Metamath Proof Explorer


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 )