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Theorem evl1scvarpw 18167
Description: Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
Hypotheses
Ref Expression
evl1varpw.q  |-  Q  =  (eval1 `  R )
evl1varpw.w  |-  W  =  (Poly1 `  R )
evl1varpw.g  |-  G  =  (mulGrp `  W )
evl1varpw.x  |-  X  =  (var1 `  R )
evl1varpw.b  |-  B  =  ( Base `  R
)
evl1varpw.e  |-  .^  =  (.g
`  G )
evl1varpw.r  |-  ( ph  ->  R  e.  CRing )
evl1varpw.n  |-  ( ph  ->  N  e.  NN0 )
evl1scvarpw.t1  |-  .X.  =  ( .s `  W )
evl1scvarpw.a  |-  ( ph  ->  A  e.  B )
evl1scvarpw.s  |-  S  =  ( R  ^s  B )
evl1scvarpw.t2  |-  .xb  =  ( .r `  S )
evl1scvarpw.m  |-  M  =  (mulGrp `  S )
evl1scvarpw.f  |-  F  =  (.g `  M )
Assertion
Ref Expression
evl1scvarpw  |-  ( ph  ->  ( Q `  ( A  .X.  ( N  .^  X ) ) )  =  ( ( B  X.  { A }
)  .xb  ( N F ( Q `  X ) ) ) )

Proof of Theorem evl1scvarpw
StepHypRef Expression
1 evl1varpw.r . . . . . 6  |-  ( ph  ->  R  e.  CRing )
2 evl1varpw.w . . . . . . 7  |-  W  =  (Poly1 `  R )
32ply1assa 18006 . . . . . 6  |-  ( R  e.  CRing  ->  W  e. AssAlg )
41, 3syl 16 . . . . 5  |-  ( ph  ->  W  e. AssAlg )
5 evl1scvarpw.a . . . . . . 7  |-  ( ph  ->  A  e.  B )
6 evl1varpw.b . . . . . . 7  |-  B  =  ( Base `  R
)
75, 6syl6eleq 2565 . . . . . 6  |-  ( ph  ->  A  e.  ( Base `  R ) )
82ply1sca 18062 . . . . . . . . 9  |-  ( R  e.  CRing  ->  R  =  (Scalar `  W ) )
98eqcomd 2475 . . . . . . . 8  |-  ( R  e.  CRing  ->  (Scalar `  W
)  =  R )
101, 9syl 16 . . . . . . 7  |-  ( ph  ->  (Scalar `  W )  =  R )
1110fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( Base `  (Scalar `  W ) )  =  ( Base `  R
) )
127, 11eleqtrrd 2558 . . . . 5  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  W )
) )
13 crngrng 16993 . . . . . . . . 9  |-  ( R  e.  CRing  ->  R  e.  Ring )
141, 13syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
152ply1rng 18057 . . . . . . . 8  |-  ( R  e.  Ring  ->  W  e. 
Ring )
1614, 15syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Ring )
17 evl1varpw.g . . . . . . . 8  |-  G  =  (mulGrp `  W )
1817rngmgp 16989 . . . . . . 7  |-  ( W  e.  Ring  ->  G  e. 
Mnd )
1916, 18syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Mnd )
20 evl1varpw.n . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
21 evl1varpw.x . . . . . . . 8  |-  X  =  (var1 `  R )
22 eqid 2467 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
2321, 2, 22vr1cl 18026 . . . . . . 7  |-  ( R  e.  Ring  ->  X  e.  ( Base `  W
) )
2414, 23syl 16 . . . . . 6  |-  ( ph  ->  X  e.  ( Base `  W ) )
2517, 22mgpbas 16934 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  G )
26 evl1varpw.e . . . . . . 7  |-  .^  =  (.g
`  G )
2725, 26mulgnn0cl 15955 . . . . . 6  |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  ( Base `  W
) )  ->  ( N  .^  X )  e.  ( Base `  W
) )
2819, 20, 24, 27syl3anc 1228 . . . . 5  |-  ( ph  ->  ( N  .^  X
)  e.  ( Base `  W ) )
29 eqid 2467 . . . . . 6  |-  (algSc `  W )  =  (algSc `  W )
30 eqid 2467 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
31 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
32 eqid 2467 . . . . . 6  |-  ( .r
`  W )  =  ( .r `  W
)
33 evl1scvarpw.t1 . . . . . 6  |-  .X.  =  ( .s `  W )
3429, 30, 31, 22, 32, 33asclmul1 17756 . . . . 5  |-  ( ( W  e. AssAlg  /\  A  e.  ( Base `  (Scalar `  W ) )  /\  ( N  .^  X )  e.  ( Base `  W
) )  ->  (
( (algSc `  W
) `  A )
( .r `  W
) ( N  .^  X ) )  =  ( A  .X.  ( N  .^  X ) ) )
354, 12, 28, 34syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( (algSc `  W ) `  A
) ( .r `  W ) ( N 
.^  X ) )  =  ( A  .X.  ( N  .^  X ) ) )
3635eqcomd 2475 . . 3  |-  ( ph  ->  ( A  .X.  ( N  .^  X ) )  =  ( ( (algSc `  W ) `  A
) ( .r `  W ) ( N 
.^  X ) ) )
3736fveq2d 5868 . 2  |-  ( ph  ->  ( Q `  ( A  .X.  ( N  .^  X ) ) )  =  ( Q `  ( ( (algSc `  W ) `  A
) ( .r `  W ) ( N 
.^  X ) ) ) )
38 evl1varpw.q . . . . 5  |-  Q  =  (eval1 `  R )
39 evl1scvarpw.s . . . . 5  |-  S  =  ( R  ^s  B )
4038, 2, 39, 6evl1rhm 18136 . . . 4  |-  ( R  e.  CRing  ->  Q  e.  ( W RingHom  S ) )
411, 40syl 16 . . 3  |-  ( ph  ->  Q  e.  ( W RingHom  S ) )
422ply1lmod 18061 . . . . . 6  |-  ( R  e.  Ring  ->  W  e. 
