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Theorem evl1expd 19421
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 15351 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
31, 2syl 15 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 evl1addd.p . . . . 5  |-  P  =  (Poly1 `  R )
54ply1rng 16326 . . . 4  |-  ( R  e.  Ring  ->  P  e. 
Ring )
6 eqid 2283 . . . . 5  |-  (mulGrp `  P )  =  (mulGrp `  P )
76rngmgp 15347 . . . 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 15331 . . . 4  |-  U  =  ( Base `  (mulGrp `  P ) )
14 evl1expd.f . . . 4  |-  .xb  =  (.g
`  (mulGrp `  P )
)
1513, 14mulgnn0cl 14583 . . 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 2283 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
19 evl1addd.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
2017, 4, 18, 19evl1rhm 19412 . . . . . . . 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 2283 . . . . . . . 8  |-  (mulGrp `  ( R  ^s  B )
)  =  (mulGrp `  ( R  ^s  B )
)
236, 22rhmmhm 15502 . . . . . . 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 2283 . . . . . . 7  |-  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
2613, 14, 25mhmmulg 14599 . . . . . 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 2283 . . . . . . 7  |-  (.g `  (
(mulGrp `  R )  ^s  B ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) )
29 eqidd 2284 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  (
Base `  (mulGrp `  ( R  ^s  B ) ) ) )
30 fvex 5539 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
3119, 30eqeltri 2353 . . . . . . . . 9  |-  B  e. 
_V
32 eqid 2283 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
33 eqid 2283 . . . . . . . . . 10  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
34 eqid 2283 . . . . . . . . . 10  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
35 eqid 2283 . . . . . . . . . 10  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
36 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )  =  ( +g  `  (mulGrp `  ( R  ^s  B ) ) )
37 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
3818, 32, 33, 22, 34, 35, 36, 37pwsmgp 15401 . . . . . . . . 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 3198 . . . . . . . 8  |-  ( Base `  (mulGrp `  ( R  ^s  B ) ) ) 
C_  _V
4241a1i 10 . . . . . . 7  |-  ( ph  ->  ( Base `  (mulGrp `  ( R  ^s  B ) ) )  C_  _V )
43 ovex 5883 . . . . . . . 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 13593 . . . . . . 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 14600 . . . . . 6  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  (.g `  ( (mulGrp `  R )  ^s  B ) ) )
4847oveqd 5875 . . . . 5  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( O `
 M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
4927, 48eqtrd 2315 . . . 4  |-  ( ph  ->  ( O `  ( N  .xb  M ) )  =  ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) )
5049fveq1d 5527 . . 3  |-  ( ph  ->  ( ( O `  ( N  .xb  M ) ) `  Y )  =  ( ( N (.g `  ( (mulGrp `  R )  ^s  B ) ) ( O `  M ) ) `  Y ) )
5132rngmgp 15347 . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
5512, 54rhmf 15504 . . . . . . . 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 5663 . . . . . . 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 15331 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  (mulGrp `  ( R  ^s  B ) ) )
6059, 40syl5eq 2327 . . . . . 6  |-  ( ph  ->  ( Base `  ( R  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
6158, 60eleqtrd 2359 . . . . 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 14609 . . . . 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 5874 . . . 4  |-  ( ph  ->  ( N  .^  (
( O `  M
) `  Y )
)  =  ( N 
.^  V ) )
6865, 67eqtrd 2315 . . 3  |-  ( ph  ->  ( ( N (.g `  ( (mulGrp `  R
)  ^s  B ) ) ( O `  M ) ) `  Y )  =  ( N  .^  V ) )
6950, 68eqtrd 2315 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   -->wf 5251   ` cfv 5255  (class class class)co 5858   NN0cn0 9965   Basecbs 13148   +g cplusg 13208    ^s cpws 13347   Mndcmnd 14361  .gcmg 14366   MndHom cmhm 14413  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   RingHom crh 15494  Poly1cpl1 16252  eval1ce1 16254
This theorem is referenced by:  plypf1  19594  lgsqrlem1  20580  idomrootle  27511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-ply1 16259  df-evl1 16261
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