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Theorem ply1divalg2 19524
Description: Reverse the order of multiplication in ply1divalg 19523 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 2283 . . 3  |-  (Poly1 `  (oppr `  R
) )  =  (Poly1 `  (oppr
`  R ) )
2 ply1divalg.d . . . 4  |-  D  =  ( deg1  `  R )
3 eqidd 2284 . . . . . 6  |-  (  T. 
->  ( Base `  R
)  =  ( Base `  R ) )
4 eqid 2283 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5 eqid 2283 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
64, 5opprbas 15411 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
76a1i 10 . . . . . 6  |-  (  T. 
->  ( Base `  R
)  =  ( Base `  (oppr
`  R ) ) )
8 eqid 2283 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
94, 8oppradd 15412 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
109oveqi 5871 . . . . . . 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 19472 . . . . 5  |-  (  T. 
->  ( deg1  `  R )  =  ( deg1  `  (oppr
`  R ) ) )
1312trud 1314 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  (oppr `  R
) )
142, 13eqtri 2303 . . 3  |-  D  =  ( deg1  `  (oppr
`  R ) )
15 ply1divalg.b . . . 4  |-  B  =  ( Base `  P
)
16 ply1divalg.p . . . . . 6  |-  P  =  (Poly1 `  R )
1716fveq2i 5528 . . . . 5  |-  ( Base `  P )  =  (
Base `  (Poly1 `  R
) )
183, 7, 11ply1baspropd 16321 . . . . . 6  |-  (  T. 
->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) ) )
1918trud 1314 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
2017, 19eqtri 2303 . . . 4  |-  ( Base `  P )  =  (
Base `  (Poly1 `  (oppr `  R
) ) )
2115, 20eqtri 2303 . . 3  |-  B  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
22 ply1divalg.m . . . 4  |-  .-  =  ( -g `  P )
2320a1i 10 . . . . . 6  |-  (  T. 
->  ( Base `  P
)  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
2416fveq2i 5528 . . . . . . . 8  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  R
) )
253, 7, 11ply1plusgpropd 16322 . . . . . . . . 9  |-  (  T. 
->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) ) )
2625trud 1314 . . . . . . . 8  |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) )
2724, 26eqtri 2303 . . . . . . 7  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  (oppr `  R
) ) )
2827a1i 10 . . . . . 6  |-  (  T. 
->  ( +g  `  P
)  =  ( +g  `  (Poly1 `  (oppr
`  R ) ) ) )
2923, 28grpsubpropd 14566 . . . . 5  |-  (  T. 
->  ( -g `  P
)  =  ( -g `  (Poly1 `  (oppr
`  R ) ) ) )
3029trud 1314 . . . 4  |-  ( -g `  P )  =  (
-g `  (Poly1 `  (oppr `  R
) ) )
3122, 30eqtri 2303 . . 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 5871 . . . . . . 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 14399 . . . . 5  |-  (  T. 
->  ( 0g `  P
)  =  ( 0g
`  (Poly1 `  (oppr
`  R ) ) ) )
3837trud 1314 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  (Poly1 `  (oppr `  R ) ) )
3932, 38eqtri 2303 . . 3  |-  .0.  =  ( 0g `  (Poly1 `  (oppr `  R
) ) )
40 eqid 2283 . . 3  |-  ( .r
`  (Poly1 `  (oppr
`  R ) ) )  =  ( .r
`  (Poly1 `  (oppr
`  R ) ) )
41 ply1divalg.r1 . . . 4  |-  ( ph  ->  R  e.  Ring )
424opprrng 15413 . . . 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 15443 . . 3  |-  U  =  (Unit `  (oppr
`  R ) )
501, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49ply1divalg 19523 . 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 16317 . . . . . . . 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 2288 . . . . . 6  |-  ( (
ph  /\  q  e.  B )  ->  (
q  .xb  G )  =  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) )
5857oveq2d 5874 . . . . 5  |-  ( (
ph  /\  q  e.  B )  ->  ( F  .-  ( q  .xb  G ) )  =  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )
5958fveq2d 5529 . . . 4  |-  ( (
ph  /\  q  e.  B )  ->  ( D `  ( F  .-  ( q  .xb  G
) ) )  =  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) ) )
6059breq1d 4033 . . 3  |-  ( (
ph  /\  q  e.  B )  ->  (
( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G )  <->  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6160reubidva 2723 . 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 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   E!wreu 2545   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    < clt 8867   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   -gcsg 14365   Ringcrg 15337  opprcoppr 15404  Unitcui 15421  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440
This theorem is referenced by:  q1peqb  19540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-subrg 15543  df-lmod 15629  df-lss 15690  df-rlreg 16024  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442
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