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 ) ) |