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Theorem evl1varpw 18165
Description: Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 18162, the proof is shorter using evls1varpw 18131 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 18135 . . . 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 5867 . . . . . . . 8  |-  (mulGrp `  W )  =  (mulGrp `  (Poly1 `  R ) )
96, 8eqtri 2496 . . . . . . 7  |-  G  =  (mulGrp `  (Poly1 `  R
) )
109fveq2i 5867 . . . . . 6  |-  (.g `  G
)  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
115, 10eqtri 2496 . . . . 5  |-  .^  =  (.g
`  (mulGrp `  (Poly1 `  R
) ) )
12 evl1varpw.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  CRing )
132ressid 14543 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( Rs  B
)  =  R )
1412, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( Rs  B )  =  R )
1514eqcomd 2475 . . . . . . . 8  |-  ( ph  ->  R  =  ( Rs  B ) )
1615fveq2d 5868 . . . . . . 7  |-  ( ph  ->  (Poly1 `  R )  =  (Poly1 `  ( Rs  B ) ) )
1716fveq2d 5868 . . . . . 6  |-  ( ph  ->  (mulGrp `  (Poly1 `  R
) )  =  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )
1817fveq2d 5868 . . . . 5  |-  ( ph  ->  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) )
1911, 18syl5eq 2520 . . . 4  |-  ( ph  -> 
.^  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) )
20 eqidd 2468 . . . 4  |-  ( ph  ->  N  =  N )
21 evl1varpw.x . . . . 5  |-  X  =  (var1 `  R )
2215fveq2d 5868 . . . . 5  |-  ( ph  ->  (var1 `  R )  =  (var1 `  ( Rs  B ) ) )
2321, 22syl5eq 2520 . . . 4  |-  ( ph  ->  X  =  (var1 `  ( Rs  B ) ) )
2419, 20, 23oveq123d 6303 . . 3  |-  ( ph  ->  ( N  .^  X
)  =  ( N (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) (var1 `  ( Rs  B ) ) ) )
254, 24fveq12d 5870 . 2  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( ( R evalSub1  B ) `  ( N (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) ) (var1 `  ( Rs  B ) ) ) ) )
26 eqid 2467 . . 3  |-  ( R evalSub1  B )  =  ( R evalSub1  B )
27 eqid 2467 . . 3  |-  ( Rs  B )  =  ( Rs  B )
28 eqid 2467 . . 3  |-  (Poly1 `  ( Rs  B ) )  =  (Poly1 `  ( Rs  B ) )
29 eqid 2467 . . 3  |-  (mulGrp `  (Poly1 `  ( Rs  B ) ) )  =  (mulGrp `  (Poly1 `  ( Rs  B ) ) )
30 eqid 2467 . . 3  |-  (var1 `  ( Rs  B ) )  =  (var1 `  ( Rs  B ) )
31 eqid 2467 . . 3  |-  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )  =  (.g `  (mulGrp `  (Poly1 `  ( Rs  B ) ) ) )
32 crngrng 16993 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Ring )
332subrgid 17211 . . . 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 18131 . 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 2480 . . . . 5  |-  ( R evalSub1  B )  =  Q
3837a1i 11 . . . 4  |-  ( ph  ->  ( R evalSub1  B )  =  Q )
3923eqcomd 2475 . . . 4  |-  ( ph  ->  (var1 `  ( Rs  B ) )  =  X )
4038, 39fveq12d 5870 . . 3  |-  ( ph  ->  ( ( R evalSub1  B ) `
 (var1 `  ( Rs  B ) ) )  =  ( Q `  X ) )
4140oveq2d 6298 . 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 2512 1  |-  ( ph  ->  ( Q `  ( N  .^  X ) )  =  ( N (.g `  (mulGrp `  ( R  ^s  B ) ) ) ( Q `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   NN0cn0 10791   Basecbs 14483   ↾s cress 14484    ^s cpws 14695  .gcmg 15724  mulGrpcmgp 16928   Ringcrg 16983   CRingccrg 16984  SubRingcsubrg 17205  var1cv1 17983  Poly1cpl1 17984   evalSub1 ces1 18118  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:  evl1scvarpw  18167
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