Description: Prove that the closure of the decimal point is RR as we have defined it. See df-dp . (Contributed by David A. Wheeler, 15-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | dpcl | |- ( ( A e. NN0 /\ B e. RR ) -> ( A . B ) e. RR ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpval | |- ( ( A e. NN0 /\ B e. RR ) -> ( A . B ) = _ A B ) |
|
2 | nn0re | |- ( A e. NN0 -> A e. RR ) |
|
3 | dp2cl | |- ( ( A e. RR /\ B e. RR ) -> _ A B e. RR ) |
|
4 | 2 3 | sylan | |- ( ( A e. NN0 /\ B e. RR ) -> _ A B e. RR ) |
5 | 1 4 | eqeltrd | |- ( ( A e. NN0 /\ B e. RR ) -> ( A . B ) e. RR ) |