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Metamath Proof Explorer

Theorem 1unit

Description: The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014)

Ref Expression
Hypotheses unit.1 
|- U = ( Unit ` R )
unit.2 
|- .1. = ( 1r ` R )
Assertion 1unit 
|- ( R e. Ring -> .1. e. U )

Proof

Step Hyp Ref Expression
1 unit.1 
 |-  U = ( Unit ` R )
2 unit.2 
 |-  .1. = ( 1r ` R )
3 eqid 
 |-  ( Base ` R ) = ( Base ` R )
4 3 2 ringidcl 
 |-  ( R e. Ring -> .1. e. ( Base ` R ) )
5 eqid 
 |-  ( ||r ` R ) = ( ||r ` R )
6 3 5 dvdsrid 
 |-  ( ( R e. Ring /\ .1. e. ( Base ` R ) ) -> .1. ( ||r ` R ) .1. )
7 4 6 mpdan 
 |-  ( R e. Ring -> .1. ( ||r ` R ) .1. )
8 eqid 
 |-  ( oppR ` R ) = ( oppR ` R )
9 8 opprring 
 |-  ( R e. Ring -> ( oppR ` R ) e. Ring )
10 8 3 opprbas 
 |-  ( Base ` R ) = ( Base ` ( oppR ` R ) )
11 eqid 
 |-  ( ||r ` ( oppR ` R ) ) = ( ||r ` ( oppR ` R ) )
12 10 11 dvdsrid 
 |-  ( ( ( oppR ` R ) e. Ring /\ .1. e. ( Base ` R ) ) -> .1. ( ||r ` ( oppR ` R ) ) .1. )
13 9 4 12 syl2anc 
 |-  ( R e. Ring -> .1. ( ||r ` ( oppR ` R ) ) .1. )
14 1 2 5 8 11 isunit 
 |-  ( .1. e. U <-> ( .1. ( ||r ` R ) .1. /\ .1. ( ||r ` ( oppR ` R ) ) .1. ) )
15 7 13 14 sylanbrc 
 |-  ( R e. Ring -> .1. e. U )