Metamath Proof Explorer

Theorem xdivid

Description: A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016)

Ref Expression
Assertion xdivid 
|- ( ( A e. RR /\ A =/= 0 ) -> ( A /e A ) = 1 )

Proof

Step Hyp Ref Expression
1 rexdiv 
 |-  ( ( A e. RR /\ A e. RR /\ A =/= 0 ) -> ( A /e A ) = ( A / A ) )
2 1 3anidm12 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( A /e A ) = ( A / A ) )
3 recn 
 |-  ( A e. RR -> A e. CC )
4 divid 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( A / A ) = 1 )
5 3 4 sylan 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( A / A ) = 1 )
6 2 5 eqtrd 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( A /e A ) = 1 )