Theorem opprmulg 14485

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 14484	. . . 4 |- ( R e. V -> .xb = tpos .x. )
6	5	oveqd 6105	. . 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 6534	. . 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 5380  (class class class)co 6088  tpos ctpos 6519   Basecbs 13410   .rcmulr 13491  opp_rcoppr 14481

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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-tpos 6520  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-sets 13417  df-mulr 13504  df-oppr 14482

This theorem is used by:  crngoppr  14486  opprrng  14491  opprrngbg  14492  opprring  14493  opprringbg  14494  oppr1g  14497  mulgass3  14500  opprunitd  14526  unitmulcl  14529  unitgrp  14532  unitpropdg  14564  rhmopp  14592  opprsubrngg  14628  subrguss  14653  subrgunit  14656  opprdomnbg  14692  isridlrng  14928  isridl  14950  2idlcpblrng  14969