Theorem opprmulg 13948

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 13947	. . . 4 |- ( R e. V -> .xb = tpos .x. )
6	5	oveqd 5984	. . 3 |- ( R e. V -> ( X .xb Y ) = ( Xtpos .x. Y ) )
7	6	3ad2ant1 1021	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Xtpos .x. Y ) )
8		ovtposg 6368	. . 3 |- ( ( X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
9	8	3adant1 1018	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
10	7, 9	eqtrd 2240	1 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Y .x. X ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967  tpos ctpos 6353   Basecbs 12947   .rcmulr 13025  opp_rcoppr 13944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-sets 12954  df-mulr 13038  df-oppr 13945

This theorem is referenced by:  crngoppr  13949  opprrng  13954  opprrngbg  13955  opprring  13956  opprringbg  13957  oppr1g  13959  mulgass3  13962  opprunitd  13987  unitmulcl  13990  unitgrp  13993  unitpropdg  14025  rhmopp  14053  opprsubrngg  14088  subrguss  14113  subrgunit  14116  opprdomnbg  14151  isridlrng  14359  isridl  14381  2idlcpblrng  14400