Step | Hyp | Ref
| Expression |
1 | | simprrl 781 |
. . . . . 6
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤) |
2 | | breq2 5034 |
. . . . . . . . 9
⊢ (𝑧 = 𝑤 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑤)) |
3 | 2 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤)) |
4 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑤)) |
5 | 3, 4 | imbi12d 348 |
. . . . . . 7
⊢ (𝑧 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 → 𝑥 ≤ 𝑤))) |
6 | | simprlr 780 |
. . . . . . 7
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) |
7 | | simplrr 778 |
. . . . . . 7
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑤 ∈ 𝐵) |
8 | 5, 6, 7 | rspcdva 3528 |
. . . . . 6
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 → 𝑥 ≤ 𝑤)) |
9 | 1, 8 | mpd 15 |
. . . . 5
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 ≤ 𝑤) |
10 | | simprll 779 |
. . . . . 6
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥) |
11 | | breq2 5034 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥)) |
12 | 11 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)) |
13 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (𝑤 ≤ 𝑧 ↔ 𝑤 ≤ 𝑥)) |
14 | 12, 13 | imbi12d 348 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 → 𝑤 ≤ 𝑥))) |
15 | | simprrr 782 |
. . . . . . 7
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)) |
16 | | simplrl 777 |
. . . . . . 7
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 ∈ 𝐵) |
17 | 14, 15, 16 | rspcdva 3528 |
. . . . . 6
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 → 𝑤 ≤ 𝑥)) |
18 | 10, 17 | mpd 15 |
. . . . 5
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑤 ≤ 𝑥) |
19 | | poslubmo.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐾) |
20 | | poslubmo.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
21 | 19, 20 | posasymb 17678 |
. . . . . . 7
⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤)) |
22 | 21 | 3expb 1121 |
. . . . . 6
⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤)) |
23 | 22 | ad4ant13 751 |
. . . . 5
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → ((𝑥 ≤ 𝑤 ∧ 𝑤 ≤ 𝑥) ↔ 𝑥 = 𝑤)) |
24 | 9, 18, 23 | mpbi2and 712 |
. . . 4
⊢ ((((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) → 𝑥 = 𝑤) |
25 | 24 | ex 416 |
. . 3
⊢ (((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤)) |
26 | 25 | ralrimivva 3103 |
. 2
⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤)) |
27 | | breq2 5034 |
. . . . 5
⊢ (𝑥 = 𝑤 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑤)) |
28 | 27 | ralbidv 3109 |
. . . 4
⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤)) |
29 | | breq1 5033 |
. . . . . 6
⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑧 ↔ 𝑤 ≤ 𝑧)) |
30 | 29 | imbi2d 344 |
. . . . 5
⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) |
31 | 30 | ralbidv 3109 |
. . . 4
⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) |
32 | 28, 31 | anbi12d 634 |
. . 3
⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧)))) |
33 | 32 | rmo4 3629 |
. 2
⊢
(∃*𝑥 ∈
𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ∧ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑤 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑤 ≤ 𝑧))) → 𝑥 = 𝑤)) |
34 | 26, 33 | sylibr 237 |
1
⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |