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Theorem opprmul 15723
Description: Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
Hypotheses
Ref Expression
opprval.1  |-  B  =  ( Base `  R
)
opprval.2  |-  .x.  =  ( .r `  R )
opprval.3  |-  O  =  (oppr
`  R )
opprmulfval.4  |-  .xb  =  ( .r `  O )
Assertion
Ref Expression
opprmul  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)

Proof of Theorem opprmul
StepHypRef Expression
1 opprval.1 . . . 4  |-  B  =  ( Base `  R
)
2 opprval.2 . . . 4  |-  .x.  =  ( .r `  R )
3 opprval.3 . . . 4  |-  O  =  (oppr
`  R )
4 opprmulfval.4 . . . 4  |-  .xb  =  ( .r `  O )
51, 2, 3, 4opprmulfval 15722 . . 3  |-  .xb  = tpos  .x.
65oveqi 6086 . 2  |-  ( X 
.xb  Y )  =  ( Xtpos  .x.  Y
)
7 ovtpos 6486 . 2  |-  ( Xtpos 
.x.  Y )  =  ( Y  .x.  X
)
86, 7eqtri 2455 1  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   ` cfv 5446  (class class class)co 6073  tpos ctpos 6470   Basecbs 13461   .rcmulr 13522  opprcoppr 15719
This theorem is referenced by:  crngoppr  15724  opprrng  15728  opprrngb  15729  oppr1  15731  mulgass3  15734  opprunit  15758  unitmulcl  15761  unitgrp  15764  unitpropd  15794  opprirred  15799  irredlmul  15805  isdrng2  15837  isdrngrd  15853  subrguss  15875  subrgunit  15878  opprsubrg  15881  srngmul  15938  issrngd  15941  2idlcpbl  16297  opprdomn  16353  psropprmul  16624  rhmopp  24249  ldualsmul  29870  lcdsmul  32337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-recs 6625  df-rdg 6660  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-sets 13467  df-mulr 13535  df-oppr 15720
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