Theorem opprmulfvalg 13173

Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

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
opprmulfvalg	|- ( R e. V -> .xb = tpos .x. )

Proof of Theorem opprmulfvalg

Step	Hyp	Ref	Expression
1		opprmulfval.4	. 2 |- .xb = ( .r ` O )
2		opprval.1	. . . . 5 |- B = ( Base ` R )
3		opprval.2	. . . . 5 |- .x. = ( .r ` R )
4		opprval.3	. . . . 5 |- O = (oppr ` R )
5	2, 3, 4	opprvalg 13172	. . . 4 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
6	5	fveq2d 5518	. . 3 |- ( R e. V -> ( .r ` O ) = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
7		mulrslid 12582	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
8	7	slotex 12481	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
9	3, 8	eqeltrid 2264	. . . . 5 |- ( R e. V -> .x. e. _V )
10		tposexg 6256	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
11	9, 10	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
12	7	setsslid 12505	. . . 4 |- ( ( R e. V /\ tpos .x. e. _V ) -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
13	11, 12	mpdan 421	. . 3 |- ( R e. V -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
14	6, 13	eqtr4d 2213	. 2 |- ( R e. V -> ( .r ` O ) = tpos .x. )
15	1, 14	eqtrid 2222	1 |- ( R e. V -> .xb = tpos .x. )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   <.cop 3595   ` cfv 5215  (class class class)co 5872  tpos ctpos 6242   ndxcnx 12451   sSet csts 12452   Basecbs 12454   .rcmulr 12529  opp_rcoppr 13170

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-sets 12461  df-mulr 12542  df-oppr 13171

This theorem is referenced by:  opprmulg  13174