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Theorem opprmul 17147
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 17146 . . 3  |-  .xb  = tpos  .x.
65oveqi 6308 . 2  |-  ( X 
.xb  Y )  =  ( Xtpos  .x.  Y
)
7 ovtpos 6982 . 2  |-  ( Xtpos 
.x.  Y )  =  ( Y  .x.  X
)
86, 7eqtri 2496 1  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   ` cfv 5594  (class class class)co 6295  tpos ctpos 6966   Basecbs 14507   .rcmulr 14573  opprcoppr 17143
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
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-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-tpos 6967  df-recs 7054  df-rdg 7088  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-sets 14513  df-mulr 14586  df-oppr 17144
This theorem is referenced by:  crngoppr  17148  opprring  17152  opprringb  17153  oppr1  17155  mulgass3  17158  opprunit  17182  unitmulcl  17185  unitgrp  17188  unitpropd  17218  opprirred  17223  irredlmul  17229  isdrng2  17277  isdrngrd  17293  subrguss  17315  subrgunit  17318  opprsubrg  17321  srngmul  17378  issrngd  17381  2idlcpbl  17752  opprdomn  17820  psropprmul  18149  invrvald  19047  rhmopp  27634  ldualsmul  34333  lcdsmul  36800
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