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Theorem ply1divalg2 22831
Description: Reverse the order of multiplication in ply1divalg 22830 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
ply1divalg.p  |-  P  =  (Poly1 `  R )
ply1divalg.d  |-  D  =  ( deg1  `  R )
ply1divalg.b  |-  B  =  ( Base `  P
)
ply1divalg.m  |-  .-  =  ( -g `  P )
ply1divalg.z  |-  .0.  =  ( 0g `  P )
ply1divalg.t  |-  .xb  =  ( .r `  P )
ply1divalg.r1  |-  ( ph  ->  R  e.  Ring )
ply1divalg.f  |-  ( ph  ->  F  e.  B )
ply1divalg.g1  |-  ( ph  ->  G  e.  B )
ply1divalg.g2  |-  ( ph  ->  G  =/=  .0.  )
ply1divalg.g3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  U )
ply1divalg.u  |-  U  =  (Unit `  R )
Assertion
Ref Expression
ply1divalg2  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( q  .xb  G ) ) )  <  ( D `  G ) )
Distinct variable groups:    ph, q    B, q    D, q    F, q    G, q    .- , q    P, q    R, q    .xb , q    .0. , q
Allowed substitution hint:    U( q)

Proof of Theorem ply1divalg2
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  (Poly1 `  (oppr `  R
) )  =  (Poly1 `  (oppr
`  R ) )
2 ply1divalg.d . . . 4  |-  D  =  ( deg1  `  R )
3 eqidd 2403 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  R ) )
4 eqid 2402 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5 eqid 2402 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
64, 5opprbas 17598 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
76a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  (oppr
`  R ) ) )
8 eqid 2402 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
94, 8oppradd 17599 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
109oveqi 6291 . . . . . . 7  |-  ( q ( +g  `  R
) r )  =  ( q ( +g  `  (oppr
`  R ) ) r )
1110a1i 11 . . . . . 6  |-  ( ( T.  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  R
) r )  =  ( q ( +g  `  (oppr
`  R ) ) r ) )
123, 7, 11deg1propd 22778 . . . . 5  |-  ( T. 
->  ( deg1  `  R )  =  ( deg1  `  (oppr
`  R ) ) )
1312trud 1414 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  (oppr `  R
) )
142, 13eqtri 2431 . . 3  |-  D  =  ( deg1  `  (oppr
`  R ) )
15 ply1divalg.b . . . 4  |-  B  =  ( Base `  P
)
16 ply1divalg.p . . . . . 6  |-  P  =  (Poly1 `  R )
1716fveq2i 5852 . . . . 5  |-  ( Base `  P )  =  (
Base `  (Poly1 `  R
) )
183, 7, 11ply1baspropd 18604 . . . . . 6  |-  ( T. 
->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) ) )
1918trud 1414 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
2017, 19eqtri 2431 . . . 4  |-  ( Base `  P )  =  (
Base `  (Poly1 `  (oppr `  R
) ) )
2115, 20eqtri 2431 . . 3  |-  B  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
22 ply1divalg.m . . . 4  |-  .-  =  ( -g `  P )
2320a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  P
)  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
2416fveq2i 5852 . . . . . . . 8  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  R
) )
253, 7, 11ply1plusgpropd 18605 . . . . . . . . 9  |-  ( T. 
->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) ) )
2625trud 1414 . . . . . . . 8  |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) )
2724, 26eqtri 2431 . . . . . . 7  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  (oppr `  R
) ) )
2827a1i 11 . . . . . 6  |-  ( T. 
->  ( +g  `  P
)  =  ( +g  `  (Poly1 `  (oppr
`  R ) ) ) )
2923, 28grpsubpropd 16464 . . . . 5  |-  ( T. 
->  ( -g `  P
)  =  ( -g `  (Poly1 `  (oppr
`  R ) ) ) )
3029trud 1414 . . . 4  |-  ( -g `  P )  =  (
-g `  (Poly1 `  (oppr `  R
) ) )
3122, 30eqtri 2431 . . 3  |-  .-  =  ( -g `  (Poly1 `  (oppr `  R
) ) )
32 ply1divalg.z . . . 4  |-  .0.  =  ( 0g `  P )
3315a1i 11 . . . . . 6  |-  ( T. 
