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Theorem opprmul 16723
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 16722 . . 3  |-  .xb  = tpos  .x.
65oveqi 6109 . 2  |-  ( X 
.xb  Y )  =  ( Xtpos  .x.  Y
)
7 ovtpos 6765 . 2  |-  ( Xtpos 
.x.  Y )  =  ( Y  .x.  X
)
86, 7eqtri 2463 1  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   ` cfv 5423  (class class class)co 6096  tpos ctpos 6749   Basecbs 14179   .rcmulr 14244  opprcoppr 16719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-tpos 6750  df-recs 6837  df-rdg 6871  df-nn 10328  df-2 10385  df-3 10386  df-ndx 14182  df-slot 14183  df-sets 14185  df-mulr 14257  df-oppr 16720
This theorem is referenced by:  crngoppr  16724  opprrng  16728  opprrngb  16729  oppr1  16731  mulgass3  16734  opprunit  16758  unitmulcl  16761  unitgrp  16764  unitpropd  16794  opprirred  16799  irredlmul  16805  isdrng2  16847  isdrngrd  16863  subrguss  16885  subrgunit  16888  opprsubrg  16891  srngmul  16948  issrngd  16951  2idlcpbl  17321  opprdomn  17378  psropprmul  17698  invrvald  18487  rhmopp  26292  ldualsmul  32785  lcdsmul  35252
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