Step |
Hyp |
Ref |
Expression |
1 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐴 ∈ ℝ* ) |
2 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐵 ∈ ℝ* ) |
3 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
4 |
|
elicc1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
5 |
4
|
biimpa |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) |
6 |
5
|
simp1d |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ℝ* ) |
7 |
1 2 3 6
|
syl21anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐶 ∈ ℝ* ) |
8 |
5
|
simp3d |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ≤ 𝐵 ) |
9 |
1 2 3 8
|
syl21anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐶 ≤ 𝐵 ) |
10 |
1 2
|
jca |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) |
11 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) |
12 |
5
|
simp2d |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 ) |
13 |
10 3 12
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐴 ≤ 𝐶 ) |
14 |
|
elico1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ) ) |
15 |
14
|
notbid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ↔ ¬ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ) ) |
16 |
15
|
biimpa |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → ¬ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ) |
17 |
|
df-3an |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) ∧ 𝐶 < 𝐵 ) ) |
18 |
17
|
notbii |
⊢ ( ¬ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ¬ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) ∧ 𝐶 < 𝐵 ) ) |
19 |
|
imnan |
⊢ ( ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ¬ 𝐶 < 𝐵 ) ↔ ¬ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) ∧ 𝐶 < 𝐵 ) ) |
20 |
18 19
|
bitr4i |
⊢ ( ¬ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ¬ 𝐶 < 𝐵 ) ) |
21 |
16 20
|
sylib |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ¬ 𝐶 < 𝐵 ) ) |
22 |
21
|
imp |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) ) → ¬ 𝐶 < 𝐵 ) |
23 |
10 11 7 13 22
|
syl22anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → ¬ 𝐶 < 𝐵 ) |
24 |
|
xeqlelt |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 = 𝐵 ↔ ( 𝐶 ≤ 𝐵 ∧ ¬ 𝐶 < 𝐵 ) ) ) |
25 |
24
|
biimpar |
⊢ ( ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐶 ≤ 𝐵 ∧ ¬ 𝐶 < 𝐵 ) ) → 𝐶 = 𝐵 ) |
26 |
7 2 9 23 25
|
syl22anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) ) → 𝐶 = 𝐵 ) |
27 |
26
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐶 = 𝐵 ) ) |
28 |
|
pm5.6 |
⊢ ( ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ ¬ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐶 = 𝐵 ) ↔ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
29 |
27 28
|
sylib |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
30 |
|
icossicc |
⊢ ( 𝐴 [,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
31 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) |
32 |
30 31
|
sselid |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
33 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 = 𝐵 ) |
34 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐵 ∈ ℝ* ) |
35 |
33 34
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 ∈ ℝ* ) |
36 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐴 ≤ 𝐵 ) |
37 |
36 33
|
breqtrrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐴 ≤ 𝐶 ) |
38 |
34
|
xrleidd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐵 ≤ 𝐵 ) |
39 |
33 38
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 ≤ 𝐵 ) |
40 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐴 ∈ ℝ* ) |
41 |
40 34 4
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
42 |
35 37 39 41
|
mpbir3and |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
43 |
32 42
|
jaodan |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
44 |
43
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∨ 𝐶 = 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
45 |
29 44
|
impbid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |