Step |
Hyp |
Ref |
Expression |
1 |
|
suprltrp.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
suprltrp.n0 |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
3 |
|
suprltrp.bnd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
4 |
|
suprltrp.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ+ ) |
5 |
|
suprcl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
6 |
1 2 3 5
|
syl3anc |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
7 |
6 4
|
ltsubrpd |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < sup ( 𝐴 , ℝ , < ) ) |
8 |
4
|
rpred |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
9 |
6 8
|
resubcld |
⊢ ( 𝜑 → ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) ∈ ℝ ) |
10 |
|
suprlub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) ∈ ℝ ) → ( ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < 𝑧 ) ) |
11 |
1 2 3 9 10
|
syl31anc |
⊢ ( 𝜑 → ( ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < 𝑧 ) ) |
12 |
7 11
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝐴 ( sup ( 𝐴 , ℝ , < ) − 𝑋 ) < 𝑧 ) |