Step |
Hyp |
Ref |
Expression |
1 |
|
sqrlem1.1 |
⊢ 𝑆 = { 𝑥 ∈ ℝ+ ∣ ( 𝑥 ↑ 2 ) ≤ 𝐴 } |
2 |
|
sqrlem1.2 |
⊢ 𝐵 = sup ( 𝑆 , ℝ , < ) |
3 |
1 2
|
sqrlem3 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( 𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 𝑦 ) ) |
4 |
|
suprcl |
⊢ ( ( 𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 𝑦 ) → sup ( 𝑆 , ℝ , < ) ∈ ℝ ) |
5 |
3 4
|
syl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → sup ( 𝑆 , ℝ , < ) ∈ ℝ ) |
6 |
2 5
|
eqeltrid |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐵 ∈ ℝ ) |
7 |
|
rpgt0 |
⊢ ( 𝐴 ∈ ℝ+ → 0 < 𝐴 ) |
8 |
7
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 0 < 𝐴 ) |
9 |
1 2
|
sqrlem2 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐴 ∈ 𝑆 ) |
10 |
|
suprub |
⊢ ( ( ( 𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 𝑦 ) ∧ 𝐴 ∈ 𝑆 ) → 𝐴 ≤ sup ( 𝑆 , ℝ , < ) ) |
11 |
3 9 10
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐴 ≤ sup ( 𝑆 , ℝ , < ) ) |
12 |
11 2
|
breqtrrdi |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐴 ≤ 𝐵 ) |
13 |
|
0re |
⊢ 0 ∈ ℝ |
14 |
|
rpre |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ ) |
15 |
|
ltletr |
⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 𝐴 ≤ 𝐵 ) → 0 < 𝐵 ) ) |
16 |
13 14 6 15
|
mp3an2ani |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( ( 0 < 𝐴 ∧ 𝐴 ≤ 𝐵 ) → 0 < 𝐵 ) ) |
17 |
8 12 16
|
mp2and |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 0 < 𝐵 ) |
18 |
6 17
|
elrpd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐵 ∈ ℝ+ ) |
19 |
1 2
|
sqrlem1 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 1 ) |
20 |
|
1re |
⊢ 1 ∈ ℝ |
21 |
|
suprleub |
⊢ ( ( ( 𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 𝑦 ) ∧ 1 ∈ ℝ ) → ( sup ( 𝑆 , ℝ , < ) ≤ 1 ↔ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 1 ) ) |
22 |
3 20 21
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( sup ( 𝑆 , ℝ , < ) ≤ 1 ↔ ∀ 𝑧 ∈ 𝑆 𝑧 ≤ 1 ) ) |
23 |
19 22
|
mpbird |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → sup ( 𝑆 , ℝ , < ) ≤ 1 ) |
24 |
2 23
|
eqbrtrid |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → 𝐵 ≤ 1 ) |
25 |
18 24
|
jca |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1 ) → ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1 ) ) |