Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
2 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
3 |
|
icossre |
⊢ ( ( 𝐵 ∈ ℝ ∧ +∞ ∈ ℝ* ) → ( 𝐵 [,) +∞ ) ⊆ ℝ ) |
4 |
1 2 3
|
sylancl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 [,) +∞ ) ⊆ ℝ ) |
5 |
|
ssrexv |
⊢ ( ( 𝐵 [,) +∞ ) ⊆ ℝ → ( ∃ 𝑗 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |
6 |
4 5
|
syl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → ( ∃ 𝑗 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |
7 |
|
simpr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → 𝑗 ∈ ℝ ) |
8 |
|
simplr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
9 |
7 8
|
ifcld |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ℝ ) |
10 |
|
max1 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ ) → 𝐵 ≤ if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ) |
11 |
10
|
adantll |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → 𝐵 ≤ if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ) |
12 |
|
elicopnf |
⊢ ( 𝐵 ∈ ℝ → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ( 𝐵 [,) +∞ ) ↔ ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ℝ ∧ 𝐵 ≤ if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ) ) ) |
13 |
12
|
ad2antlr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ( 𝐵 [,) +∞ ) ↔ ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ℝ ∧ 𝐵 ≤ if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ) ) ) |
14 |
9 11 13
|
mpbir2and |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ( 𝐵 [,) +∞ ) ) |
15 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
16 |
|
simplr |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑗 ∈ ℝ ) |
17 |
|
simpll |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → 𝐴 ⊆ ℝ ) |
18 |
17
|
sselda |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ ℝ ) |
19 |
|
maxle |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 ↔ ( 𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘 ) ) ) |
20 |
15 16 18 19
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 ↔ ( 𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘 ) ) ) |
21 |
|
simpr |
⊢ ( ( 𝐵 ≤ 𝑘 ∧ 𝑗 ≤ 𝑘 ) → 𝑗 ≤ 𝑘 ) |
22 |
20 21
|
syl6bi |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 → 𝑗 ≤ 𝑘 ) ) |
23 |
22
|
imim1d |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑗 ≤ 𝑘 → 𝜑 ) → ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 → 𝜑 ) ) ) |
24 |
23
|
ralimdva |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∀ 𝑘 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 → 𝜑 ) ) ) |
25 |
|
breq1 |
⊢ ( 𝑛 = if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) → ( 𝑛 ≤ 𝑘 ↔ if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 ) ) |
26 |
25
|
rspceaimv |
⊢ ( ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ∈ ( 𝐵 [,) +∞ ) ∧ ∀ 𝑘 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑗 , 𝑗 , 𝐵 ) ≤ 𝑘 → 𝜑 ) ) → ∃ 𝑛 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑛 ≤ 𝑘 → 𝜑 ) ) |
27 |
14 24 26
|
syl6an |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) ∧ 𝑗 ∈ ℝ ) → ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∃ 𝑛 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑛 ≤ 𝑘 → 𝜑 ) ) ) |
28 |
27
|
rexlimdva |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∃ 𝑛 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑛 ≤ 𝑘 → 𝜑 ) ) ) |
29 |
|
breq1 |
⊢ ( 𝑛 = 𝑗 → ( 𝑛 ≤ 𝑘 ↔ 𝑗 ≤ 𝑘 ) ) |
30 |
29
|
imbi1d |
⊢ ( 𝑛 = 𝑗 → ( ( 𝑛 ≤ 𝑘 → 𝜑 ) ↔ ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |
31 |
30
|
ralbidv |
⊢ ( 𝑛 = 𝑗 → ( ∀ 𝑘 ∈ 𝐴 ( 𝑛 ≤ 𝑘 → 𝜑 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |
32 |
31
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑛 ≤ 𝑘 → 𝜑 ) ↔ ∃ 𝑗 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) |
33 |
28 32
|
syl6ib |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) → ∃ 𝑗 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |
34 |
6 33
|
impbid |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ) → ( ∃ 𝑗 ∈ ( 𝐵 [,) +∞ ) ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ↔ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) ) |