Theorem tpr2uni

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	tpr2uni	⊢ ∪ ( 𝐽 ×t 𝐽 ) = ( ℝ × ℝ )

Proof

Step	Hyp	Ref	Expression
1		tpr2tp.0	⊢ 𝐽 = ( topGen ‘ ran (,) )
2	1	tpr2tp	⊢ ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) )
3	2	toponunii	⊢ ( ℝ × ℝ ) = ∪ ( 𝐽 ×t 𝐽 )
4	3	eqcomi	⊢ ∪ ( 𝐽 ×t 𝐽 ) = ( ℝ × ℝ )