Step |
Hyp |
Ref |
Expression |
1 |
|
eliccelioc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
eliccelioc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
eliccelioc.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ* ) |
4 |
|
iocssicc |
⊢ ( 𝐴 (,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
5 |
4
|
sseli |
⊢ ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
7 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
8 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
9 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ* ) |
10 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐵 ∈ ℝ ) |
11 |
10
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐵 ∈ ℝ* ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) |
13 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < 𝐶 ) |
14 |
9 11 12 13
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < 𝐶 ) |
15 |
7 14
|
gtned |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐶 ≠ 𝐴 ) |
16 |
6 15
|
jca |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) |
17 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐴 ∈ ℝ* ) |
18 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐵 ∈ ℝ* ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐶 ∈ ℝ* ) |
21 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐴 ∈ ℝ ) |
22 |
1 2
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
23 |
22
|
sselda |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ℝ ) |
24 |
23
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐶 ∈ ℝ ) |
25 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ* ) |
26 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ* ) |
27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
28 |
|
iccgelb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 ) |
29 |
25 26 27 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 ) |
30 |
29
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐴 ≤ 𝐶 ) |
31 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐶 ≠ 𝐴 ) |
32 |
21 24 30 31
|
leneltd |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐴 < 𝐶 ) |
33 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ≤ 𝐵 ) |
34 |
25 26 27 33
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ≤ 𝐵 ) |
35 |
34
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐶 ≤ 𝐵 ) |
36 |
17 19 20 32 35
|
eliocd |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) → 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) |
37 |
16 36
|
impbida |
⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐶 ≠ 𝐴 ) ) ) |