Metamath Proof Explorer


Theorem unitgrpid

Description: The identity of the multiplicative group is 1r . (Contributed by Mario Carneiro, 2-Dec-2014)

Ref Expression
Hypotheses unitmulcl.1
|- U = ( Unit ` R )
unitgrp.2
|- G = ( ( mulGrp ` R ) |`s U )
unitgrp.3
|- .1. = ( 1r ` R )
Assertion unitgrpid
|- ( R e. Ring -> .1. = ( 0g ` G ) )

Proof

Step Hyp Ref Expression
1 unitmulcl.1
 |-  U = ( Unit ` R )
2 unitgrp.2
 |-  G = ( ( mulGrp ` R ) |`s U )
3 unitgrp.3
 |-  .1. = ( 1r ` R )
4 1 3 1unit
 |-  ( R e. Ring -> .1. e. U )
5 eqid
 |-  ( Base ` R ) = ( Base ` R )
6 5 1 unitss
 |-  U C_ ( Base ` R )
7 2 5 3 ringidss
 |-  ( ( R e. Ring /\ U C_ ( Base ` R ) /\ .1. e. U ) -> .1. = ( 0g ` G ) )
8 6 7 mp3an2
 |-  ( ( R e. Ring /\ .1. e. U ) -> .1. = ( 0g ` G ) )
9 4 8 mpdan
 |-  ( R e. Ring -> .1. = ( 0g ` G ) )