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Theorem evl1varpw 17793
Description: Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 17790, the proof is shorter using evls1varpw 17759 instead of proving it directly. (Contributed by AV, 15-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 )
Assertion
Ref Expression
evl1varpw  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `  X
) ) )

Proof of Theorem evl1varpw
StepHypRef Expression
1 evl1varpw.q . . . . 5  |-  Q  =  (eval1 `  R )
2 evl1varpw.b . . . . 5  |-  B  =  ( Base `  R
)
31, 2evl1fval1 17763 . . . 4  |-  Q  =  ( R evalSub1  B )
43a1i 11 . . 3  |-  ( ph  ->  Q  =  ( R evalSub1  B ) )
5 evl1varpw.e . . . . . 6  |-  .^  =  (.g
`  G )
6 evl1varpw.g . . . . . . . 8  |-  G  =  (mulGrp `  W )
7 evl1varpw.w . . . . . . . . 9  |-  W  =  (Poly1 `  R )
87fveq2i 5692 . . . . . . . 8  |-  (mulGrp `  W )  =  (mulGrp `  (Poly1 `  R ) )
96, 8eqtri 2461 . . . . . . 7  |-  G  =  (mulGrp `  (Poly1 `  R
) )
109fveq2i 5692 . . . . . 6  |-  (.g `  G
)  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
115, 10eqtri 2461 . . . . 5  |-  .^  =  (.g
`  (mulGrp `  (Poly1 `  R
) ) )
12 evl1varpw.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  CRing )
132ressid 14231 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( Rs  B
)  =  R )
1412, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( Rs  B )  =  R )
1514eqcomd 2446 . . . . . . . 8  |-  ( ph  ->  R  =  ( Rs  B ) )
1615fveq2d 5693 . . . . . . 7  |-  ( ph  ->  (Poly1 `  R )  =  (Poly1 `  ( Rs  B ) ) )
1716fveq2d 5693 . . . . . 6  |-  ( ph  ->  (mulGrp `  (Poly1 `  R
) )  =  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )
1817fveq2d 5693 . . . . 5  |-  ( ph  ->  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) )
1911, 18syl5eq 2485 . . . 4  |-  ( ph  -> 
.^  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) )
20 eqidd 2442 . . . 4  |-  ( ph  ->  N  =  N )
21 evl1varpw.x . . . . 5  |-  X  =  (var1 `  R )
2215fveq2d 5693 . . . . 5  |-  ( ph  ->  (var1 `  R )  =  (var1 `  ( Rs  B ) ) )
2321, 22syl5eq 2485 . . . 4  |-  ( ph  ->  X  =  (var1 `  ( Rs  B ) ) )
2419, 20, 23oveq123d 6110 . . 3  |-  ( ph  ->  ( N  .^  X
)  =  ( N (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) (var1 `  ( Rs  B ) ) ) )
254, 24fveq12d 5695 . 2  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( ( R evalSub1  B ) `  ( N (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) (var1 `  ( Rs  B ) ) ) ) )
26 eqid 2441 . . 3  |-  ( R evalSub1  B )  =  ( R evalSub1  B )
27 eqid 2441 . . 3  |-  ( Rs  B )  =  ( Rs  B )
28 eqid 2441 . . 3  |-  (Poly1 `  ( Rs  B ) )  =  (Poly1 `  ( Rs  B ) )
29 eqid 2441 . . 3  |-  (mulGrp `  (Poly1 `  ( Rs  B ) ) )  =  (mulGrp `  (Poly1 `  ( Rs  B ) ) )
30 eqid 2441 . . 3  |-  (var1 `  ( Rs  B ) )  =  (var1 `  ( Rs  B ) )
31 eqid 2441 . . 3  |-  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )
32 crngrng 16653 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
332subrgid 16865 . . . 4  |-  ( R  e.  Ring  ->  B  e.  (SubRing `  R )
)
3412, 32, 333syl 20 . . 3  |-  ( ph  ->  B  e.  (SubRing `  R
) )
35 evl1varpw.n . . 3  |-  ( ph  ->  N  e.  NN0 )
3626, 27, 28, 29, 30, 2, 31, 12, 34, 35evls1varpw 17759 . 2  |-  ( ph  ->  ( ( R evalSub1  B ) `
 ( N (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) (var1 `  ( Rs  B ) ) ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( ( R evalSub1  B ) `  (var1 `  ( Rs  B ) ) ) ) )
373eqcomi 2445 . . . . 5  |-  ( R evalSub1  B )  =  Q
3837a1i 11 . . . 4  |-  ( ph  ->  ( R evalSub1  B )  =  Q )
3923eqcomd 2446 . . . 4  |-  ( ph  ->  (var1 `  ( Rs  B ) )  =  X )
4038, 39fveq12d 5695 . . 3  |-  ( ph  ->  ( ( R evalSub1  B ) `
 (var1 `  ( Rs  B ) ) )  =  ( Q `  X ) )
4140oveq2d 6105 . 2  |-  ( ph  ->  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( ( R evalSub1  B ) `  (var1 `  ( Rs  B ) ) ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `  X
) ) )
4225, 36, 413eqtrd 2477 1  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   NN0cn0 10577   Basecbs 14172   ↾s cress 14173    ^s cpws 14383  .gcmg 15412  mulGrpcmgp 16589   Ringcrg 16643   CRingccrg 16644  SubRingcsubrg 16859  var1cv1 17630  Poly1cpl1 17631   evalSub1 ces1 17746  eval1ce1 17747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-ofr 6319  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-hom 14260  df-cco 14261  df-0g 14378  df-gsum 14379  df-prds 14384  df-pws 14386  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-ghm 15743  df-cntz 15833  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-srg 16606  df-rng 16645  df-cring 16646  df-rnghom 16804  df-subrg 16861  df-lmod 16948  df-lss 17012  df-lsp 17051  df-assa 17382  df-asp 17383  df-ascl 17384  df-psr 17421  df-mvr 17422  df-mpl 17423  df-opsr 17425  df-evls 17586  df-evl 17587  df-psr1 17634  df-vr1 17635  df-ply1 17636  df-evls1 17748  df-evl1 17749
This theorem is referenced by:  evl1scvarpw  17795
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