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Theorem opprmul 15736
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 15735 . . 3  |-  .xb  = tpos  .x.
65oveqi 6097 . 2  |-  ( X 
.xb  Y )  =  ( Xtpos  .x.  Y
)
7 ovtpos 6497 . 2  |-  ( Xtpos 
.x.  Y )  =  ( Y  .x.  X
)
86, 7eqtri 2458 1  |-  ( X 
.xb  Y )  =  ( Y  .x.  X
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653   ` cfv 5457  (class class class)co 6084  tpos ctpos 6481   Basecbs 13474   .rcmulr 13535  opprcoppr 15732
This theorem is referenced by:  crngoppr  15737  opprrng  15741  opprrngb  15742  oppr1  15744  mulgass3  15747  opprunit  15771  unitmulcl  15774  unitgrp  15777  unitpropd  15807  opprirred  15812  irredlmul  15818  isdrng2  15850  isdrngrd  15866  subrguss  15888  subrgunit  15891  opprsubrg  15894  srngmul  15951  issrngd  15954  2idlcpbl  16310  opprdomn  16366  psropprmul  16637  rhmopp  24262  ldualsmul  30007  lcdsmul  32474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rrecex 9067  ax-cnre 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tpos 6482  df-recs 6636  df-rdg 6671  df-nn 10006  df-2 10063  df-3 10064  df-ndx 13477  df-slot 13478  df-sets 13480  df-mulr 13548  df-oppr 15733
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