) -> (deg ` F ) e. ZZ )
9		uzid 11702	. . . . . . 7 |- ( (deg ` F ) e. ZZ -> (deg ` F ) e. ( ZZ>= ` (deg ` F ) ) )
10		peano2uz 11741	. . . . . . 7 |- ( (deg ` F ) e. ( ZZ>= ` (deg ` F ) ) -> ( (deg ` F ) + 1 ) e. ( ZZ>= ` (deg ` F ) ) )
11	8, 9, 10	3syl 18	. . . . . 6 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ a e. CC ) -> ( (deg ` F ) + 1 ) e. ( ZZ>= ` (deg ` F ) ) )
12		simpr 477	. . . . . 6 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ a e. CC ) -> a e. CC )
13		eqid 2622	. . . . . . 7 |- (coeff ` F ) = (coeff ` F )
14		eqid 2622	. . . . . . 7 |- (deg ` F ) = (deg ` F )
15	13, 14	coeid3 23996	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ ( (deg ` F ) + 1 ) e. ( ZZ>= ` (deg ` F ) ) /\ a e. CC ) -> ( F ` a ) = sum_ b e. ( 0 ... ( (deg ` F ) + 1 ) ) ( ( (coeff ` F ) ` b ) x. ( a ^ b ) ) )
16	4, 11, 12, 15	syl3anc 1326	. . . . 5 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ a e. CC ) -> ( F ` a ) = sum_ b e. ( 0 ... ( (deg ` F ) + 1 ) ) ( ( (coeff ` F ) ` b ) x. ( a ^ b ) ) )
17	16	mpteq2dva 4744	. . . 4 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( a e. CC |-> ( F ` a ) ) = ( a e. CC |-> sum_ b e. ( 0 ... ( (deg ` F ) + 1 ) ) ( ( (coeff ` F ) ` b ) x. ( a ^ b ) ) ) )
18	3, 17	eqtrd 2656	. . 3 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F = ( a e. CC |-> sum_ b e. ( 0 ... ( (deg ` F ) + 1 ) ) ( ( (coeff ` F ) ` b ) x. ( a ^ b ) ) ) )
19	6	nn0cnd 11353	. . . . . . . 8 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> (deg ` F ) e. CC )
20		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
21		pncan 10287	. . . . . . . 8 |- ( ( (deg ` F ) e. CC /\ 1 e. CC ) -> ( ( (deg ` F ) + 1 ) - 1 ) = (deg ` F ) )
22	19, 20, 21	sylancl 694	. . . . . . 7 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( ( (deg ` F ) + 1 ) - 1 ) = (deg ` F ) )
23	22	eqcomd 2628	. . . . . 6 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> (deg ` F ) = ( ( (deg ` F ) + 1 ) - 1 ) )
24	23	oveq2d 6666	. . . . 5 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( 0 ... (deg ` F ) ) = ( 0 ... ( ( (deg ` F ) + 1 ) - 1 ) ) )
25	24	sumeq1d 14431	. . . 4 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> sum_ b e. ( 0 ... (deg ` F ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) = sum_ b e. ( 0 ... ( ( (deg ` F ) + 1 ) - 1 ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) )
26	25	mpteq2dv 4745	. . 3 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( a e. CC |-> sum_ b e. ( 0 ... (deg ` F ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) = ( a e. CC |-> sum_ b e. ( 0 ... ( ( (deg ` F ) + 1 ) - 1 ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) )
27	13	coef3 23988	. . . 4 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
28	27	adantl 482	. . 3 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> (coeff ` F ) : NN0 --> CC )
29		oveq1 6657	. . . . 5 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
30	29	fveq2d 6195	. . . . 5 |- ( c = b -> ( (coeff ` F ) ` ( c + 1 ) ) = ( (coeff ` F ) ` ( b + 1 ) ) )
31	29, 30	oveq12d 6668	. . . 4 |- ( c = b -> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( (coeff ` F ) ` ( b + 1 ) ) ) )
32	31	cbvmptv 4750	. . 3 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( (coeff ` F ) ` ( b + 1 ) ) ) )
33		peano2nn0 11333	. . . 4 |- ( (deg ` F ) e. NN0 -> ( (deg ` F ) + 1 ) e. NN0 )
34	6, 33	syl 17	. . 3 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( (deg ` F ) + 1 ) e. NN0 )
