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Theorem opprmulg 13055
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
opprmulg  |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( X  .xb  Y
)  =  ( Y 
.x.  X ) )

Proof of Theorem opprmulg
StepHypRef Expression
1 opprval.1 . . . . 5  |-  B  =  ( Base `  R
)
2 opprval.2 . . . . 5  |-  .x.  =  ( .r `  R )
3 opprval.3 . . . . 5  |-  O  =  (oppr
`  R )
4 opprmulfval.4 . . . . 5  |-  .xb  =  ( .r `  O )
51, 2, 3, 4opprmulfvalg 13054 . . . 4  |-  ( R  e.  V  ->  .xb  = tpos  .x.  )
65oveqd 5885 . . 3  |-  ( R  e.  V  ->  ( X  .xb  Y )  =  ( Xtpos  .x.  Y
) )
763ad2ant1 1018 . 2  |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( X  .xb  Y
)  =  ( Xtpos 
.x.  Y ) )
8 ovtposg 6253 . . 3  |-  ( ( X  e.  W  /\  Y  e.  U )  ->  ( Xtpos  .x.  Y
)  =  ( Y 
.x.  X ) )
983adant1 1015 . 2  |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( Xtpos  .x.  Y
)  =  ( Y 
.x.  X ) )
107, 9eqtrd 2210 1  |-  ( ( R  e.  V  /\  X  e.  W  /\  Y  e.  U )  ->  ( X  .xb  Y
)  =  ( Y 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868  tpos ctpos 6238   Basecbs 12432   .rcmulr 12506  opprcoppr 13051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-tpos 6239  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-sets 12439  df-mulr 12519  df-oppr 13052
This theorem is referenced by:  crngoppr  13056  opprring  13061  opprringbg  13062  oppr1g  13064  mulgass3  13066  opprunitd  13091  unitmulcl  13094  unitgrp  13097
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