Step |
Hyp |
Ref |
Expression |
1 |
|
rexr |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ* ) |
3 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) |
4 |
|
df-ico |
⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } ) |
5 |
|
xrletr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 ≤ 𝑤 ) ) |
6 |
4 4 5
|
ixxss1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 [,) +∞ ) ⊆ ( 𝐴 [,) +∞ ) ) |
7 |
2 3 6
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 [,) +∞ ) ⊆ ( 𝐴 [,) +∞ ) ) |
8 |
|
imass2 |
⊢ ( ( 𝐵 [,) +∞ ) ⊆ ( 𝐴 [,) +∞ ) → ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ⊆ ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ) |
9 |
|
ssrin |
⊢ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ⊆ ( 𝐹 “ ( 𝐴 [,) +∞ ) ) → ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) ) |
10 |
7 8 9
|
3syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) ) |
11 |
|
inss2 |
⊢ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* |
12 |
|
supxrcl |
⊢ ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
13 |
11 12
|
ax-mp |
⊢ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
14 |
|
xrleid |
⊢ ( sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* → sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
15 |
13 14
|
ax-mp |
⊢ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) |
16 |
|
supxrleub |
⊢ ( ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ∧ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) → ( sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ↔ ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
17 |
11 13 16
|
mp2an |
⊢ ( sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ↔ ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
18 |
15 17
|
mpbi |
⊢ ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) |
19 |
|
ssralv |
⊢ ( ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) → ( ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) → ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
20 |
10 18 19
|
mpisyl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
21 |
|
inss2 |
⊢ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* |
22 |
|
supxrleub |
⊢ ( ( ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ∧ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) → ( sup ( ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ↔ ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
23 |
21 13 22
|
mp2an |
⊢ ( sup ( ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ↔ ∀ 𝑥 ∈ ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) 𝑥 ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
24 |
20 23
|
sylibr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → sup ( ( ( 𝐹 “ ( 𝐵 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝐴 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |