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Metamath Proof Explorer

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 ) )