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Theorem ply1divalg2 22274
Description: Reverse the order of multiplication in ply1divalg 22273 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 2467 . . 3  |-  (Poly1 `  (oppr `  R
) )  =  (Poly1 `  (oppr
`  R ) )
2 ply1divalg.d . . . 4  |-  D  =  ( deg1  `  R )
3 eqidd 2468 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  R ) )
4 eqid 2467 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5 eqid 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
64, 5opprbas 17062 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
76a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  (oppr
`  R ) ) )
8 eqid 2467 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
94, 8oppradd 17063 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
109oveqi 6295 . . . . . . 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 22221 . . . . 5  |-  ( T. 
->  ( deg1  `  R )  =  ( deg1  `  (oppr
`  R ) ) )
1312trud 1388 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  (oppr `  R
) )
142, 13eqtri 2496 . . 3  |-  D  =  ( deg1  `  (oppr
`  R ) )
15 ply1divalg.b . . . 4  |-  B  =  ( Base `  P
)
16 ply1divalg.p . . . . . 6  |-  P  =  (Poly1 `  R )
1716fveq2i 5867 . . . . 5  |-  ( Base `  P )  =  (
Base `  (Poly1 `  R
) )
183, 7, 11ply1baspropd 18055 . . . . . 6  |-  ( T. 
->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) ) )
1918trud 1388 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
2017, 19eqtri 2496 . . . 4  |-  ( Base `  P )  =  (
Base `  (Poly1 `  (oppr `  R
) ) )
2115, 20eqtri 2496 . . 3  |-  B  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
22 ply1divalg.m . . . 4  |-  .-  =  ( -g `  P )
2320a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  P
)  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
2416fveq2i 5867 . . . . . . . 8  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  R
) )
253, 7, 11ply1plusgpropd 18056 . . . . . . . . 9  |-  ( T. 
->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) ) )
2625trud 1388 . . . . . . . 8  |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) )
2724, 26eqtri 2496 . . . . . . 7  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  (oppr `  R
) ) )
2827a1i 11 . . . . . 6  |-  ( T. 
->  ( +g  `  P
)  =  ( +g  `  (Poly1 `  (oppr
`  R ) ) ) )
2923, 28grpsubpropd 15941 . . . . 5  |-  ( T. 
->  ( -g `  P
)  =  ( -g `  (Poly1 `  (oppr
`  R ) ) ) )
3029trud 1388 . . . 4  |-  ( -g `  P )  =  (
-g `  (Poly1 `  (oppr `  R
) ) )
3122, 30eqtri 2496 . . 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 6295 . . . . . . 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 15760 . . . . 5  |-  ( T. 
->  ( 0g `  P
)  =  ( 0g
`  (Poly1 `  (oppr
`  R ) ) ) )
3837trud 1388 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  (Poly1 `  (oppr `  R ) ) )
3932, 38eqtri 2496 . . 3  |-  .0.  =  ( 0g `  (Poly1 `  (oppr `  R
) ) )
40 eqid 2467 . . 3  |-  ( .r
`  (Poly1 `  (oppr
`  R ) ) )  =  ( .r
`  (Poly1 `  (oppr
`  R ) ) )
41 ply1divalg.r1 . . . 4  |-  ( ph  ->  R  e.  Ring )
424opprrng 17064 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
4341, 42syl 16 . . 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 17094 . . 3  |-  U  =  (Unit `  (oppr
`  R ) )
501, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49ply1divalg 22273 . 2  |-  ( ph  ->  E! q  e.  B  ( D `  ( F 
.-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) )  < 
( D `  G
) )
5141adantr 465 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  R  e.  Ring )
5245adantr 465 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  G  e.  B )
53 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  q  e.  B )  ->  q  e.  B )
54 ply1divalg.t . . . . . . . . 9  |-  .xb  =  ( .r `  P )
5516, 4, 1, 54, 40, 15ply1opprmul 18051 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5651, 52, 53, 55syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5756eqcomd 2475 . . . . . 6  |-  ( (
ph  /\  q  e.  B )  ->  (
q  .xb  G )  =  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) )
5857oveq2d 6298 . . . . 5  |-  ( (
ph  /\  q  e.  B )  ->  ( F  .-  ( q  .xb  G ) )  =  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )
5958fveq2d 5868 . . . 4  |-  ( (
ph  /\  q  e.  B )  ->  ( D `  ( F  .-  ( q  .xb  G
) ) )  =  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) ) )
6059breq1d 4457 . . 3  |-  ( (
ph  /\  q  e.  B )  ->  (
( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G )  <->  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6160reubidva 3045 . 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 369    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   E!wreu 2816   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    < clt 9624   Basecbs 14486   +g cplusg 14551   .rcmulr 14552   0gc0g 14691   -gcsg 15726   Ringcrg 16986  opprcoppr 17055  Unitcui 17072  Poly1cpl1 17987  coe1cco1 17988   deg1 cdg1 22187
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  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
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-tpos 6952  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 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-subrg 17210  df-lmod 17297  df-lss 17362  df-rlreg 17702  df-psr 17776  df-mvr 17777  df-mpl 17778  df-opsr 17780  df-psr1 17990  df-vr1 17991  df-ply1 17992  df-coe1 17993  df-cnfld 18192  df-mdeg 22188  df-deg1 22189
This theorem is referenced by:  q1peqb  22290
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