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Theorem dvply2g 22975
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 ` fld )  /\  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.
StepHypRef Expression
1 plyf 22889 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
21adantl 466 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
32feqmptd 5904 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  =  ( a  e.  CC  |->  ( F `  a ) ) )
4 simplr 756 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  F  e.  (Poly `  S ) )
5 dgrcl 22924 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
65adantl 466 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
76nn0zd 11008 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  ZZ )
87adantr 465 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  (deg `  F )  e.  ZZ )
9 uzid 11143 . . . . . . 7  |-  ( (deg
`  F )  e.  ZZ  ->  (deg `  F
)  e.  ( ZZ>= `  (deg `  F ) ) )
10 peano2uz 11182 . . . . . . 7  |-  ( (deg
`  F )  e.  ( ZZ>= `  (deg `  F
) )  ->  (
(deg `  F )  +  1 )  e.  ( ZZ>= `  (deg `  F
) ) )
118, 9, 103syl 18 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  ( (deg `  F
)  +  1 )  e.  ( ZZ>= `  (deg `  F ) ) )
12 simpr 461 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  a  e.  CC )
13 eqid 2404 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
14 eqid 2404 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
1513, 14coeid3 22931 . . . . . 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
) ) )
164, 11, 12, 15syl3anc 1232 . . . . 5  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  ( F `  a
)  =  sum_ b  e.  ( 0 ... (
(deg `  F )  +  1 ) ) ( ( (coeff `  F ) `  b
)  x.  ( a ^ b ) ) )
1716mpteq2dva 4483 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  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 ) ) ) )
183, 17eqtrd 2445 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  =  ( a  e.  CC  |->  sum_ b  e.  ( 0 ... ( (deg `  F )  +  1 ) ) ( ( (coeff `  F ) `  b )  x.  (
a ^ b ) ) ) )
196nn0cnd 10897 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  CC )
20 ax-1cn 9582 . . . . . . . 8  |-  1  e.  CC
21 pncan 9864 . . . . . . . 8  |-  ( ( (deg `  F )  e.  CC  /\  1  e.  CC )  ->  (
( (deg `  F
)  +  1 )  -  1 )  =  (deg `  F )
)
2219, 20, 21sylancl 662 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (
(deg `  F )  +  1 )  - 
1 )  =  (deg
`  F ) )
2322eqcomd 2412 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  =  ( ( (deg `  F )  +  1 )  - 
1 ) )
2423oveq2d 6296 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... ( ( (deg `  F )  +  1 )  - 
1 ) ) )
2524sumeq1d 13674 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  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 ) ) )
2625mpteq2dv 4484 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  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 ) ) ) )
2713coef3 22923 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2827adantl 466 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
29 oveq1 6287 . . . . 5  |-  ( c  =  b  ->  (
c  +  1 )  =  ( b  +  1 ) )
3029fveq2d 5855 . . . . 5  |-  ( c  =  b  ->  (
(coeff `  F ) `  ( c  +  1 ) )  =  ( (coeff `  F ) `  ( b  +  1 ) ) )
3129, 30oveq12d 6298 . . . 4  |-  ( c  =  b  ->  (
( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  =  ( ( b  +  1 )  x.  ( (coeff `  F ) `  (
b  +  1 ) ) ) )
3231cbvmptv 4489 . . 3  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( b  e.  NN0  |->  ( ( b  +  1 )  x.  (
(coeff `  F ) `  ( b  +  1 ) ) ) )
33 peano2nn0 10879 . . . 4  |-  ( (deg
`  F )  e. 
NN0  ->  ( (deg `  F )  +  1 )  e.  NN0 )
346, 33syl 17 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  1 )  e.  NN0 )
3518, 26, 28, 32, 34dvply1 22974 . 2  |-  ( ( S  e.  (SubRing ` fld )  /\  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 18746 . . . . 5  |-  CC  =  ( Base ` fld )
3736subrgss 17752 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
3837adantr 465 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  S  C_  CC )
39 elfznn0 11828 . . . 4  |-  ( b  e.  ( 0 ... (deg `  F )
)  ->  b  e.  NN0 )
40 simpll 754 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  S  e.  (SubRing ` fld ) )
41 zsssubrg 18798 . . . . . . . . 9  |-  ( S  e.  (SubRing ` fld )  ->  ZZ  C_  S )
4241ad2antrr 726 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  ZZ  C_  S )
43 peano2nn0 10879 . . . . . . . . . 10  |-  ( c  e.  NN0  ->  ( c  +  1 )  e. 
NN0 )
4443adantl 466 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  NN0 )
4544nn0zd 11008 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  ZZ )
4642, 45sseldd 3445 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  S )
47 simplr 756 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  F  e.  (Poly `  S
) )
48 subrgsubg 17757 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
49 cnfld0 18764 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
5049subg0cl 16535 . . . . . . . . . . 11  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
5148, 50syl 17 . . . . . . . . . 10  |-  ( S  e.  (SubRing ` fld )  ->  0  e.  S )
5251ad2antrr 726 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
0  e.  S )
5313coef2 22922 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  (coeff `  F ) : NN0 --> S )
5447, 52, 53syl2anc 661 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> S )
5554, 44ffvelrnd 6012 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( (coeff `  F
) `  ( c  +  1 ) )  e.  S )
56 cnfldmul 18748 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5756subrgmcl 17763 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  (
c  +  1 )  e.  S  /\  (
(coeff `  F ) `  ( c  +  1 ) )  e.  S
)  ->  ( (
c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  e.  S
)
5840, 46, 55, 57syl3anc 1232 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) )  e.  S )
59 eqid 2404 . . . . . 6  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) ) )
6058, 59fmptd 6035 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) : NN0 --> S )
6160ffvelrnda 6011 . . . 4  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  b  e.  NN0 )  -> 
( ( c  e. 
NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) `  b
)  e.  S )
6239, 61sylan2 474 . . 3  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  b  e.  ( 0 ... (deg `  F
) ) )  -> 
( ( c  e. 
NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) `  b
)  e.  S )
6338, 6, 62elplyd 22893 . 2  |-  ( ( S  e.  (SubRing ` fld )  /\  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 ) )
6435, 63eqeltrd 2492 1  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( CC  _D  F )  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    C_ wss 3416    |-> cmpt 4455   -->wf 5567   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    - cmin 9843   NN0cn0 10838   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728   ^cexp 12212   sum_csu 13659  SubGrpcsubg 16521  SubRingcsubrg 17747  ℂfldccnfld 18742    _D cdv 22561  Polycply 22875  coeffccoe 22877  degcdgr 22878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-rlim 13463  df-sum 13660  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-grp 16383  df-minusg 16384  df-mulg 16386  df-subg 16524  df-cntz 16681  df-cmn 17126  df-mgp 17464  df-ur 17476  df-ring 17522  df-cring 17523  df-subrg 17749  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-fbas 18738  df-fg 18739  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-lp 19932  df-perf 19933  df-cn 20023  df-cnp 20024  df-haus 20111  df-tx 20357  df-hmeo 20550  df-fil 20641  df-fm 20733  df-flim 20734  df-flf 20735  df-xms 21117  df-ms 21118  df-tms 21119  df-cncf 21676  df-0p 22371  df-limc 22564  df-dv 22565  df-ply 22879  df-coe 22881  df-dgr 22882
This theorem is referenced by:  dvply2  22976  dvnply2  22977
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