35	18, 26, 28, 32, 34	dvply1 24039	. 2 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) = ( a e. CC |-> sum_ b e. ( 0 ... (deg ` F ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) )
36		cnfldbas 19750	. . . . 5 |- CC = ( Base ` ℂfld )
37	36	subrgss 18781	. . . 4 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
38	37	adantr 481	. . 3 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> S C_ CC )
39		elfznn0 12433	. . . 4 |- ( b e. ( 0 ... (deg ` F ) ) -> b e. NN0 )
40		simpll 790	. . . . . . 7 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> S e. (SubRing ` ℂfld ) )
41		zsssubrg 19804	. . . . . . . . 9 |- ( S e. (SubRing ` ℂfld ) -> ZZ C_ S )
42	41	ad2antrr 762	. . . . . . . 8 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ZZ C_ S )
43		peano2nn0 11333	. . . . . . . . . 10 |- ( c e. NN0 -> ( c + 1 ) e. NN0 )
44	43	adantl 482	. . . . . . . . 9 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ( c + 1 ) e. NN0 )
45	44	nn0zd 11480	. . . . . . . 8 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ( c + 1 ) e. ZZ )
46	42, 45	sseldd 3604	. . . . . . 7 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ( c + 1 ) e. S )
47		simplr 792	. . . . . . . . 9 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> F e. (Poly ` S ) )
48		subrgsubg 18786	. . . . . . . . . . 11 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
49		cnfld0 19770	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
50	49	subg0cl 17602	. . . . . . . . . . 11 |- ( S e. (SubGrp ` ℂfld ) -> 0 e. S )
51	48, 50	syl 17	. . . . . . . . . 10 |- ( S e. (SubRing ` ℂfld ) -> 0 e. S )
52	51	ad2antrr 762	. . . . . . . . 9 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> 0 e. S )
53	13	coef2 23987	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ 0 e. S ) -> (coeff ` F ) : NN0 --> S )
54	47, 52, 53	syl2anc 693	. . . . . . . 8 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> (coeff ` F ) : NN0 --> S )
55	54, 44	ffvelrnd 6360	. . . . . . 7 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ( (coeff ` F ) ` ( c + 1 ) ) e. S )
56		cnfldmul 19752	. . . . . . . 8 |- x. = ( .r ` ℂfld )
57	56	subrgmcl 18792	. . . . . . 7 |- ( ( S e. (SubRing ` ℂfld ) /\ ( c + 1 ) e. S /\ ( (coeff ` F ) ` ( c + 1 ) ) e. S ) -> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) e. S )
58	40, 46, 55, 57	syl3anc 1326	. . . . . 6 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ c e. NN0 ) -> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) e. S )
59		eqid 2622	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) = ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) )
60	58, 59	fmptd 6385	. . . . 5 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) : NN0 --> S )
61	60	ffvelrnda 6359	. . . 4 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ b e. NN0 ) -> ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) e. S )
62	39, 61	sylan2 491	. . 3 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ b e. ( 0 ... (deg ` F ) ) ) -> ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) e. S )
63	38, 6, 62	elplyd 23958	. 2 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( a e. CC |-> sum_ b e. ( 0 ... (deg ` F ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( (coeff ` F ) ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) e. (Poly ` S ) )
64	35, 63	eqeltrd 2701	1 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416  SubGrpcsubg 17588  SubRingcsubrg 18776  ℂ_fldccnfld 19746   _D cdv 23627  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dvply2  24041  dvnply2  24042