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 ) |
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 ) |