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Theorem evl1expd 18249
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 17077 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 evl1addd.p . . . . 5  |-  P  =  (Poly1 `  R )
54ply1ring 18157 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
6 eqid 2441 . . . . 5  |-  (mulGrp `  P )  =  (mulGrp `  P )
76ringmgp 17072 . . . 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 17015 . . . 4  |-  U  =  ( Base `  (mulGrp `  P ) )
14 evl1expd.f . . . 4  |-  .xb  =  (.g
`  (mulGrp `  P )
)
1513, 14mulgnn0cl 16027 . . 3  |-  ( ( (mulGrp `  P )  e.  Mnd  /\  N  e. 
NN0  /\  M  e.  U )  ->  ( N  .xb  M )  e.  U )
168, 9, 11, 15syl3anc 1227 . 2  |-  ( ph  ->  ( N  .xb  M
)  e.  U )
17 evl1addd.q . . . . . . . . 9  |-  O  =  (eval1 `  R )
18 eqid 2441 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
19 evl1addd.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
2017, 4, 18, 19evl1rhm 18236 . . . . . . . 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 2441 . . . . . . . 8  |-  (mulGrp `  ( R  ^s  B )
)  =  (mulGrp `  ( R  ^s  B )
)
236, 22rhmmhm 17239 . . . . . . 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 2441 . . . . . . 7  |-  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
2613, 14, 25mhmmulg 16043 . . . . . 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 1227 . . . . 5  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `  M
) ) )
28 eqid 2441 . . . . . . 7  |-  (.g `  (
(mulGrp `  R )  ^s  B ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) )
29 eqidd 2442 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  (mulGrp `  ( R  ^s  B ) ) ) )
30 fvex 5862 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
3119, 30eqeltri 2525 . . . . . . . . 9  |-  B  e. 
_V
32 eqid 2441 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
33 eqid 2441 . . . . . . . . . 10  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
34 eqid 2441 . . . . . . . . . 10  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
35 eqid 2441 . . . . . . . . . 10  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
36 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )
37 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
3818, 32, 33, 22, 34, 35, 36, 37pwsmgp 17135 . . . . . . . . 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 3506 . . . . . . . 8  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) ) 
C_  _V
4241a1i 11 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  C_  _V )
43 ovex 6305 . . . . . . . 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 ) ) )
4645oveqdr 6301 . . . . . . 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 16044 . . . . . 6  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) ) )
4847oveqd 6294 . . . . 5  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
4927, 48eqtrd 2482 . . . 4  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
5049fveq1d 5854 . . 3  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y ) )
5132ringmgp 17072 . . . . . 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 2441 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
5512, 54rhmf 17243 . . . . . . . 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 6013 . . . . . 6  |-  ( ph  ->  ( O `  M
)  e.  ( Base `  ( R  ^s  B ) ) )
5822, 54mgpbas 17015 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
5958, 40syl5eq 2494 . . . . . 6  |-  ( ph  ->  ( Base `  ( R  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
6057, 59eleqtrd 2531 . . . . 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 16053 . . . . 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 1234 . . . 4  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  ( ( O `  M ) `  Y
) ) )
6510simprd 463 . . . . 5  |-  ( ph  ->  ( ( O `  M ) `  Y
)  =  V )
6665oveq2d 6293 . . . 4  |-  ( ph  ->  ( N  .^  (
( O `  M
) `  Y )
)  =  ( N 
.^  V ) )
6764, 66eqtrd 2482 . . 3  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  V ) )
6850, 67eqtrd 2482 . 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 1381    e. wcel 1802   _Vcvv 3093    C_ wss 3458   -->wf 5570   ` cfv 5574  (class class class)co 6277   NN0cn0 10796   Basecbs 14504   +g cplusg 14569    ^s cpws 14716   Mndcmnd 15788   MndHom cmhm 15833  .gcmg 15925  mulGrpcmgp 17009   Ringcrg 17066   CRingccrg 17067   RingHom crh 17229  Poly1cpl1 18084  eval1ce1 18219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-ofr 6522  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-hom 14593  df-cco 14594  df-0g 14711  df-gsum 14712  df-prds 14717  df-pws 14719  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-mulg 15929  df-subg 16067  df-ghm 16134  df-cntz 16224  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-srg 17026  df-ring 17068  df-cring 17069  df-rnghom 17232  df-subrg 17295  df-lmod 17382  df-lss 17447  df-lsp 17486  df-assa 17829  df-asp 17830  df-ascl 17831  df-psr 17873  df-mvr 17874  df-mpl 17875  df-opsr 17877  df-evls 18039  df-evl 18040  df-psr1 18087  df-ply1 18089  df-evl1 18221
This theorem is referenced by:  evl1varpwval  18266  plypf1  22475  lgsqrlem1  23481  idomrootle  31121
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