Step |
Hyp |
Ref |
Expression |
1 |
|
sqrlem1.1 |
⊢ 𝑆 = { 𝑥 ∈ ℝ+ ∣ ( 𝑥 ↑ 2 ) ≤ 𝐴 } |
2 |
|
sqrlem1.2 |
⊢ 𝐵 = sup ( 𝑆 , ℝ , < ) |
3 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) |
4 |
3
|
breq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ↑ 2 ) ≤ 𝐴 ↔ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) |
5 |
4 1
|
elrab2 |
⊢ ( 𝑦 ∈ 𝑆 ↔ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) |
6 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( 𝑦 ↑ 2 ) ≤ 𝐴 ) |
7 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → 𝐴 ≤ 1 ) |
8 |
|
rpre |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ ) |
9 |
8
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → 𝑦 ∈ ℝ ) |
10 |
9
|
resqcld |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( 𝑦 ↑ 2 ) ∈ ℝ ) |
11 |
|
rpre |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → 𝐴 ∈ ℝ ) |
13 |
|
1re |
⊢ 1 ∈ ℝ |
14 |
|
letr |
⊢ ( ( ( 𝑦 ↑ 2 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( ( 𝑦 ↑ 2 ) ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝑦 ↑ 2 ) ≤ 1 ) ) |
15 |
13 14
|
mp3an3 |
⊢ ( ( ( 𝑦 ↑ 2 ) ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( ( 𝑦 ↑ 2 ) ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝑦 ↑ 2 ) ≤ 1 ) ) |
16 |
10 12 15
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( ( ( 𝑦 ↑ 2 ) ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝑦 ↑ 2 ) ≤ 1 ) ) |
17 |
6 7 16
|
mp2and |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( 𝑦 ↑ 2 ) ≤ 1 ) |
18 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
19 |
17 18
|
breqtrrdi |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( 𝑦 ↑ 2 ) ≤ ( 1 ↑ 2 ) ) |
20 |
|
rpge0 |
⊢ ( 𝑦 ∈ ℝ+ → 0 ≤ 𝑦 ) |
21 |
20
|
ad2antrl |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → 0 ≤ 𝑦 ) |
22 |
|
0le1 |
⊢ 0 ≤ 1 |
23 |
|
le2sq |
⊢ ( ( ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) ∧ ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ) → ( 𝑦 ≤ 1 ↔ ( 𝑦 ↑ 2 ) ≤ ( 1 ↑ 2 ) ) ) |
24 |
13 22 23
|
mpanr12 |
⊢ ( ( 𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ) → ( 𝑦 ≤ 1 ↔ ( 𝑦 ↑ 2 ) ≤ ( 1 ↑ 2 ) ) ) |
25 |
9 21 24
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → ( 𝑦 ≤ 1 ↔ ( 𝑦 ↑ 2 ) ≤ ( 1 ↑ 2 ) ) ) |
26 |
19 25
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) ∧ ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) ) → 𝑦 ≤ 1 ) |
27 |
26
|
ex |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( ( 𝑦 ∈ ℝ+ ∧ ( 𝑦 ↑ 2 ) ≤ 𝐴 ) → 𝑦 ≤ 1 ) ) |
28 |
5 27
|
syl5bi |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( 𝑦 ∈ 𝑆 → 𝑦 ≤ 1 ) ) |
29 |
28
|
ralrimiv |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 1 ) |