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 ( 𝜑𝐴 ∈ ℝ )
fourierdlem17.b ( 𝜑𝐵 ∈ ℝ )
fourierdlem17.altb ( 𝜑𝐴 < 𝐵 )
fourierdlem17.l 𝐿 = ( 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) )
Assertion fourierdlem17 ( 𝜑𝐿 : ( 𝐴 (,] 𝐵 ) ⟶ ( 𝐴 [,] 𝐵 ) )

Proof

Step Hyp Ref Expression
1 fourierdlem17.a ( 𝜑𝐴 ∈ ℝ )
2 fourierdlem17.b ( 𝜑𝐵 ∈ ℝ )
3 fourierdlem17.altb ( 𝜑𝐴 < 𝐵 )
4 fourierdlem17.l 𝐿 = ( 𝑥 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) )
5 1 leidd ( 𝜑𝐴𝐴 )
6 1 2 3 ltled ( 𝜑𝐴𝐵 )
7 1 2 1 5 6 eliccd ( 𝜑𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
8 7 ad2antrr ( ( ( 𝜑𝑥 ∈ ( 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 ( ( ( 𝜑𝑥 ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
12 8 11 ifclda ( ( 𝜑𝑥 ∈ ( 𝐴 (,] 𝐵 ) ) → if ( 𝑥 = 𝐵 , 𝐴 , 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) )
13 12 4 fmptd ( 𝜑𝐿 : ( 𝐴 (,] 𝐵 ) ⟶ ( 𝐴 [,] 𝐵 ) )