This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dvrid

Description: A ring element divided by itself is the ring unity. ( divid analog.) (Contributed by Mario Carneiro, 18-Jun-2015)

Ref Expression
Hypotheses unitdvcl.o 
|- U = ( Unit ` R )
unitdvcl.d 
|- ./ = ( /r ` R )
dvrid.o 
|- .1. = ( 1r ` R )
Assertion dvrid 
|- ( ( R e. Ring /\ X e. U ) -> ( X ./ X ) = .1. )

Proof

Step Hyp Ref Expression
1 unitdvcl.o 
 |-  U = ( Unit ` R )
2 unitdvcl.d 
 |-  ./ = ( /r ` R )
3 dvrid.o 
 |-  .1. = ( 1r ` R )
4 eqid 
 |-  ( Base ` R ) = ( Base ` R )
5 4 1 unitcl 
 |-  ( X e. U -> X e. ( Base ` R ) )
6 5 adantl 
 |-  ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
7 eqid 
 |-  ( .r ` R ) = ( .r ` R )
8 eqid 
 |-  ( invr ` R ) = ( invr ` R )
9 4 7 1 8 2 dvrval 
 |-  ( ( X e. ( Base ` R ) /\ X e. U ) -> ( X ./ X ) = ( X ( .r ` R ) ( ( invr ` R ) ` X ) ) )
10 6 9 sylancom 
 |-  ( ( R e. Ring /\ X e. U ) -> ( X ./ X ) = ( X ( .r ` R ) ( ( invr ` R ) ` X ) ) )
11 1 8 7 3 unitrinv 
 |-  ( ( R e. Ring /\ X e. U ) -> ( X ( .r ` R ) ( ( invr ` R ) ` X ) ) = .1. )
12 10 11 eqtrd 
 |-  ( ( R e. Ring /\ X e. U ) -> ( X ./ X ) = .1. )