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Theorem evl1expd 19437
Description: Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
evl1addd.q  |-  O  =  (eval1 `  R )
evl1addd.p  |-  P  =  (Poly1 `  R )
evl1addd.b  |-  B  =  ( Base `  R
)
evl1addd.u  |-  U  =  ( Base `  P
)
evl1addd.1  |-  ( ph  ->  R  e.  CRing )
evl1addd.2  |-  ( ph  ->  Y  e.  B )
evl1addd.3  |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y
)  =  V ) )
evl1expd.f  |-  .xb  =  (.g
`  (mulGrp `  P )
)
evl1expd.e  |-  .^  =  (.g
`  (mulGrp `  R )
)
evl1expd.4  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
evl1expd  |-  ( ph  ->  ( ( N  .xb  M )  e.  U  /\  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) ) )

Proof of Theorem evl1expd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evl1addd.1 . . . . 5  |-  ( ph  ->  R  e.  CRing )
2 crngrng 15367 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
31, 2syl 15 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 evl1addd.p . . . . 5  |-  P  =  (Poly1 `  R )
54ply1rng 16342 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
6 eqid 2296 . . . . 5  |-  (mulGrp `  P )  =  (mulGrp `  P )
76rngmgp 15363 . . . 4  |-  ( P  e.  Ring  ->  (mulGrp `  P )  e.  Mnd )
83, 5, 73syl 18 . . 3  |-  ( ph  ->  (mulGrp `  P )  e.  Mnd )
9 evl1expd.4 . . 3  |-  ( ph  ->  N  e.  NN0 )
10 evl1addd.3 . . . 4  |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y
)  =  V ) )
1110simpld 445 . . 3  |-  ( ph  ->  M  e.  U )
12 evl1addd.u . . . . 5  |-  U  =  ( Base `  P
)
136, 12mgpbas 15347 . . . 4  |-  U  =  ( Base `  (mulGrp `  P ) )
14 evl1expd.f . . . 4  |-  .xb  =  (.g
`  (mulGrp `  P )
)
1513, 14mulgnn0cl 14599 . . 3  |-  ( ( (mulGrp `  P )  e.  Mnd  /\  N  e. 
NN0  /\  M  e.  U )  ->  ( N  .xb  M )  e.  U )
168, 9, 11, 15syl3anc 1182 . 2  |-  ( ph  ->  ( N  .xb  M
)  e.  U )
17 evl1addd.q . . . . . . . . 9  |-  O  =  (eval1 `  R )
18 eqid 2296 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
19 evl1addd.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
2017, 4, 18, 19evl1rhm 19428 . . . . . . . 8  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  B
) ) )
211, 20syl 15 . . . . . . 7  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  B ) ) )
22 eqid 2296 . . . . . . . 8  |-  (mulGrp `  ( R  ^s  B )
)  =  (mulGrp `  ( R  ^s  B )
)
236, 22rhmmhm 15518 . . . . . . 7  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
2421, 23syl 15 . . . . . 6  |-  ( ph  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
25 eqid 2296 . . . . . . 7  |-  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
2613, 14, 25mhmmulg 14615 . . . . . 6  |-  ( ( O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) )  /\  N  e.  NN0  /\  M  e.  U )  ->  ( O `  ( N  .xb 
M ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) ) )
2724, 9, 11, 26syl3anc 1182 . . . . 5  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `  M
) ) )
28 eqid 2296 . . . . . . 7  |-  (.g `  (
(mulGrp `  R )  ^s  B ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) )
29 eqidd 2297 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  (mulGrp `  ( R  ^s  B ) ) ) )
30 fvex 5555 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
3119, 30eqeltri 2366 . . . . . . . . 9  |-  B  e. 
