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Theorem dvply2g 21867
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 21782 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
21adantl 466 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
32feqmptd 5843 . . . 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 21817 . . . . . . . . . 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 10846 . . . . . . . 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 10976 . . . . . . 7  |-  ( (deg
`  F )  e.  ZZ  ->  (deg `  F
)  e.  ( ZZ>= `  (deg `  F ) ) )
10 peano2uz 11009 . . . . . . 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 2451 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
14 eqid 2451 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
1513, 14coeid3 21824 . . . . . 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 1219 . . . . 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 4476 . . . 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 2492 . . 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 10739 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  CC )
20 ax-1cn 9441 . . . . . . . 8  |-  1  e.  CC
21 pncan 9717 . . . . . . . 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 2459 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  (deg `  F
)  =  ( ( (deg `  F )  +  1 )  - 
1 ) )
2423oveq2d 6206 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... ( ( (deg `  F )  +  1 )  - 
1 ) ) )
2524sumeq1d 13280 . . . 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 4477 . . 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 21816 . . . 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 6197 . . . . 5  |-  ( c  =  b  ->  (
c  +  1 )  =  ( b  +  1 ) )
3029fveq2d 5793 . . . . 5  |-  ( c  =  b  ->  (
(coeff `  F ) `  ( c  +  1 ) )  =  ( (coeff `  F ) `  ( b  +  1 ) ) )
3129, 30oveq12d 6208 . . . 4  |-  ( c  =  b  ->  (
( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) )  =  ( ( b  +  1 )  x.  ( (coeff `  F ) `  (
b  +  1 ) ) ) )
3231cbvmptv 4481 . . 3  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( b  e.  NN0  |->  ( ( b  +  1 )  x.  (
(coeff `  F ) `  ( b  +  1 ) ) ) )
33 peano2nn0 10721 . . . 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 21866 . 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 17931 . . . . 5  |-  CC  =  ( Base ` fld )
3736subrgss 16972 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
3837adantr 465 . . 3  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  S  C_  CC )
39 elfznn0 11582 . . . 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 17980 . . . . . . . . 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 10721 . . . . . . . . . 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 10846 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( c  +  1 )  e.  ZZ )
4642, 45sseldd 3455 . . . . . . 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 16977 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
49 cnfld0 17949 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
5049subg0cl 15791 . . . . . . . . . . 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 21815 . . . . . . . . 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 5943 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( (coeff `  F
) `  ( c  +  1 ) )  e.  S )
56 cnfldmul 17933 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5756subrgmcl 16983 . . . . . . 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 1219 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  c  e.  NN0 )  -> 
( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) )  e.  S )
59 eqid 2451 . . . . . 6  |-  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F ) `  (
c  +  1 ) ) ) )  =  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  (
(coeff `  F ) `  ( c  +  1 ) ) ) )
6058, 59fmptd 5966 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( c  e.  NN0  |->  ( ( c  +  1 )  x.  ( (coeff `  F
) `  ( c  +  1 ) ) ) ) : NN0 --> S )
6160ffvelrnda 5942 . . . 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 21786 . 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 2539 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 1370    e. wcel 1758    C_ wss 3426    |-> cmpt 4448   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   0cc0 9383   1c1 9384    + caddc 9386    x. cmul 9388    - cmin 9696   NN0cn0 10680   ZZcz 10747   ZZ>=cuz 10962   ...cfz 11538   ^cexp 11966   sum_csu 13265  SubGrpcsubg 15777  SubRingcsubrg 16967  ℂfldccnfld 17927    _D cdv 21454  Polycply 21768  coeffccoe 21770  degcdgr 21771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069  df-sum 13266  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-grp 15647  df-minusg 15648  df-mulg 15650  df-subg 15780  df-cntz 15937  df-cmn 16383  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-subrg 16969  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-0p 21264  df-limc 21457  df-dv 21458  df-ply 21772  df-coe 21774  df-dgr 21775
This theorem is referenced by:  dvply2  21868  dvnply2  21869
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