Theorem fourierdlem17

Description: The defined L is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem17.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	fourierdlem17.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	fourierdlem17.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
	fourierdlem17.l	⊢ 𝐿 = ( 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) )
Assertion	fourierdlem17	⊢ ( 𝜑 → 𝐿 : ( 𝐴 (,] 𝐵 ) ⟶ ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem17.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		fourierdlem17.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		fourierdlem17.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
4		fourierdlem17.l	⊢ 𝐿 = ( 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) )
5	1	leidd	⊢ ( 𝜑 → 𝐴 ≤ 𝐴 )
6	1 2 3	ltled	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
7	1 2 1 5 6	eliccd	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
8	7	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑥 = 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
9		iocssicc	⊢ ( 𝐴 (,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )
10	9	sseli	⊢ ( 𝑥 ∈ ( 𝐴 (,] 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
11	10	ad2antlr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
12	8 11	ifclda	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ) → if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) )
13	12 4	fmptd	⊢ ( 𝜑 → 𝐿 : ( 𝐴 (,] 𝐵 ) ⟶ ( 𝐴 [,] 𝐵 ) )