Theorem smf2id

Description: Twice the identity function is Borel sigma-measurable (just an example, to test previous general theorems). (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	smf2id.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
	smf2id.b	⊢ 𝐵 = ( SalGen ‘ 𝐽 )
	smf2id.a	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
Assertion	smf2id	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 2 · 𝑥 ) ) ∈ ( SMblFn ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		smf2id.j	⊢ 𝐽 = ( topGen ‘ ran (,) )
2		smf2id.b	⊢ 𝐵 = ( SalGen ‘ 𝐽 )
3		smf2id.a	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
4		nfv	⊢ Ⅎ 𝑥 𝜑
5		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
6	1 5	eqeltri	⊢ 𝐽 ∈ Top
7	6	a1i	⊢ ( 𝜑 → 𝐽 ∈ Top )
8	7 2	salgencld	⊢ ( 𝜑 → 𝐵 ∈ SAlg )
9		reex	⊢ ℝ ∈ V
10	9	a1i	⊢ ( 𝜑 → ℝ ∈ V )
11	10 3	ssexd	⊢ ( 𝜑 → 𝐴 ∈ V )
12	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ⊆ ℝ )
13		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
14	12 13	sseldd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ )
15		2re	⊢ 2 ∈ ℝ
16	15	a1i	⊢ ( 𝜑 → 2 ∈ ℝ )
17	1 2 3	smfid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( SMblFn ‘ 𝐵 ) )
18	4 8 11 14 16 17	smfmulc1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 2 · 𝑥 ) ) ∈ ( SMblFn ‘ 𝐵 ) )