Metamath Proof Explorer


Theorem xmulmnf2

Description: Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmulmnf2
|- ( ( A e. RR* /\ 0 < A ) -> ( -oo *e A ) = -oo )

Proof

Step Hyp Ref Expression
1 mnfxr
 |-  -oo e. RR*
2 xmulcom
 |-  ( ( -oo e. RR* /\ A e. RR* ) -> ( -oo *e A ) = ( A *e -oo ) )
3 1 2 mpan
 |-  ( A e. RR* -> ( -oo *e A ) = ( A *e -oo ) )
4 3 adantr
 |-  ( ( A e. RR* /\ 0 < A ) -> ( -oo *e A ) = ( A *e -oo ) )
5 xmulmnf1
 |-  ( ( A e. RR* /\ 0 < A ) -> ( A *e -oo ) = -oo )
6 4 5 eqtrd
 |-  ( ( A e. RR* /\ 0 < A ) -> ( -oo *e A ) = -oo )