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Theorem evl1expd 18228
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 crngring 17058 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 evl1addd.p . . . . 5  |-  P  =  (Poly1 `  R )
54ply1ring 18136 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
6 eqid 2467 . . . . 5  |-  (mulGrp `  P )  =  (mulGrp `  P )
76ringmgp 17053 . . . 4  |-  ( P  e.  Ring  ->  (mulGrp `  P )  e.  Mnd )
83, 5, 73syl 20 . . 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 459 . . 3  |-  ( ph  ->  M  e.  U )
12 evl1addd.u . . . . 5  |-  U  =  ( Base `  P
)
136, 12mgpbas 16996 . . . 4  |-  U  =  ( Base `  (mulGrp `  P ) )
14 evl1expd.f . . . 4  |-  .xb  =  (.g
`  (mulGrp `  P )
)
1513, 14mulgnn0cl 16007 . . 3  |-  ( ( (mulGrp `  P )  e.  Mnd  /\  N  e. 
NN0  /\  M  e.  U )  ->  ( N  .xb  M )  e.  U )
168, 9, 11, 15syl3anc 1228 . 2  |-  ( ph  ->  ( N  .xb  M
)  e.  U )
17 evl1addd.q . . . . . . . . 9  |-  O  =  (eval1 `  R )
18 eqid 2467 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
19 evl1addd.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
2017, 4, 18, 19evl1rhm 18215 . . . . . . . 8  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  B
) ) )
211, 20syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  B ) ) )
22 eqid 2467 . . . . . . . 8  |-  (mulGrp `  ( R  ^s  B )
)  =  (mulGrp `  ( R  ^s  B )
)
236, 22rhmmhm 17220 . . . . . . 7  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
2421, 23syl 16 . . . . . 6  |-  ( ph  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
25 eqid 2467 . . . . . . 7  |-  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
2613, 14, 25mhmmulg 16023 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `  M
) ) )
28 eqid 2467 . . . . . . 7  |-  (.g `  (
(mulGrp `  R )  ^s  B ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) )
29 eqidd 2468 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  (mulGrp `  ( R  ^s  B ) ) ) )
30 fvex 5881 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
3119, 30eqeltri 2551 . . . . . . . . 9  |-  B  e. 
_V
32 eqid 2467 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
33 eqid 2467 . . . . . . . . . 10  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
34 eqid 2467 . . . . . . . . . 10  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
35 eqid 2467 . . . . . . . . . 10  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
36 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )
37 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
3818, 32, 33, 22, 34, 35, 36, 37pwsmgp 17116 . . . . . . . . 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 662 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4039simpld 459 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
41 ssv 3529 . . . . . . . 8  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) ) 
C_  _V
4241a1i 11 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  C_  _V )
43 ovex 6319 . . . . . . . 8  |-  ( x ( +g  `  (mulGrp `  ( R  ^s  B ) ) ) y )  e.  _V
4443a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  _V  /\  y  e. 
_V ) )  -> 
( x ( +g  `  (mulGrp `  ( R  ^s  B ) ) ) y )  e.  _V )
4539simprd 463 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  ( (mulGrp `  R )  ^s  B ) ) )
4645proplem3 14958 . . . . . . 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 16024 . . . . . 6  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) ) )
4847oveqd 6311 . . . . 5  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
4927, 48eqtrd 2508 . . . 4  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
5049fveq1d 5873 . . 3  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y ) )
5132ringmgp 17053 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
523, 51syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
5331a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
54 eqid 2467 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
5512, 54rhmf 17224 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
5621, 55syl 16 . . . . . . 7  |-  ( ph  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
5756, 11ffvelrnd 6032 . . . . . 6  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( R  ^s  B ) ) )
5822, 54mgpbas 16996 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
5958, 40syl5eq 2520 . . . . . 6  |-  ( ph  ->  ( Base `  ( R  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
6057, 59eleqtrd 2557 . . . . 5  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
61 evl1addd.2 . . . . 5  |-  ( ph  ->  Y  e.  B )
62 evl1expd.e . . . . . 6  |-  .^  =  (.g
`  (mulGrp `  R )
)
6333, 35, 28, 62pwsmulg 16033 . . . . 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 ) ) )
6452, 53, 9, 60, 61, 63syl23anc 1235 . . . 4  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  ( ( O `  M ) `  Y
) ) )
6510simprd 463 . . . . 5  |-  ( ph  ->  ( ( O `  M ) `  Y
)  =  V )
6665oveq2d 6310 . . . 4  |-  ( ph  ->  ( N  .^  (
( O `  M
) `  Y )
)  =  ( N 
.^  V ) )
6764, 66eqtrd 2508 . . 3  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  V ) )
6850, 67eqtrd 2508 . 2  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) )
6916, 68jca 532 1  |-  ( ph  ->  ( ( N  .xb  M )  e.  U  /\  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   -->wf 5589   ` cfv 5593  (class class class)co 6294   NN0cn0 10805   Basecbs 14502   +g cplusg 14567    ^s cpws 14714   Mndcmnd 15772   MndHom cmhm 15817  .gcmg 15905  mulGrpcmgp 16990   Ringcrg 17047   CRingccrg 17048   RingHom crh 17210  Poly1cpl1 18063  eval1ce1 18198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-ofr 6535  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-hash 12384  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-hom 14591  df-cco 14592  df-0g 14709  df-gsum 14710  df-prds 14715  df-pws 14717  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-mulg 15909  df-subg 16047  df-ghm 16114  df-cntz 16204  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-srg 17007  df-ring 17049  df-cring 17050  df-rnghom 17213  df-subrg 17275  df-lmod 17362  df-lss 17427  df-lsp 17466  df-assa 17808  df-asp 17809  df-ascl 17810  df-psr 17852  df-mvr 17853  df-mpl 17854  df-opsr 17856  df-evls 18018  df-evl 18019  df-psr1 18066  df-ply1 18068  df-evl1 18200
This theorem is referenced by:  evl1varpwval  18245  plypf1  22454  lgsqrlem1  23459  idomrootle  31049
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