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Metamath Proof Explorer

Theorem 1rinv

Description: The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015)

Ref Expression
Hypotheses 1rinv.1 
|- I = ( invr ` R )
1rinv.2 
|- .1. = ( 1r ` R )
Assertion 1rinv 
|- ( R e. Ring -> ( I ` .1. ) = .1. )

Proof

Step Hyp Ref Expression
1 1rinv.1 
 |-  I = ( invr ` R )
2 1rinv.2 
 |-  .1. = ( 1r ` R )
3 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
4 3 2 1unit 
 |-  ( R e. Ring -> .1. e. ( Unit ` R ) )
5 eqid 
 |-  ( Base ` R ) = ( Base ` R )
6 3 1 5 ringinvcl 
 |-  ( ( R e. Ring /\ .1. e. ( Unit ` R ) ) -> ( I ` .1. ) e. ( Base ` R ) )
7 4 6 mpdan 
 |-  ( R e. Ring -> ( I ` .1. ) e. ( Base ` R ) )
8 eqid 
 |-  ( .r ` R ) = ( .r ` R )
9 5 8 2 ringlidm 
 |-  ( ( R e. Ring /\ ( I ` .1. ) e. ( Base ` R ) ) -> ( .1. ( .r ` R ) ( I ` .1. ) ) = ( I ` .1. ) )
10 7 9 mpdan 
 |-  ( R e. Ring -> ( .1. ( .r ` R ) ( I ` .1. ) ) = ( I ` .1. ) )
11 3 1 8 2 unitrinv 
 |-  ( ( R e. Ring /\ .1. e. ( Unit ` R ) ) -> ( .1. ( .r ` R ) ( I ` .1. ) ) = .1. )
12 4 11 mpdan 
 |-  ( R e. Ring -> ( .1. ( .r ` R ) ( I ` .1. ) ) = .1. )
13 10 12 eqtr3d 
 |-  ( R e. Ring -> ( I ` .1. ) = .1. )