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 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) )

Proof

Step	Hyp	Ref	Expression
1		tpr2tp.0	⊢ 𝐽 = ( topGen ‘ ran (,) )
2		retopon	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
3	1 2	eqeltri	⊢ 𝐽 ∈ ( TopOn ‘ ℝ )
4		txtopon	⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝐽 ∈ ( TopOn ‘ ℝ ) ) → ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) )
5	3 3 4	mp2an	⊢ ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) )