Theorem reperf

Description: The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016)

	Ref	Expression
Hypothesis	recld2.1	⊢ 𝐽 = ( TopOpen ‘ ℂfld )
Assertion	reperf	⊢ ( 𝐽 ↾t ℝ ) ∈ Perf

Proof

Step	Hyp	Ref	Expression
1		recld2.1	⊢ 𝐽 = ( TopOpen ‘ ℂfld )
2		readdcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
3		ax-resscn	⊢ ℝ ⊆ ℂ
4	1 2 3	reperflem	⊢ ( 𝐽 ↾t ℝ ) ∈ Perf