MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1divalg2 Structured version   Unicode version

Theorem ply1divalg2 21726
Description: Reverse the order of multiplication in ply1divalg 21725 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 2451 . . 3  |-  (Poly1 `  (oppr `  R
) )  =  (Poly1 `  (oppr
`  R ) )
2 ply1divalg.d . . . 4  |-  D  =  ( deg1  `  R )
3 eqidd 2452 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  R ) )
4 eqid 2451 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5 eqid 2451 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
64, 5opprbas 16827 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
76a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  R
)  =  ( Base `  (oppr
`  R ) ) )
8 eqid 2451 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
94, 8oppradd 16828 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
109oveqi 6203 . . . . . . 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 21673 . . . . 5  |-  ( T. 
->  ( deg1  `  R )  =  ( deg1  `  (oppr
`  R ) ) )
1312trud 1379 . . . 4  |-  ( deg1  `  R
)  =  ( deg1  `  (oppr `  R
) )
142, 13eqtri 2480 . . 3  |-  D  =  ( deg1  `  (oppr
`  R ) )
15 ply1divalg.b . . . 4  |-  B  =  ( Base `  P
)
16 ply1divalg.p . . . . . 6  |-  P  =  (Poly1 `  R )
1716fveq2i 5792 . . . . 5  |-  ( Base `  P )  =  (
Base `  (Poly1 `  R
) )
183, 7, 11ply1baspropd 17805 . . . . . 6  |-  ( T. 
->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) ) )
1918trud 1379 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
2017, 19eqtri 2480 . . . 4  |-  ( Base `  P )  =  (
Base `  (Poly1 `  (oppr `  R
) ) )
2115, 20eqtri 2480 . . 3  |-  B  =  ( Base `  (Poly1 `  (oppr `  R ) ) )
22 ply1divalg.m . . . 4  |-  .-  =  ( -g `  P )
2320a1i 11 . . . . . 6  |-  ( T. 
->  ( Base `  P
)  =  ( Base `  (Poly1 `  (oppr
`  R ) ) ) )
2416fveq2i 5792 . . . . . . . 8  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  R
) )
253, 7, 11ply1plusgpropd 17806 . . . . . . . . 9  |-  ( T. 
->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) ) )
2625trud 1379 . . . . . . . 8  |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (oppr `  R ) ) )
2724, 26eqtri 2480 . . . . . . 7  |-  ( +g  `  P )  =  ( +g  `  (Poly1 `  (oppr `  R
) ) )
2827a1i 11 . . . . . 6  |-  ( T. 
->  ( +g  `  P
)  =  ( +g  `  (Poly1 `  (oppr
`  R ) ) ) )
2923, 28grpsubpropd 15728 . . . . 5  |-  ( T. 
->  ( -g `  P
)  =  ( -g `  (Poly1 `  (oppr
`  R ) ) ) )
3029trud 1379 . . . 4  |-  ( -g `  P )  =  (
-g `  (Poly1 `  (oppr `  R
) ) )
3122, 30eqtri 2480 . . 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 6203 . . . . . . 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 15549 . . . . 5  |-  ( T. 
->  ( 0g `  P
)  =  ( 0g
`  (Poly1 `  (oppr
`  R ) ) ) )
3837trud 1379 . . . 4  |-  ( 0g
`  P )  =  ( 0g `  (Poly1 `  (oppr `  R ) ) )
3932, 38eqtri 2480 . . 3  |-  .0.  =  ( 0g `  (Poly1 `  (oppr `  R
) ) )
40 eqid 2451 . . 3  |-  ( .r
`  (Poly1 `  (oppr
`  R ) ) )  =  ( .r
`  (Poly1 `  (oppr
`  R ) ) )
41 ply1divalg.r1 . . . 4  |-  ( ph  ->  R  e.  Ring )
424opprrng 16829 . . . 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 16859 . . 3  |-  U  =  (Unit `  (oppr
`  R ) )
501, 14, 21, 31, 39, 40, 43, 44, 45, 46, 47, 49ply1divalg 21725 . 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 17801 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  G  e.  B  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5651, 52, 53, 55syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  q  e.  B )  ->  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q )  =  ( q  .xb  G
) )
5756eqcomd 2459 . . . . . 6  |-  ( (
ph  /\  q  e.  B )  ->  (
q  .xb  G )  =  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) )
5857oveq2d 6206 . . . . 5  |-  ( (
ph  /\  q  e.  B )  ->  ( F  .-  ( q  .xb  G ) )  =  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )
5958fveq2d 5793 . . . 4  |-  ( (
ph  /\  q  e.  B )  ->  ( D `  ( F  .-  ( q  .xb  G
) ) )  =  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr `  R
) ) ) q ) ) ) )
6059breq1d 4400 . . 3  |-  ( (
ph  /\  q  e.  B )  ->  (
( D `  ( F  .-  ( q  .xb  G ) ) )  <  ( D `  G )  <->  ( D `  ( F  .-  ( G ( .r `  (Poly1 `  (oppr
`  R ) ) ) q ) ) )  <  ( D `
 G ) ) )
6160reubidva 3000 . 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 1370   T. wtru 1371    e. wcel 1758    =/= wne 2644   E!wreu 2797   class class class wbr 4390   ` cfv 5516  (class class class)co 6190    < clt 9519   Basecbs 14276   +g cplusg 14340   .rcmulr 14341   0gc0g 14480   -gcsg 15515   Ringcrg 16751  opprcoppr 16820  Unitcui 16837  Poly1cpl1 17740  coe1cco1 17741   deg1 cdg1 21639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-ofr 6421  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-0g 14482  df-gsum 14483  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-subrg 16969  df-lmod 17056  df-lss 17120  df-rlreg 17460  df-psr 17529  df-mvr 17530  df-mpl 17531  df-opsr 17533  df-psr1 17743  df-vr1 17744  df-ply1 17745  df-coe1 17746  df-cnfld 17928  df-mdeg 21640  df-deg1 21641
This theorem is referenced by:  q1peqb  21742
  Copyright terms: Public domain W3C validator