Description: An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022)
Ref | Expression | ||
---|---|---|---|
Hypotheses | xrnpnfmnf.1 | |- ( ph -> A e. RR* ) |
|
xrnpnfmnf.2 | |- ( ph -> -. A e. RR ) |
||
xrnpnfmnf.3 | |- ( ph -> A =/= +oo ) |
||
Assertion | xrnpnfmnf | |- ( ph -> A = -oo ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnpnfmnf.1 | |- ( ph -> A e. RR* ) |
|
2 | xrnpnfmnf.2 | |- ( ph -> -. A e. RR ) |
|
3 | xrnpnfmnf.3 | |- ( ph -> A =/= +oo ) |
|
4 | 1 3 | jca | |- ( ph -> ( A e. RR* /\ A =/= +oo ) ) |
5 | xrnepnf | |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) ) |
|
6 | 4 5 | sylib | |- ( ph -> ( A e. RR \/ A = -oo ) ) |
7 | pm2.53 | |- ( ( A e. RR \/ A = -oo ) -> ( -. A e. RR -> A = -oo ) ) |
|
8 | 6 2 7 | sylc | |- ( ph -> A = -oo ) |