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Theorem dvply2g 19665
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 19580 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
21adantl 452 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
32feqmptd 5575 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  =  ( a  e.  CC  |->  ( F `  a ) ) )
4 simplr 731 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  F  e.  (Poly `  S ) )
5 dgrcl 19615 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
65adantl 452 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
76nn0zd 10115 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  ZZ )
87adantr 451 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  (deg `  F )  e.  ZZ )
9 uzid 10242 . . . . . . 7  |-  ( (deg
`  F )  e.  ZZ  ->  (deg `  F
)  e.  ( ZZ>= `  (deg `  F ) ) )
10 peano2uz 10272 . . . . . . 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 447 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  a  e.  CC )
13 eqid 2283 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
14 eqid 2283 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
1513, 14coeid3 19622 . . . . . 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 1182 . . . . 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 4106 . . . 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 2315 . . 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 10020 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  CC )
20 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
21 pncan 9057 . . . . . . . 8  |-  ( ( (deg `  F )  e.  CC  /\  1  e.  CC )  ->  (
( (deg `  F
)  +  1 )  -  1 )  =  (deg `  F )
)
2219, 20, 21sylancl 643 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (
(deg `  F )  +  1 )  - 
1 )  =  (deg
`  F ) )
2322eqcomd 2288 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  =  ( ( (deg `  F )  +  1 )  - 
1 ) )
2423oveq2d 5874 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... ( ( (deg `  F )  +  1 )  - 
1 ) ) )
2524sumeq1d 12174 . . . 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 4107 . . 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 19614 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2827adantl 452 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
29 oveq1 5865 . . . . 5  |-  ( c  =  b  ->  (
c  +  1 )  =  ( b  +  1 ) )
3029fveq2d 5529 . . . . 5  |-  ( c  =  b  ->  (
(coeff `  F ) `  ( c  +  1 ) )  =  ( (coeff `  F ) `  ( b  +  1 ) ) )
3129, 30oveq12d 5876 . . . 4  |-  ( c  =  b  ->  (
( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  =  ( ( b  +  1 )  x.  ( (coeff `  F ) `  (
b  +  1 ) ) ) )
3231cbvmptv 4111 . . 3  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( b  e.  NN0  |->  ( ( b  +  1 )  x.  (
(coeff `  F ) `  ( b  +  1 ) ) ) )
33 peano2nn0 10004 . . . 4  |-  ( (deg
`  F )  e. 
NN0  ->  ( (deg `  F )  +  1 )  e.  NN0 )
346, 33syl 15 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  1 )  e.  NN0 )
3518, 26, 28, 32, 34dvply1 19664 . 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 16383 . . . . 5  |-  CC  =  ( Base ` fld )
3736subrgss 15546 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
3837adantr 451 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  S  C_  CC )
39 elfznn0 10822 . . . 4  |-  ( b  e.  ( 0 ... (deg `  F )
)  ->  b  e.  NN0 )
40 simpll 730 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  S  e.  (SubRing ` fld ) )
41 zsssubrg 16430 . . . . . . . . 9  |-  ( S  e.  (SubRing ` fld )  ->  ZZ  C_  S )
4241ad2antrr 706 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  ZZ  C_  S )
43 peano2nn0 10004 . . . . . . . . . 10  |-  ( c  e.  NN0  ->  ( c  +  1 )  e. 
NN0 )
4443adantl 452 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  NN0 )
4544nn0zd 10115 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  ZZ )
4642, 45sseldd 3181 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  S )
47 simplr 731 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  F  e.  (Poly `  S
) )
48 subrgsubg 15551 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
49 cnfld0 16398 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
5049subg0cl 14629 . . . . . . . . . . 11  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
5148, 50syl 15 . . . . . . . . . 10  |-  ( S  e.  (SubRing ` fld )  ->  0  e.  S )
5251ad2antrr 706 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
0  e.  S )
5313coef2 19613 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  (coeff `  F ) : NN0 --> S )
5447, 52, 53syl2anc 642 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> S )
5554, 44ffvelrnd 5666 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( (coeff `  F
) `  ( c  +  1 ) )  e.  S )
56 cnfldmul 16385 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5756subrgmcl 15557 . . . . . . 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 1182 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) )  e.  S )
59 eqid 2283 . . . . . 6  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) ) )
6058, 59fmptd 5684 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) : NN0 --> S )
6160ffvelrnda 5665 . . . 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 460 . . 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 19584 . 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 2357 1  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( CC  _D  F )  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158  SubGrpcsubg 14615  SubRingcsubrg 15541  ℂfldccnfld 16377    _D cdv 19213  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dvply2  19666  dvnply2  19667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-cntz 14793  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-0p 19025  df-limc 19216  df-dv 19217  df-ply 19570  df-coe 19572  df-dgr 19573
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