->  B  =  ( Base `  P ) )
3421a1i 11 . . . . . 6  |-  ( T. 
->  B  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
3527oveqi 6291 . . . . . . 7  |-  ( q ( +g  `  P
) r )  =  ( q ( +g  `  (Poly1 `  (oppr
`  R ) ) ) r )
3635a1i 11 . . . . . 6  |-  ( ( T.  /\  ( q  e.  B  /\  r  e.  B ) )  -> 
( q ( +g  `  P ) r )  =  ( q ( +g  `  (Poly1 `  (oppr `  R
) ) ) r ) )
3733, 34, 36grpidpropd 16212 . . . . 5  |-  ( T. 
->  ( 0g `  P
)  =  ( 0g
`  (Poly1 `  (oppr
`  R ) ) ) )
3837trud 1414 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  (Poly1 `  (oppr `  R ) ) )
3932, 38eqtri 2431 . . 3  |-  .0.  =  ( 0g `  (Poly1 `  (oppr `  R
) ) )
40 eqid 2402 . . 3  |-  ( .r
`  (Poly1 `  (oppr
`  R ) ) )  =  ( .r
`  (Poly1 `  (oppr
`  R ) ) )
41 ply1divalg.r1 . . . 4  |-  ( ph  ->  R  e.  Ring )
424opprring 17600 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
4341, 42syl 17 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
44 ply1divalg.f . . 3  |-  ( ph  ->  F  e.  B )
45 ply1divalg.g1 . . 3  |-  ( ph  ->  G  e.  B )
46 ply1divalg.g2 . . 3  |-  ( ph  ->  G  =/=  .0.  )
47 ply1divalg.g3 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  U )
48 ply1divalg.u . . . 4  |-  U  =  (Unit `  R )
4948, 4opprunit 17630 . . 3  |-  U  =  (Unit `  (oppr
`  R ) )
501, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49ply1divalg 22830 . 2  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) )  < 
( D `  G
) )
5141adantr 463 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  R  e.  Ring )
5245adantr 463 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  G  e.  B )
53 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  q  e.  B )
54 ply1divalg.t . . . . . . . . 9  |-  .xb  =  ( .r `  P )
5516, 4, 1, 54, 40, 15ply1opprmul 18600 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5651, 52, 53, 55syl3anc 1230 . . . . . . 7  |-  ( (
ph  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5756eqcomd 2410 . . . . . 6  |-  ( (
ph  /\  q  e.  B )  ->  (
q  .xb  G )  =  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) )
5857oveq2d 6294 . . . . 5  |-  ( (
ph  /\  q  e.  B )  ->  ( F  .-  ( q  .xb  G ) )  =  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )
5958fveq2d 5853 . . . 4  |-  ( (
ph  /\  q  e.  B )  ->  ( D `  ( F  .-  ( q  .xb  G
) ) )  =  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) ) )
6059breq1d 4405 . . 3  |-  ( (
ph  /\  q  e.  B )  ->  (
( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G )  <->  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6160reubidva 2991 . 2  |-  ( ph  ->  ( E! q  e.  B  ( D `  ( F  .-  ( q 
.xb  G ) ) )  <  ( D `
 G )  <->  E! q  e.  B  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6250, 61mpbird 232 1  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( q  .xb  G ) ) )  <  ( D `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842    =/= wne 2598   E!wreu 2756   class class class wbr 4395   ` cfv 5569  (class class class)co 6278    < clt 9658   Basecbs 14841   +g cplusg 14909   .rcmulr 14910   0gc0g 15054   -gcsg 16379   Ringcrg 17518  opprcoppr 17591  Unitcui 17608  Poly1cpl1 18536  coe1cco1 18537   deg1 cdg1 22744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-ofr 6522  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-0g 15056  df-gsum 15057  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-ghm 16589  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-subrg 17747  df-lmod 17834  df-lss 17899  df-rlreg 18251  df-psr 18325  df-mvr 18326  df-mpl 18327  df-opsr 18329  df-psr1 18539  df-vr1 18540  df-ply1 18541  df-coe1 18542  df-cnfld 18741  df-mdeg 22745  df-deg1 22746
This theorem is referenced by:  q1peqb  22847
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