Theorem opprmulfvalg 13626

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 13625	. . . 4 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
6	5	fveq2d 5562	. . 3 |- ( R e. V -> ( .r ` O ) = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
7		mulrslid 12809	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
8	7	slotex 12705	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
9	3, 8	eqeltrid 2283	. . . . 5 |- ( R e. V -> .x. e. _V )
10		tposexg 6316	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
11	9, 10	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
12	7	setsslid 12729	. . . 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 2232	. 2 |- ( R e. V -> ( .r ` O ) = tpos .x. )
15	1, 14	eqtrid 2241	1 |- ( R e. V -> .xb = tpos .x. )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3625   ` cfv 5258  (class class class)co 5922  tpos ctpos 6302   ndxcnx 12675   sSet csts 12676   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:  opprmulg  13627