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Theorem ply1divalg2 19628
Description: Reverse the order of multiplication in ply1divalg 19627 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 2358 . . 3  |-  (Poly1 `  (oppr `  R
) )  =  (Poly1 `  (oppr
`  R ) )
2 ply1divalg.d . . . 4  |-  D  =  ( deg1  `  R )
3 eqidd 2359 . . . . . 6  |-  (  T. 
->  ( Base `  R
)  =  ( Base `  R ) )
4 eqid 2358 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5 eqid 2358 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
64, 5opprbas 15510 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
76a1i 10 . . . . . 6  |-  (  T. 
->  ( Base `  R
)  =  ( Base `  (oppr
`  R ) ) )
8 eqid 2358 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
94, 8oppradd 15511 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
109oveqi 5958 . . . . . . 7  |-  ( q ( +g  `  R
) r )  =  ( q ( +g  `  (oppr
`  R ) ) r )
1110a1i 10 . . . . . 6  |-  ( (  T.  /\  ( q  e.  ( Base `  R
)  /\  r  e.  ( Base `  R )
) )  ->  (
q ( +g  `  R
) r )  =  ( q ( +g  `  (oppr
`  R ) ) r ) )
123, 7, 11deg1propd 19576 . . . . 5  |-  (  T. 
->  ( deg1  `  R )  =  ( deg1  `  (oppr
`  R ) ) )
1312trud 1323 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  (oppr `  R
) )
142, 13eqtri 2378 . . 3  |-  D  =  ( deg1  `  (oppr
`  R ) )
15 ply1divalg.b . . . 4  |-  B  =  ( Base `  P
)
16 ply1divalg.p . . . . . 6  |-  P  =  (Poly1 `  R )
1716fveq2i 5611 . . . . 5  |-  ( Base `  P )  =  (
Base `  (Poly1 `  R
) )
183, 7, 11ply1baspropd 16420 . . . . . 6  |-  (  T. 
->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) ) )
1918trud 1323 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
2017, 19eqtri 2378 . . . 4  |-  ( Base `  P )  =  (
Base `  (Poly1 `  (oppr `  R
) ) )
2115, 20eqtri 2378 . . 3  |-  B  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
22 ply1divalg.m . . . 4  |-  .-  =  ( -g `  P )
2320a1i 10 . . . . . 6  |-  (  T. 
->  ( Base `  P
)  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
2416fveq2i 5611 . . . . . . . 8  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  R
) )
253, 7, 11ply1plusgpropd 16421 . . . . . . . . 9  |-  (  T. 
->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) ) )
2625trud 1323 . . . . . . . 8  |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) )
2724, 26eqtri 2378 . . . . . . 7  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  (oppr `  R
) ) )
2827a1i 10 . . . . . 6  |-  (  T. 
->  ( +g  `  P
)  =  ( +g  `  (Poly1 `  (oppr
`  R ) ) ) )
2923, 28grpsubpropd 14665 . . . . 5  |-  (  T. 
->  ( -g `  P
)  =  ( -g `  (Poly1 `  (oppr
`  R ) ) ) )
3029trud 1323 . . . 4  |-  ( -g `  P )  =  (
-g `  (Poly1 `  (oppr `  R
) ) )
3122, 30eqtri 2378 . . 3  |-  .-  =  ( -g `  (Poly1 `  (oppr `  R
) ) )
32 ply1divalg.z . . . 4  |-  .0.  =  ( 0g `  P )
3315a1i 10 . . . . . 6  |-  (  T. 
->  B  =  ( Base `  P ) )
3421a1i 10 . . . . . 6  |-  (  T. 
->  B  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
3527oveqi 5958 . . . . . . 7  |-  ( q ( +g  `  P
) r )  =  ( q ( +g  `  (Poly1 `  (oppr
`  R ) ) ) r )
3635a1i 10 . . . . . 6  |-  ( (  T.  /\  ( q  e.  B  /\  r  e.  B ) )  -> 
( q ( +g  `  P ) r )  =  ( q ( +g  `  (Poly1 `  (oppr `  R
) ) ) r ) )
3733, 34, 36grpidpropd 14498 . . . . 5  |-  (  T. 
->  ( 0g `  P
)  =  ( 0g
`  (Poly1 `  (oppr
`  R ) ) ) )
3837trud 1323 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  (Poly1 `  (oppr `  R ) ) )
3932, 38eqtri 2378 . . 3  |-  .0.  =  ( 0g `  (Poly1 `  (oppr `  R
) ) )
40 eqid 2358 . . 3  |-  ( .r
`  (Poly1 `  (oppr
`  R ) ) )  =  ( .r
`  (Poly1 `  (oppr
`  R ) ) )
41 ply1divalg.r1 . . . 4  |-  ( ph  ->  R  e.  Ring )
424opprrng 15512 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
4341, 42syl 15 . . 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 15542 . . 3  |-  U  =  (Unit `  (oppr
`  R ) )
501, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49ply1divalg 19627 . 2  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) )  < 
( D `  G
) )
5141adantr 451 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  R  e.  Ring )
5245adantr 451 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  G  e.  B )
53 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  q  e.  B )
54 ply1divalg.t . . . . . . . . 9  |-  .xb  =  ( .r `  P )
5516, 4, 1, 54, 40, 15ply1opprmul 16416 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5651, 52, 53, 55syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5756eqcomd 2363 . . . . . 6  |-  ( (
ph  /\  q  e.  B )  ->  (
q  .xb  G )  =  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) )
5857oveq2d 5961 . . . . 5  |-  ( (
ph  /\  q  e.  B )  ->  ( F  .-  ( q  .xb  G ) )  =  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )
5958fveq2d 5612 . . . 4  |-  ( (
ph  /\  q  e.  B )  ->  ( D `  ( F  .-  ( q  .xb  G
) ) )  =  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) ) )
6059breq1d 4114 . . 3  |-  ( (
ph  /\  q  e.  B )  ->  (
( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G )  <->  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6160reubidva 2799 . 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 223 1  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( q  .xb  G ) ) )  <  ( D `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    T. wtru 1316    = wceq 1642    e. wcel 1710    =/= wne 2521   E!wreu 2621   class class class wbr 4104   ` cfv 5337  (class class class)co 5945    < clt 8957   Basecbs 13245   +g cplusg 13305   .rcmulr 13306   0gc0g 13499   -gcsg 14464   Ringcrg 15436  opprcoppr 15503  Unitcui 15520  Poly1cpl1 16351  coe1cco1 16354   deg1 cdg1 19544
This theorem is referenced by:  q1peqb  19644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-subrg 15642  df-lmod 15728  df-lss 15789  df-rlreg 16123  df-psr 16197  df-mvr 16198  df-mpl 16199  df-opsr 16205  df-psr1 16356  df-vr1 16357  df-ply1 16358  df-coe1 16361  df-cnfld 16483  df-mdeg 19545  df-deg1 19546
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