Metamath Proof Explorer

Theorem oppr1

Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014)

Ref Expression
Hypotheses opprbas.1 
|- O = ( oppR ` R )
oppr1.2 
|- .1. = ( 1r ` R )
Assertion oppr1 
|- .1. = ( 1r ` O )

Proof

Step Hyp Ref Expression
1 opprbas.1 
 |-  O = ( oppR ` R )
2 oppr1.2 
 |-  .1. = ( 1r ` R )
3 eqid 
 |-  ( Base ` R ) = ( Base ` R )
4 eqid 
 |-  ( .r ` R ) = ( .r ` R )
5 eqid 
 |-  ( .r ` O ) = ( .r ` O )
6 3 4 1 5 opprmul 
 |-  ( x ( .r ` O ) y ) = ( y ( .r ` R ) x )
7 6 eqeq1i 
 |-  ( ( x ( .r ` O ) y ) = y <-> ( y ( .r ` R ) x ) = y )
8 3 4 1 5 opprmul 
 |-  ( y ( .r ` O ) x ) = ( x ( .r ` R ) y )
9 8 eqeq1i 
 |-  ( ( y ( .r ` O ) x ) = y <-> ( x ( .r ` R ) y ) = y )
10 7 9 anbi12ci 
 |-  ( ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) )
11 10 ralbii 
 |-  ( A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) )
12 11 anbi2i 
 |-  ( ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) <-> ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
13 12 iotabii 
 |-  ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
14 eqid 
 |-  ( mulGrp ` O ) = ( mulGrp ` O )
15 1 3 opprbas 
 |-  ( Base ` R ) = ( Base ` O )
16 14 15 mgpbas 
 |-  ( Base ` R ) = ( Base ` ( mulGrp ` O ) )
17 14 5 mgpplusg 
 |-  ( .r ` O ) = ( +g ` ( mulGrp ` O ) )
18 eqid 
 |-  ( 0g ` ( mulGrp ` O ) ) = ( 0g ` ( mulGrp ` O ) )
19 16 17 18 grpidval 
 |-  ( 0g ` ( mulGrp ` O ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) )
20 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
21 20 3 mgpbas 
 |-  ( Base ` R ) = ( Base ` ( mulGrp ` R ) )
22 20 4 mgpplusg 
 |-  ( .r ` R ) = ( +g ` ( mulGrp ` R ) )
23 eqid 
 |-  ( 0g ` ( mulGrp ` R ) ) = ( 0g ` ( mulGrp ` R ) )
24 21 22 23 grpidval 
 |-  ( 0g ` ( mulGrp ` R ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
25 13 19 24 3eqtr4i 
 |-  ( 0g ` ( mulGrp ` O ) ) = ( 0g ` ( mulGrp ` R ) )
26 eqid 
 |-  ( 1r ` O ) = ( 1r ` O )
27 14 26 ringidval 
 |-  ( 1r ` O ) = ( 0g ` ( mulGrp ` O ) )
28 20 2 ringidval 
 |-  .1. = ( 0g ` ( mulGrp ` R ) )
29 25 27 28 3eqtr4ri 
 |-  .1. = ( 1r ` O )