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Theorem dvply2g 21720
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 21635 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
21adantl 466 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
32feqmptd 5737 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  =  ( a  e.  CC  |->  ( F `  a ) ) )
4 simplr 754 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  a  e.  CC )  ->  F  e.  (Poly `  S ) )
5 dgrcl 21670 . . . . . . . . . 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 10737 . . . . . . . 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 10867 . . . . . . 7  |-  ( (deg
`  F )  e.  ZZ  ->  (deg `  F
)  e.  ( ZZ>= `  (deg `  F ) ) )
10 peano2uz 10900 . . . . . . 7  |-  ( (deg
`  F )  e.  ( ZZ>= `  (deg `  F
) )  ->  (
(deg `  F )  +  1 )  e.  ( ZZ>= `  (deg `  F
) ) )
118, 9, 103syl 20 . . . . . 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 2437 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
14 eqid 2437 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
1513, 14coeid3 21677 . . . . . 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 1218 . . . . 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 4371 . . . 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 2469 . . 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 10630 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  CC )
20 ax-1cn 9332 . . . . . . . 8  |-  1  e.  CC
21 pncan 9608 . . . . . . . 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 2442 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  =  ( ( (deg `  F )  +  1 )  - 
1 ) )
2423oveq2d 6102 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... ( ( (deg `  F )  +  1 )  - 
1 ) ) )
2524sumeq1d 13170 . . . 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 4372 . . 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 21669 . . . 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 6093 . . . . 5  |-  ( c  =  b  ->  (
c  +  1 )  =  ( b  +  1 ) )
3029fveq2d 5688 . . . . 5  |-  ( c  =  b  ->  (
(coeff `  F ) `  ( c  +  1 ) )  =  ( (coeff `  F ) `  ( b  +  1 ) ) )
3129, 30oveq12d 6104 . . . 4  |-  ( c  =  b  ->  (
( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  =  ( ( b  +  1 )  x.  ( (coeff `  F ) `  (
b  +  1 ) ) ) )
3231cbvmptv 4376 . . 3  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( b  e.  NN0  |->  ( ( b  +  1 )  x.  (
(coeff `  F ) `  ( b  +  1 ) ) ) )
33 peano2nn0 10612 . . . 4  |-  ( (deg
`  F )  e. 
NN0  ->  ( (deg `  F )  +  1 )  e.  NN0 )
346, 33syl 16 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  1 )  e.  NN0 )
3518, 26, 28, 32, 34dvply1 21719 . 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 17791 . . . . 5  |-  CC  =  ( Base ` fld )
3736subrgss 16842 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
3837adantr 465 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  S  C_  CC )
39 elfznn0 11473 . . . 4  |-  ( b  e.  ( 0 ... (deg `  F )
)  ->  b  e.  NN0 )
40 simpll 753 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  S  e.  (SubRing ` fld ) )
41 zsssubrg 17840 . . . . . . . . 9  |-  ( S  e.  (SubRing ` fld )  ->  ZZ  C_  S )
4241ad2antrr 725 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  ZZ  C_  S )
43 peano2nn0 10612 . . . . . . . . . 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 10737 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  ZZ )
4642, 45sseldd 3350 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  S )
47 simplr 754 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  ->  F  e.  (Poly `  S
) )
48 subrgsubg 16847 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
49 cnfld0 17809 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
5049subg0cl 15678 . . . . . . . . . . 11  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
5148, 50syl 16 . . . . . . . . . 10  |-  ( S  e.  (SubRing ` fld )  ->  0  e.  S )
5251ad2antrr 725 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
0  e.  S )
5313coef2 21668 . . . . . . . . 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 5837 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( (coeff `  F
) `  ( c  +  1 ) )  e.  S )
56 cnfldmul 17793 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5756subrgmcl 16853 . . . . . . 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 1218 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) )  e.  S )
59 eqid 2437 . . . . . 6  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) ) )
6058, 59fmptd 5860 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) : NN0 --> S )
6160ffvelrnda 5836 . . . 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 21639 . 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 2511 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 1369    e. wcel 1756    C_ wss 3321    e. cmpt 4343   -->wf 5407   ` cfv 5411  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857   sum_csu 13155  SubGrpcsubg 15664  SubRingcsubrg 16837  ℂfldccnfld 17787    _D cdv 21307  Polycply 21621  coeffccoe 21623  degcdgr 21624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-iin 4167  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-grp 15534  df-minusg 15535  df-mulg 15537  df-subg 15667  df-cntz 15824  df-cmn 16268  df-mgp 16578  df-ur 16590  df-rng 16633  df-cring 16634  df-subrg 16839  df-psmet 17778  df-xmet 17779  df-met 17780  df-bl 17781  df-mopn 17782  df-fbas 17783  df-fg 17784  df-cnfld 17788  df-top 18472  df-bases 18474  df-topon 18475  df-topsp 18476  df-cld 18592  df-ntr 18593  df-cls 18594  df-nei 18671  df-lp 18709  df-perf 18710  df-cn 18800  df-cnp 18801  df-haus 18888  df-tx 19104  df-hmeo 19297  df-fil 19388  df-fm 19480  df-flim 19481  df-flf 19482  df-xms 19864  df-ms 19865  df-tms 19866  df-cncf 20423  df-0p 21117  df-limc 21310  df-dv 21311  df-ply 21625  df-coe 21627  df-dgr 21628
This theorem is referenced by:  dvply2  21721  dvnply2  21722
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