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Theorem dvply2g 19718
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 19633 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
21adantl 452 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
32feqmptd 5613 . . . 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 19668 . . . . . . . . . 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 10162 . . . . . . . 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 10289 . . . . . . 7  |-  ( (deg
`  F )  e.  ZZ  ->  (deg `  F
)  e.  ( ZZ>= `  (deg `  F ) ) )
10 peano2uz 10319 . . . . . . 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 2316 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
14 eqid 2316 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
1513, 14coeid3 19675 . . . . . 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 4143 . . . 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 2348 . . 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 10067 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  CC )
20 ax-1cn 8840 . . . . . . . 8  |-  1  e.  CC
21 pncan 9102 . . . . . . . 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 2321 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  =  ( ( (deg `  F )  +  1 )  - 
1 ) )
2423oveq2d 5916 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... ( ( (deg `  F )  +  1 )  - 
1 ) ) )
2524sumeq1d 12221 . . . 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 4144 . . 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 19667 . . . 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 5907 . . . . 5  |-  ( c  =  b  ->  (
c  +  1 )  =  ( b  +  1 ) )
3029fveq2d 5567 . . . . 5  |-  ( c  =  b  ->  (
(coeff `  F ) `  ( c  +  1 ) )  =  ( (coeff `  F ) `  ( b  +  1 ) ) )
3129, 30oveq12d 5918 . . . 4  |-  ( c  =  b  ->  (
( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  =  ( ( b  +  1 )  x.  ( (coeff `  F ) `  (
b  +  1 ) ) ) )
3231cbvmptv 4148 . . 3  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( b  e.  NN0  |->  ( ( b  +  1 )  x.  (
(coeff `  F ) `  ( b  +  1 ) ) ) )
33 peano2nn0 10051 . . . 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 19717 . 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 16436 . . . . 5  |-  CC  =  ( Base ` fld )
3736subrgss 15595 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
3837adantr 451 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  S  C_  CC )
39 elfznn0 10869 . . . 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 16486 . . . . . . . . 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 10051 . . . . . . . . . 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 10162 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  ZZ )
4642, 45sseldd 3215 . . . . . . 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 15600 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
49 cnfld0 16454 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
5049subg0cl 14678 . . . . . . . . . . 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 19666 . . . . . . . . 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 5704 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( (coeff `  F
) `  ( c  +  1 ) )  e.  S )
56 cnfldmul 16438 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5756subrgmcl 15606 . . . . . . 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 2316 . . . . . 6  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) ) )
6058, 59fmptd 5722 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) : NN0 --> S )
6160ffvelrnda 5703 . . . 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 19637 . 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 2390 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 1633    e. wcel 1701    C_ wss 3186    e. cmpt 4114   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082   NN0cn0 10012   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829   ^cexp 11151   sum_csu 12205  SubGrpcsubg 14664  SubRingcsubrg 15590  ℂfldccnfld 16432    _D cdv 19266  Polycply 19619  coeffccoe 19621  degcdgr 19622
This theorem is referenced by:  dvply2  19719  dvnply2  19720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-rlim 12010  df-sum 12206  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-mulg 14541  df-subg 14667  df-cntz 14842  df-cmn 15140  df-mgp 15375  df-rng 15389  df-cring 15390  df-ur 15391  df-subrg 15592  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-0p 19078  df-limc 19269  df-dv 19270  df-ply 19623  df-coe 19625  df-dgr 19626
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