Step |
Hyp |
Ref |
Expression |
1 |
|
ltord.1 |
⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 ) |
2 |
|
ltord.2 |
⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝑀 ) |
3 |
|
ltord.3 |
⊢ ( 𝑥 = 𝐷 → 𝐴 = 𝑁 ) |
4 |
|
ltord.4 |
⊢ 𝑆 ⊆ ℝ |
5 |
|
ltord.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ ℝ ) |
6 |
|
ltord.6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) ) |
7 |
1 3 2 4 5 6
|
ltord1 |
⊢ ( ( 𝜑 ∧ ( 𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐷 < 𝐶 ↔ 𝑁 < 𝑀 ) ) |
8 |
7
|
ancom2s |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐷 < 𝐶 ↔ 𝑁 < 𝑀 ) ) |
9 |
8
|
notbid |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( ¬ 𝐷 < 𝐶 ↔ ¬ 𝑁 < 𝑀 ) ) |
10 |
4
|
sseli |
⊢ ( 𝐶 ∈ 𝑆 → 𝐶 ∈ ℝ ) |
11 |
4
|
sseli |
⊢ ( 𝐷 ∈ 𝑆 → 𝐷 ∈ ℝ ) |
12 |
|
lenlt |
⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) → ( 𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶 ) ) |
13 |
10 11 12
|
syl2an |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) → ( 𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶 ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶 ) ) |
15 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 𝐴 ∈ ℝ ) |
16 |
2
|
eleq1d |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ ) ) |
17 |
16
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆 ) → 𝑀 ∈ ℝ ) |
18 |
15 17
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝑆 ) → 𝑀 ∈ ℝ ) |
19 |
18
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → 𝑀 ∈ ℝ ) |
20 |
3
|
eleq1d |
⊢ ( 𝑥 = 𝐷 → ( 𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ ) ) |
21 |
20
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆 ) → 𝑁 ∈ ℝ ) |
22 |
15 21
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ 𝑆 ) → 𝑁 ∈ ℝ ) |
23 |
22
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → 𝑁 ∈ ℝ ) |
24 |
19 23
|
lenltd |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀 ) ) |
25 |
9 14 24
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐶 ≤ 𝐷 ↔ 𝑀 ≤ 𝑁 ) ) |