Metamath Proof Explorer

Theorem divgt0

Description: The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999)

Ref Expression
Assertion divgt0 
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) )

Proof

Step Hyp Ref Expression
1 gt0div 
 |-  ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 < A <-> 0 < ( A / B ) ) )
2 1 biimpd 
 |-  ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 < A -> 0 < ( A / B ) ) )
3 2 3exp 
 |-  ( A e. RR -> ( B e. RR -> ( 0 < B -> ( 0 < A -> 0 < ( A / B ) ) ) ) )
4 3 com34 
 |-  ( A e. RR -> ( B e. RR -> ( 0 < A -> ( 0 < B -> 0 < ( A / B ) ) ) ) )
5 4 com23 
 |-  ( A e. RR -> ( 0 < A -> ( B e. RR -> ( 0 < B -> 0 < ( A / B ) ) ) ) )
6 5 imp43 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) )