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 )