Theorem opprmulg 13627

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 13626	. . . 4 |- ( R e. V -> .xb = tpos .x. )
6	5	oveqd 5939	. . 3 |- ( R e. V -> ( X .xb Y ) = ( Xtpos .x. Y ) )
7	6	3ad2ant1 1020	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Xtpos .x. Y ) )
8		ovtposg 6317	. . 3 |- ( ( X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
9	8	3adant1 1017	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
10	7, 9	eqtrd 2229	1 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Y .x. X ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922  tpos ctpos 6302   Basecbs 12678   .rcmulr 12756  opp_rcoppr 13623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-sets 12685  df-mulr 12769  df-oppr 13624

This theorem is referenced by:  crngoppr  13628  opprrng  13633  opprrngbg  13634  opprring  13635  opprringbg  13636  oppr1g  13638  mulgass3  13641  opprunitd  13666  unitmulcl  13669  unitgrp  13672  unitpropdg  13704  rhmopp  13732  opprsubrngg  13767  subrguss  13792  subrgunit  13795  opprdomnbg  13830  isridlrng  14038  isridl  14060  2idlcpblrng  14079