_V
32 eqid 2296 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
33 eqid 2296 . . . . . . . . . 10  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
34 eqid 2296 . . . . . . . . . 10  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
35 eqid 2296 . . . . . . . . . 10  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
36 eqid 2296 . . . . . . . . . 10  |-  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )
37 eqid 2296 . . . . . . . . . 10  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
3818, 32, 33, 22, 34, 35, 36, 37pwsmgp 15417 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  B  e.  _V )  ->  (
( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  ( R  ^s  B )
) )  =  ( +g  `  ( (mulGrp `  R )  ^s  B ) ) ) )
391, 31, 38sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4039simpld 445 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
41 ssv 3211 . . . . . . . 8  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) ) 
C_  _V
4241a1i 10 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  C_  _V )
43 ovex 5899 . . . . . . . 8  |-  ( x ( +g  `  (mulGrp `  ( R  ^s  B ) ) ) y )  e.  _V
4443a1i 10 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  _V  /\  y  e. 
_V ) )  -> 
( x ( +g  `  (mulGrp `  ( R  ^s  B ) ) ) y )  e.  _V )
4539simprd 449 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  ( (mulGrp `  R )  ^s  B ) ) )
4645proplem3 13609 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  _V  /\  y  e. 
_V ) )  -> 
( x ( +g  `  (mulGrp `  ( R  ^s  B ) ) ) y )  =  ( x ( +g  `  (
(mulGrp `  R )  ^s  B ) ) y ) )
4725, 28, 29, 40, 42, 44, 46mulgpropd 14616 . . . . . 6  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) ) )
4847oveqd 5891 . . . . 5  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
4927, 48eqtrd 2328 . . . 4  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
5049fveq1d 5543 . . 3  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y ) )
5132rngmgp 15363 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
523, 51syl 15 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
5331a1i 10 . . . . 5  |-  ( ph  ->  B  e.  _V )
54 eqid 2296 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
5512, 54rhmf 15520 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
5621, 55syl 15 . . . . . . 7  |-  ( ph  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
57 ffvelrn 5679 . . . . . . 7  |-  ( ( O : U --> ( Base `  ( R  ^s  B ) )  /\  M  e.  U )  ->  ( O `  M )  e.  ( Base `  ( R  ^s  B ) ) )
5856, 11, 57syl2anc 642 . . . . . 6  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( R  ^s  B ) ) )
5922, 54mgpbas 15347 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
6059, 40syl5eq 2340 . . . . . 6  |-  ( ph  ->  ( Base `  ( R  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
6158, 60eleqtrd 2372 . . . . 5  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
62 evl1addd.2 . . . . 5  |-  ( ph  ->  Y  e.  B )
63 evl1expd.e . . . . . 6  |-  .^  =  (.g
`  (mulGrp `  R )
)
6433, 35, 28, 63pwsmulg 14625 . . . . 5  |-  ( ( ( (mulGrp `  R
)  e.  Mnd  /\  B  e.  _V )  /\  ( N  e.  NN0  /\  ( O `  M
)  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  /\  Y  e.  B )
)  ->  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  ( ( O `  M ) `  Y ) ) )
6552, 53, 9, 61, 62, 64syl23anc 1189 . . . 4  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  ( ( O `  M ) `  Y
) ) )
6610simprd 449 . . . . 5  |-  ( ph  ->  ( ( O `  M ) `  Y
)  =  V )
6766oveq2d 5890 . . . 4  |-  ( ph  ->  ( N  .^  (
( O `  M
) `  Y )
)  =  ( N 
.^  V ) )
6865, 67eqtrd 2328 . . 3  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  V ) )
6950, 68eqtrd 2328 . 2  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) )
7016, 69jca 518 1  |-  ( ph  ->  ( ( N  .xb  M )  e.  U  /\  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   -->wf 5267   ` cfv 5271  (class class class)co 5874   NN0cn0 9981   Basecbs 13164   +g cplusg 13224    ^s cpws 13363   Mndcmnd 14377  .gcmg 14382   MndHom cmhm 14429  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   RingHom crh 15510  Poly1cpl1 16268  eval1ce1 16270
This theorem is referenced by:  plypf1  19610  lgsqrlem1  20596  idomrootle  27614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117  df-evl 16118  df-opsr 16122  df-psr1 16273  df-ply1 16275  df-evl1 16277
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