Theorem opprmulg 14460

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

Step	Hyp	Ref	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 )
5	1, 2, 3, 4	opprmulfvalg 14459	. . . 4 |- ( R e. V -> .xb = tpos .x. )
6	5	oveqd 6102	. . 3 |- ( R e. V -> ( X .xb Y ) = ( Xtpos .x. Y ) )
7	6	3ad2ant1 1049	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Xtpos .x. Y ) )
8		ovtposg 6530	. . 3 |- ( ( X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
9	8	3adant1 1046	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
10	7, 9	eqtrd 2271	1 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Y .x. X ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085  tpos ctpos 6515   Basecbs 13404   .rcmulr 13485  opp_rcoppr 14456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-inn 9308  df-2 9366  df-3 9367  df-ndx 13407  df-slot 13408  df-sets 13411  df-mulr 13498  df-oppr 14457

This theorem is used by:  crngoppr  14461  opprrng  14466  opprrngbg  14467  opprring  14468  opprringbg  14469  oppr1g  14472  mulgass3  14475  opprunitd  14501  unitmulcl  14504  unitgrp  14507  unitpropdg  14539  rhmopp  14567  opprsubrngg  14603  subrguss  14628  subrgunit  14631  opprdomnbg  14667  isridlrng  14903  isridl  14925  2idlcpblrng  14944