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Theorem evl1expd 18874
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 17732 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
31, 2syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 evl1addd.p . . . . 5  |-  P  =  (Poly1 `  R )
54ply1ring 18782 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
6 eqid 2420 . . . . 5  |-  (mulGrp `  P )  =  (mulGrp `  P )
76ringmgp 17727 . . . 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 460 . . 3  |-  ( ph  ->  M  e.  U )
12 evl1addd.u . . . . 5  |-  U  =  ( Base `  P
)
136, 12mgpbas 17670 . . . 4  |-  U  =  ( Base `  (mulGrp `  P ) )
14 evl1expd.f . . . 4  |-  .xb  =  (.g
`  (mulGrp `  P )
)
1513, 14mulgnn0cl 16726 . . 3  |-  ( ( (mulGrp `  P )  e.  Mnd  /\  N  e. 
NN0  /\  M  e.  U )  ->  ( N  .xb  M )  e.  U )
168, 9, 11, 15syl3anc 1264 . 2  |-  ( ph  ->  ( N  .xb  M
)  e.  U )
17 evl1addd.q . . . . . . . . 9  |-  O  =  (eval1 `  R )
18 eqid 2420 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
19 evl1addd.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
2017, 4, 18, 19evl1rhm 18861 . . . . . . . 8  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  B
) ) )
211, 20syl 17 . . . . . . 7  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  B ) ) )
22 eqid 2420 . . . . . . . 8  |-  (mulGrp `  ( R  ^s  B )
)  =  (mulGrp `  ( R  ^s  B )
)
236, 22rhmmhm 17891 . . . . . . 7  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
2421, 23syl 17 . . . . . 6  |-  ( ph  ->  O  e.  ( (mulGrp `  P ) MndHom  (mulGrp `  ( R  ^s  B )
) ) )
25 eqid 2420 . . . . . . 7  |-  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
2613, 14, 25mhmmulg 16742 . . . . . 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 1264 . . . . 5  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `  M
) ) )
28 eqid 2420 . . . . . . 7  |-  (.g `  (
(mulGrp `  R )  ^s  B ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) )
29 eqidd 2421 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  (mulGrp `  ( R  ^s  B ) ) ) )
30 fvex 5882 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
3119, 30eqeltri 2504 . . . . . . . . 9  |-  B  e. 
_V
32 eqid 2420 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
33 eqid 2420 . . . . . . . . . 10  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
34 eqid 2420 . . . . . . . . . 10  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
35 eqid 2420 . . . . . . . . . 10  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
36 eqid 2420 . . . . . . . . . 10  |-  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )
37 eqid 2420 . . . . . . . . . 10  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
3818, 32, 33, 22, 34, 35, 36, 37pwsmgp 17787 . . . . . . . . 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 666 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4039simpld 460 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
41 ssv 3481 . . . . . . . 8  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) ) 
C_  _V
4241a1i 11 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  C_  _V )
43 ovex 6324 . . . . . . . 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 464 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  ( (mulGrp `  R )  ^s  B ) ) )
4645oveqdr 6320 . . . . . . 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 16743 . . . . . 6  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) ) )
4847oveqd 6313 . . . . 5  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
4927, 48eqtrd 2461 . . . 4  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
5049fveq1d 5874 . . 3  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y ) )
5132ringmgp 17727 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
523, 51syl 17 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
5331a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
54 eqid 2420 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
5512, 54rhmf 17895 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  B ) )  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
5621, 55syl 17 . . . . . . 7  |-  ( ph  ->  O : U --> ( Base `  ( R  ^s  B ) ) )
5756, 11ffvelrnd 6029 . . . . . 6  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( R  ^s  B ) ) )
5822, 54mgpbas 17670 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
5958, 40syl5eq 2473 . . . . . 6  |-  ( ph  ->  ( Base `  ( R  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
6057, 59eleqtrd 2510 . . . . 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 16752 . . . . 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 1271 . . . 4  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  ( ( O `  M ) `  Y
) ) )
6510simprd 464 . . . . 5  |-  ( ph  ->  ( ( O `  M ) `  Y
)  =  V )
6665oveq2d 6312 . . . 4  |-  ( ph  ->  ( N  .^  (
( O `  M
) `  Y )
)  =  ( N 
.^  V ) )
6764, 66eqtrd 2461 . . 3  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  V ) )
6850, 67eqtrd 2461 . 2  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( N  .^  V ) )
6916, 68jca 534 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 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   -->wf 5588   ` cfv 5592  (class class class)co 6296   NN0cn0 10858   Basecbs 15081   +g cplusg 15150    ^s cpws 15305   Mndcmnd 16487   MndHom cmhm 16532  .gcmg 16624  mulGrpcmgp 17664   Ringcrg 17721   CRingccrg 17722   RingHom crh 17881  Poly1cpl1 18711  eval1ce1 18844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-fzo 11903  df-seq 12200  df-hash 12502  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-hom 15174  df-cco 15175  df-0g 15300  df-gsum 15301  df-prds 15306  df-pws 15308  df-mre 15444  df-mrc 15445  df-acs 15447  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-mhm 16534  df-submnd 16535  df-grp 16625  df-minusg 16626  df-sbg 16627  df-mulg 16628  df-subg 16766  df-ghm 16833  df-cntz 16923  df-cmn 17373  df-abl 17374  df-mgp 17665  df-ur 17677  df-srg 17681  df-ring 17723  df-cring 17724  df-rnghom 17884  df-subrg 17947  df-lmod 18034  df-lss 18097  df-lsp 18136  df-assa 18477  df-asp 18478  df-ascl 18479  df-psr 18521  df-mvr 18522  df-mpl 18523  df-opsr 18525  df-evls 18670  df-evl 18671  df-psr1 18714  df-ply1 18716  df-evl1 18846
This theorem is referenced by:  evl1varpwval  18891  plypf1  23073  lgsqrlem1  24171  idomrootle  35816
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