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Theorem opprmul 15424
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 15423 . . 3  |-  .xb  = tpos  .x.
65oveqi 5887 . 2  |-  ( X 
.xb  Y )  =  ( Xtpos  .x.  Y
)
7 ovtpos 6265 . 2  |-  ( Xtpos 
.x.  Y )  =  ( Y  .x.  X
)
86, 7eqtri 2316 1  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   Basecbs 13164   .rcmulr 13225  opprcoppr 15420
This theorem is referenced by:  crngoppr  15425  opprrng  15429  opprrngb  15430  oppr1  15432  mulgass3  15435  opprunit  15459  unitmulcl  15462  unitgrp  15465  unitpropd  15495  opprirred  15500  irredlmul  15506  isdrng2  15538  isdrngrd  15554  subrguss  15576  subrgunit  15579  opprsubrg  15582  srngmul  15639  issrngd  15642  2idlcpbl  16002  opprdomn  16058  psropprmul  16332  ldualsmul  29947  lcdsmul  32414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-recs 6404  df-rdg 6439  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-sets 13170  df-mulr 13238  df-oppr 15421
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