Metamath Proof Explorer

Theorem xdiv0

Description: Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016)

Ref Expression
Assertion xdiv0 
|- ( ( A e. RR /\ A =/= 0 ) -> ( 0 /e A ) = 0 )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 rexdiv 
 |-  ( ( 0 e. RR /\ A e. RR /\ A =/= 0 ) -> ( 0 /e A ) = ( 0 / A ) )
3 1 2 mp3an1 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( 0 /e A ) = ( 0 / A ) )
4 recn 
 |-  ( A e. RR -> A e. CC )
5 div0 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( 0 / A ) = 0 )
6 4 5 sylan 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( 0 / A ) = 0 )
7 3 6 eqtrd 
 |-  ( ( A e. RR /\ A =/= 0 ) -> ( 0 /e A ) = 0 )