Theorem tpr2tp

Description: The usual topology on ( RR X. RR ) is the product topology of the usual topology on RR . (Contributed by Thierry Arnoux, 21-Sep-2017)

	Ref	Expression
Hypothesis	tpr2tp.0	|- J = ( topGen ` ran (,) )
Assertion	tpr2tp	|- ( J tX J ) e. ( TopOn ` ( RR X. RR ) )

Proof

Step	Hyp	Ref	Expression
1		tpr2tp.0	|- J = ( topGen ` ran (,) )
2		retopon	|- ( topGen ` ran (,) ) e. ( TopOn ` RR )
3	1 2	eqeltri	|- J e. ( TopOn ` RR )
4		txtopon	|- ( ( J e. ( TopOn ` RR ) /\ J e. ( TopOn ` RR ) ) -> ( J tX J ) e. ( TopOn ` ( RR X. RR ) ) )
5	3 3 4	mp2an	|- ( J tX J ) e. ( TopOn ` ( RR X. RR ) )