Theorem rrxunitopnfi

Description: The base set of the standard topology on the space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Assertion	rrxunitopnfi	⊢ ( 𝑋 ∈ Fin → ∪ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) = ( ℝ ↑m 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		eqidd	⊢ ( 𝑋 ∈ Fin → ( ℝ ↑m 𝑋 ) = ( ℝ ↑m 𝑋 ) )
2		eqid	⊢ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) = ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) )
3	2	rrxtoponfi	⊢ ( 𝑋 ∈ Fin → ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ∈ ( TopOn ‘ ( ℝ ↑m 𝑋 ) ) )
4		toponuni	⊢ ( ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) ∈ ( TopOn ‘ ( ℝ ↑m 𝑋 ) ) → ( ℝ ↑m 𝑋 ) = ∪ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) )
5	3 4	syl	⊢ ( 𝑋 ∈ Fin → ( ℝ ↑m 𝑋 ) = ∪ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) )
6	1 5	eqtr2d	⊢ ( 𝑋 ∈ Fin → ∪ ( TopOpen ‘ ( ℝ^ ‘ 𝑋 ) ) = ( ℝ ↑m 𝑋 ) )