| Step | Hyp | Ref
| Expression |
| 1 | | equivbnd.2 |
. 2
⊢ (𝜑 → 𝑁 ∈ (Met‘𝑋)) |
| 2 | | equivbnd.1 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ (Bnd‘𝑋)) |
| 3 | | isbnd3b 37792 |
. . . . 5
⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟)) |
| 4 | 3 | simprbi 496 |
. . . 4
⊢ (𝑀 ∈ (Bnd‘𝑋) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟) |
| 5 | 2, 4 | syl 17 |
. . 3
⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟) |
| 6 | | equivbnd.3 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
| 7 | 6 | rpred 13077 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ ℝ) |
| 8 | | remulcl 11240 |
. . . . . 6
⊢ ((𝑅 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (𝑅 · 𝑟) ∈ ℝ) |
| 9 | 7, 8 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (𝑅 · 𝑟) ∈ ℝ) |
| 10 | | bndmet 37788 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) |
| 11 | 2, 10 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (Met‘𝑋)) |
| 12 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → 𝑀 ∈ (Met‘𝑋)) |
| 13 | | metcl 24342 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) ∈ ℝ) |
| 14 | 13 | 3expb 1121 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑀𝑦) ∈ ℝ) |
| 15 | 12, 14 | sylan 580 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑀𝑦) ∈ ℝ) |
| 16 | | simplr 769 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑟 ∈ ℝ) |
| 17 | 6 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 ∈
ℝ+) |
| 18 | 15, 16, 17 | lemul2d 13121 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝑀𝑦) ≤ 𝑟 ↔ (𝑅 · (𝑥𝑀𝑦)) ≤ (𝑅 · 𝑟))) |
| 19 | | equivbnd.4 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦))) |
| 20 | 19 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦))) |
| 21 | 1 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → 𝑁 ∈ (Met‘𝑋)) |
| 22 | | metcl 24342 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑁𝑦) ∈ ℝ) |
| 23 | 22 | 3expb 1121 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ (Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ∈ ℝ) |
| 24 | 21, 23 | sylan 580 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ∈ ℝ) |
| 25 | 7 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 ∈ ℝ) |
| 26 | 25, 15 | remulcld 11291 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑅 · (𝑥𝑀𝑦)) ∈ ℝ) |
| 27 | 9 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑅 · 𝑟) ∈ ℝ) |
| 28 | | letr 11355 |
. . . . . . . . 9
⊢ (((𝑥𝑁𝑦) ∈ ℝ ∧ (𝑅 · (𝑥𝑀𝑦)) ∈ ℝ ∧ (𝑅 · 𝑟) ∈ ℝ) → (((𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)) ∧ (𝑅 · (𝑥𝑀𝑦)) ≤ (𝑅 · 𝑟)) → (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 29 | 24, 26, 27, 28 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)) ∧ (𝑅 · (𝑥𝑀𝑦)) ≤ (𝑅 · 𝑟)) → (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 30 | 20, 29 | mpand 695 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑅 · (𝑥𝑀𝑦)) ≤ (𝑅 · 𝑟) → (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 31 | 18, 30 | sylbid 240 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝑀𝑦) ≤ 𝑟 → (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 32 | 31 | ralimdvva 3206 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 33 | | breq2 5147 |
. . . . . . 7
⊢ (𝑠 = (𝑅 · 𝑟) → ((𝑥𝑁𝑦) ≤ 𝑠 ↔ (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 34 | 33 | 2ralbidv 3221 |
. . . . . 6
⊢ (𝑠 = (𝑅 · 𝑟) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟))) |
| 35 | 34 | rspcev 3622 |
. . . . 5
⊢ (((𝑅 · 𝑟) ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ (𝑅 · 𝑟)) → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠) |
| 36 | 9, 32, 35 | syl6an 684 |
. . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠)) |
| 37 | 36 | rexlimdva 3155 |
. . 3
⊢ (𝜑 → (∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) ≤ 𝑟 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠)) |
| 38 | 5, 37 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠) |
| 39 | | isbnd3b 37792 |
. 2
⊢ (𝑁 ∈ (Bnd‘𝑋) ↔ (𝑁 ∈ (Met‘𝑋) ∧ ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑁𝑦) ≤ 𝑠)) |
| 40 | 1, 38, 39 | sylanbrc 583 |
1
⊢ (𝜑 → 𝑁 ∈ (Bnd‘𝑋)) |