Step |
Hyp |
Ref |
Expression |
1 |
|
exmid |
⊢ ( 𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵 ) |
2 |
|
xrltle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) ) |
3 |
2
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐵 ) |
4 |
3
|
3adantl3 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐵 ) |
5 |
|
iocinioc2 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) |
6 |
4 5
|
syldan |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) |
7 |
6
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐵 → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ) |
8 |
7
|
ancld |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐵 → ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ) ) |
9 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* ) |
10 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* ) |
11 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → ¬ 𝐴 < 𝐵 ) |
12 |
|
xrlenlt |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) ) |
13 |
12
|
biimpar |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐴 ) |
14 |
9 10 11 13
|
syl21anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐴 ) |
15 |
|
3ancoma |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ↔ ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ) |
16 |
|
incom |
⊢ ( ( 𝐵 (,] 𝐶 ) ∩ ( 𝐴 (,] 𝐶 ) ) = ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) |
17 |
|
iocinioc2 |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐵 (,] 𝐶 ) ∩ ( 𝐴 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) |
18 |
16 17
|
eqtr3id |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) |
19 |
15 18
|
sylanbr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) |
20 |
14 19
|
syldan |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ¬ 𝐴 < 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) |
21 |
20
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ¬ 𝐴 < 𝐵 → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) |
22 |
21
|
ancld |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ¬ 𝐴 < 𝐵 → ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) ) |
23 |
8 22
|
orim12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵 ) → ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) ) ) |
24 |
1 23
|
mpi |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) ) |
25 |
|
eqif |
⊢ ( ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 (,] 𝐶 ) , ( 𝐴 (,] 𝐶 ) ) ↔ ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) ) |
26 |
24 25
|
sylibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 (,] 𝐶 ) , ( 𝐴 (,] 𝐶 ) ) ) |