Theorem xmulpnf2

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

	Ref	Expression
Assertion	xmulpnf2	|- ( ( A e. RR* /\ 0 < A ) -> ( +oo *e A ) = +oo )

Proof

Step	Hyp	Ref	Expression
1		pnfxr	|- +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		xmulpnf1	|- ( ( A e. RR* /\ 0 < A ) -> ( A *e +oo ) = +oo )
6	4 5	eqtrd	|- ( ( A e. RR* /\ 0 < A ) -> ( +oo *e A ) = +oo )