Proof of Theorem lhpat3
Step | Hyp | Ref
| Expression |
1 | | simpl3r 1110 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 ≤ (𝑃 ∨ 𝑄)) |
2 | | simpr 476 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 ≤ 𝑊) |
3 | | simp1ll 1117 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL) |
4 | | hllat 33668 |
. . . . . . . . . 10
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
5 | 3, 4 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat) |
6 | | simp2r 1081 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ 𝐴) |
7 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝐾) =
(Base‘𝐾) |
8 | | lhpat.a |
. . . . . . . . . . 11
⊢ 𝐴 = (Atoms‘𝐾) |
9 | 7, 8 | atbase 33594 |
. . . . . . . . . 10
⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾)) |
10 | 6, 9 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ (Base‘𝐾)) |
11 | | simp1rl 1119 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴) |
12 | | simp2l 1080 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴) |
13 | | lhpat.j |
. . . . . . . . . . 11
⊢ ∨ =
(join‘𝐾) |
14 | 7, 13, 8 | hlatjcl 33671 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
15 | 3, 11, 12, 14 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
16 | | simp1lr 1118 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻) |
17 | | lhpat.h |
. . . . . . . . . . 11
⊢ 𝐻 = (LHyp‘𝐾) |
18 | 7, 17 | lhpbase 34302 |
. . . . . . . . . 10
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
19 | 16, 18 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ (Base‘𝐾)) |
20 | | lhpat.l |
. . . . . . . . . 10
⊢ ≤ =
(le‘𝐾) |
21 | | lhpat.m |
. . . . . . . . . 10
⊢ ∧ =
(meet‘𝐾) |
22 | 7, 20, 21 | latlem12 16901 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ 𝑊) ↔ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑊))) |
23 | 5, 10, 15, 19, 22 | syl13anc 1320 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ 𝑊) ↔ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑊))) |
24 | 23 | adantr 480 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ 𝑊) ↔ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑊))) |
25 | 1, 2, 24 | mpbi2and 958 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑊)) |
26 | | lhpat2.r |
. . . . . 6
⊢ 𝑅 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
27 | 25, 26 | syl6breqr 4625 |
. . . . 5
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 ≤ 𝑅) |
28 | 3 | adantr 480 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝐾 ∈ HL) |
29 | | hlatl 33665 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
30 | 28, 29 | syl 17 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝐾 ∈ AtLat) |
31 | | simpl2r 1108 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 ∈ 𝐴) |
32 | | simpl1l 1105 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
33 | | simpl1r 1106 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
34 | | simpl2l 1107 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑄 ∈ 𝐴) |
35 | | simpl3l 1109 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑃 ≠ 𝑄) |
36 | 20, 13, 21, 8, 17, 26 | lhpat2 34349 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑅 ∈ 𝐴) |
37 | 32, 33, 34, 35, 36 | syl112anc 1322 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑅 ∈ 𝐴) |
38 | 20, 8 | atcmp 33616 |
. . . . . 6
⊢ ((𝐾 ∈ AtLat ∧ 𝑆 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑆 ≤ 𝑅 ↔ 𝑆 = 𝑅)) |
39 | 30, 31, 37, 38 | syl3anc 1318 |
. . . . 5
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → (𝑆 ≤ 𝑅 ↔ 𝑆 = 𝑅)) |
40 | 27, 39 | mpbid 221 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑆 ≤ 𝑊) → 𝑆 = 𝑅) |
41 | 40 | ex 449 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑆 ≤ 𝑊 → 𝑆 = 𝑅)) |
42 | 7, 20, 21 | latmle2 16900 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) |
43 | 5, 15, 19, 42 | syl3anc 1318 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) |
44 | 26, 43 | syl5eqbr 4618 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ≤ 𝑊) |
45 | | breq1 4586 |
. . . 4
⊢ (𝑆 = 𝑅 → (𝑆 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊)) |
46 | 44, 45 | syl5ibrcom 236 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑆 = 𝑅 → 𝑆 ≤ 𝑊)) |
47 | 41, 46 | impbid 201 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑆 ≤ 𝑊 ↔ 𝑆 = 𝑅)) |
48 | 47 | necon3bbid 2819 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (¬ 𝑆 ≤ 𝑊 ↔ 𝑆 ≠ 𝑅)) |