Proof of Theorem mreclatBAD
Step | Hyp | Ref
| Expression |
1 | | mreclat.i |
. . . 4
⊢ 𝐼 = (toInc‘𝐶) |
2 | 1 | ipopos 17762 |
. . 3
⊢ 𝐼 ∈ Poset |
3 | 2 | a1i 11 |
. 2
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ Poset) |
4 | | eqid 2798 |
. . . . . . . 8
⊢
(mrCls‘𝐶) =
(mrCls‘𝐶) |
5 | | eqid 2798 |
. . . . . . . 8
⊢
(lub‘𝐼) =
(lub‘𝐼) |
6 | 1, 4, 5 | mrelatlub 17788 |
. . . . . . 7
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) = ((mrCls‘𝐶)‘∪ 𝑥)) |
7 | | uniss 4808 |
. . . . . . . . . 10
⊢ (𝑥 ⊆ 𝐶 → ∪ 𝑥 ⊆ ∪ 𝐶) |
8 | 7 | adantl 485 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ ∪ 𝐶) |
9 | | mreuni 16863 |
. . . . . . . . . 10
⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 =
𝑋) |
10 | 9 | adantr 484 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝐶 = 𝑋) |
11 | 8, 10 | sseqtrd 3955 |
. . . . . . . 8
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ∪ 𝑥 ⊆ 𝑋) |
12 | 4 | mrccl 16874 |
. . . . . . . 8
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∪ 𝑥
⊆ 𝑋) →
((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶) |
13 | 11, 12 | syldan 594 |
. . . . . . 7
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((mrCls‘𝐶)‘∪ 𝑥) ∈ 𝐶) |
14 | 6, 13 | eqeltrd 2890 |
. . . . . 6
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((lub‘𝐼)‘𝑥) ∈ 𝐶) |
15 | | fveq2 6645 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ →
((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅)) |
16 | 15 | adantl 485 |
. . . . . . . . 9
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = ((glb‘𝐼)‘∅)) |
17 | | eqid 2798 |
. . . . . . . . . . 11
⊢
(glb‘𝐼) =
(glb‘𝐼) |
18 | 1, 17 | mrelatglb0 17787 |
. . . . . . . . . 10
⊢ (𝐶 ∈ (Moore‘𝑋) → ((glb‘𝐼)‘∅) = 𝑋) |
19 | 18 | ad2antrr 725 |
. . . . . . . . 9
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘∅) = 𝑋) |
20 | 16, 19 | eqtrd 2833 |
. . . . . . . 8
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) = 𝑋) |
21 | | mre1cl 16857 |
. . . . . . . . 9
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) |
22 | 21 | ad2antrr 725 |
. . . . . . . 8
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → 𝑋 ∈ 𝐶) |
23 | 20, 22 | eqeltrd 2890 |
. . . . . . 7
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 = ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶) |
24 | 1, 17 | mrelatglb 17786 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) = ∩ 𝑥) |
25 | | mreintcl 16858 |
. . . . . . . . 9
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥
∈ 𝐶) |
26 | 24, 25 | eqeltrd 2890 |
. . . . . . . 8
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶) |
27 | 26 | 3expa 1115 |
. . . . . . 7
⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) ∧ 𝑥 ≠ ∅) → ((glb‘𝐼)‘𝑥) ∈ 𝐶) |
28 | 23, 27 | pm2.61dane 3074 |
. . . . . 6
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → ((glb‘𝐼)‘𝑥) ∈ 𝐶) |
29 | 14, 28 | jca 515 |
. . . . 5
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)) |
30 | 29 | ex 416 |
. . . 4
⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶))) |
31 | 1 | ipobas 17757 |
. . . . 5
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 = (Base‘𝐼)) |
32 | | sseq2 3941 |
. . . . . 6
⊢ (𝐶 = (Base‘𝐼) → (𝑥 ⊆ 𝐶 ↔ 𝑥 ⊆ (Base‘𝐼))) |
33 | | eleq2 2878 |
. . . . . . 7
⊢ (𝐶 = (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼))) |
34 | | eleq2 2878 |
. . . . . . 7
⊢ (𝐶 = (Base‘𝐼) → (((glb‘𝐼)‘𝑥) ∈ 𝐶 ↔ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))) |
35 | 33, 34 | anbi12d 633 |
. . . . . 6
⊢ (𝐶 = (Base‘𝐼) → ((((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶) ↔ (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))) |
36 | 32, 35 | imbi12d 348 |
. . . . 5
⊢ (𝐶 = (Base‘𝐼) → ((𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)) ↔ (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))) |
37 | 31, 36 | syl 17 |
. . . 4
⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑥 ⊆ 𝐶 → (((lub‘𝐼)‘𝑥) ∈ 𝐶 ∧ ((glb‘𝐼)‘𝑥) ∈ 𝐶)) ↔ (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))) |
38 | 30, 37 | mpbid 235 |
. . 3
⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))) |
39 | 38 | alrimiv 1928 |
. 2
⊢ (𝐶 ∈ (Moore‘𝑋) → ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))) |
40 | | isclatBAD. |
. 2
⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧
∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼))))) |
41 | 3, 39, 40 | sylanbrc 586 |
1
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat) |