Theorem opprmulg 14319

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 14318	. . . 4 |- ( R e. V -> .xb = tpos .x. )
6	5	oveqd 6076	. . 3 |- ( R e. V -> ( X .xb Y ) = ( Xtpos .x. Y ) )
7	6	3ad2ant1 1045	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Xtpos .x. Y ) )
8		ovtposg 6504	. . 3 |- ( ( X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
9	8	3adant1 1042	. 2 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( Xtpos .x. Y ) = ( Y .x. X ) )
10	7, 9	eqtrd 2267	1 |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Y .x. X ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5358  (class class class)co 6059  tpos ctpos 6489   Basecbs 13301   .rcmulr 13380  opp_rcoppr 14315

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1re 8238  ax-addrcl 8241

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-tpos 6490  df-inn 9259  df-2 9317  df-3 9318  df-ndx 13304  df-slot 13305  df-sets 13308  df-mulr 13393  df-oppr 14316

This theorem is referenced by:  crngoppr  14320  opprrng  14325  opprrngbg  14326  opprring  14327  opprringbg  14328  oppr1g  14331  mulgass3  14334  opprunitd  14360  unitmulcl  14363  unitgrp  14366  unitpropdg  14398  rhmopp  14426  opprsubrngg  14462  subrguss  14487  subrgunit  14490  opprdomnbg  14526  isridlrng  14761  isridl  14783  2idlcpblrng  14802