Metamath Proof Explorer


Theorem opprmulfval

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

Ref Expression
Hypotheses opprval.1
|- B = ( Base ` R )
opprval.2
|- .x. = ( .r ` R )
opprval.3
|- O = ( oppR ` R )
opprmulfval.4
|- .xb = ( .r ` O )
Assertion opprmulfval
|- .xb = tpos .x.

Proof

Step Hyp Ref Expression
1 opprval.1
 |-  B = ( Base ` R )
2 opprval.2
 |-  .x. = ( .r ` R )
3 opprval.3
 |-  O = ( oppR ` R )
4 opprmulfval.4
 |-  .xb = ( .r ` O )
5 1 2 3 opprval
 |-  O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. )
6 5 fveq2i
 |-  ( .r ` O ) = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
7 2 fvexi
 |-  .x. e. _V
8 7 tposex
 |-  tpos .x. e. _V
9 mulrid
 |-  .r = Slot ( .r ` ndx )
10 9 setsid
 |-  ( ( R e. _V /\ tpos .x. e. _V ) -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
11 8 10 mpan2
 |-  ( R e. _V -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
12 6 11 eqtr4id
 |-  ( R e. _V -> ( .r ` O ) = tpos .x. )
13 tpos0
 |-  tpos (/) = (/)
14 9 str0
 |-  (/) = ( .r ` (/) )
15 13 14 eqtr2i
 |-  ( .r ` (/) ) = tpos (/)
16 fvprc
 |-  ( -. R e. _V -> ( oppR ` R ) = (/) )
17 3 16 eqtrid
 |-  ( -. R e. _V -> O = (/) )
18 17 fveq2d
 |-  ( -. R e. _V -> ( .r ` O ) = ( .r ` (/) ) )
19 fvprc
 |-  ( -. R e. _V -> ( .r ` R ) = (/) )
20 2 19 eqtrid
 |-  ( -. R e. _V -> .x. = (/) )
21 20 tposeqd
 |-  ( -. R e. _V -> tpos .x. = tpos (/) )
22 15 18 21 3eqtr4a
 |-  ( -. R e. _V -> ( .r ` O ) = tpos .x. )
23 12 22 pm2.61i
 |-  ( .r ` O ) = tpos .x.
24 4 23 eqtri
 |-  .xb = tpos .x.