LMod )
4314, 42syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
4429, 30, 16, 43, 31, 22asclf 17754 . . . 4  |-  ( ph  ->  (algSc `  W ) : ( Base `  (Scalar `  W ) ) --> (
Base `  W )
)
4544, 12ffvelrnd 6020 . . 3  |-  ( ph  ->  ( (algSc `  W
) `  A )  e.  ( Base `  W
) )
46 evl1scvarpw.t2 . . . 4  |-  .xb  =  ( .r `  S )
4722, 32, 46rhmmul 17157 . . 3  |-  ( ( Q  e.  ( W RingHom  S )  /\  (
(algSc `  W ) `  A )  e.  (
Base `  W )  /\  ( N  .^  X
)  e.  ( Base `  W ) )  -> 
( Q `  (
( (algSc `  W
) `  A )
( .r `  W
) ( N  .^  X ) ) )  =  ( ( Q `
 ( (algSc `  W ) `  A
) )  .xb  ( Q `  ( N  .^  X ) ) ) )
4841, 45, 28, 47syl3anc 1228 . 2  |-  ( ph  ->  ( Q `  (
( (algSc `  W
) `  A )
( .r `  W
) ( N  .^  X ) ) )  =  ( ( Q `
 ( (algSc `  W ) `  A
) )  .xb  ( Q `  ( N  .^  X ) ) ) )
4938, 2, 6, 29evl1sca 18138 . . . 4  |-  ( ( R  e.  CRing  /\  A  e.  B )  ->  ( Q `  ( (algSc `  W ) `  A
) )  =  ( B  X.  { A } ) )
501, 5, 49syl2anc 661 . . 3  |-  ( ph  ->  ( Q `  (
(algSc `  W ) `  A ) )  =  ( B  X.  { A } ) )
5138, 2, 17, 21, 6, 26, 1, 20evl1varpw 18165 . . . 4  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `  X
) ) )
52 evl1scvarpw.f . . . . . . . 8  |-  F  =  (.g `  M )
53 evl1scvarpw.m . . . . . . . . . 10  |-  M  =  (mulGrp `  S )
5439fveq2i 5867 . . . . . . . . . 10  |-  (mulGrp `  S )  =  (mulGrp `  ( R  ^s  B ) )
5553, 54eqtri 2496 . . . . . . . . 9  |-  M  =  (mulGrp `  ( R  ^s  B ) )
5655fveq2i 5867 . . . . . . . 8  |-  (.g `  M
)  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
5752, 56eqtri 2496 . . . . . . 7  |-  F  =  (.g `  (mulGrp `  ( R  ^s  B ) ) )
5857a1i 11 . . . . . 6  |-  ( ph  ->  F  =  (.g `  (mulGrp `  ( R  ^s  B ) ) ) )
5958eqcomd 2475 . . . . 5  |-  ( ph  ->  (.g `  (mulGrp `  ( R  ^s  B ) ) )  =  F )
6059oveqd 6299 . . . 4  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `
 X ) )  =  ( N F ( Q `  X
) ) )
6151, 60eqtrd 2508 . . 3  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N F ( Q `  X
) ) )
6250, 61oveq12d 6300 . 2  |-  ( ph  ->  ( ( Q `  ( (algSc `  W ) `  A ) )  .xb  ( Q `  ( N 
.^  X ) ) )  =  ( ( B  X.  { A } )  .xb  ( N F ( Q `  X ) ) ) )
6337, 48, 623eqtrd 2512 1  |-  ( ph  ->  ( Q `  ( A  .X.  ( N  .^  X ) ) )  =  ( ( B  X.  { A }
)  .xb  ( N F ( Q `  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {csn 4027    X. cxp 4997   ` cfv 5586  (class class class)co 6282   NN0cn0 10791   Basecbs 14483   .rcmulr 14549  Scalarcsca 14551   .scvsca 14552    ^s cpws 14695   Mndcmnd 15719  .gcmg 15724  mulGrpcmgp 16928   Ringcrg 16983   CRingccrg 16984   RingHom crh 17142   LModclmod 17292  AssAlgcasa 17726  algSccascl 17728  var1cv1 17983  Poly1cpl1 17984  eval1ce1 18119
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12071  df-hash 12368  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-hom 14572  df-cco 14573  df-0g 14690  df-gsum 14691  df-prds 14696  df-pws 14698  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-mhm 15774  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mulg 15858  df-subg 15990  df-ghm 16057  df-cntz 16147  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-srg 16945  df-rng 16985  df-cring 16986  df-rnghom 17145  df-subrg 17207  df-lmod 17294  df-lss 17359  df-lsp 17398  df-assa 17729  df-asp 17730  df-ascl 17731  df-psr 17773  df-mvr 17774  df-mpl 17775  df-opsr 17777  df-evls 17939  df-evl 17940  df-psr1 17987  df-vr1 17988  df-ply1 17989  df-evls1 18120  df-evl1 18121
This theorem is referenced by: (None)
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