Step | Hyp | Ref
| Expression |
1 | | simpl1 1064 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simpl3 1066 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) |
3 | | dihord3.b |
. . . 4
⊢ 𝐵 = (Base‘𝐾) |
4 | | dihord3.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
5 | | eqid 2622 |
. . . 4
⊢
(join‘𝐾) =
(join‘𝐾) |
6 | | eqid 2622 |
. . . 4
⊢
(meet‘𝐾) =
(meet‘𝐾) |
7 | | eqid 2622 |
. . . 4
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) |
8 | | dihord3.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
9 | 3, 4, 5, 6, 7, 8 | lhpmcvr2 35310 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) |
10 | 1, 2, 9 | syl2anc 693 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) |
11 | | simp1r 1086 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ 𝑌) |
12 | | simpl2r 1115 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑊) |
13 | 12 | 3ad2ant1 1082 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ 𝑊) |
14 | | simpl1l 1112 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL) |
15 | 14 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ HL) |
16 | | hllat 34650 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ Lat) |
18 | | simpl2l 1114 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵) |
19 | 18 | 3ad2ant1 1082 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ∈ 𝐵) |
20 | | simpl3l 1116 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵) |
21 | 20 | 3ad2ant1 1082 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑌 ∈ 𝐵) |
22 | | simpl1r 1113 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑊 ∈ 𝐻) |
23 | 22 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊 ∈ 𝐻) |
24 | 3, 8 | lhpbase 35284 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊 ∈ 𝐵) |
26 | 3, 4, 6 | latlem12 17078 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊))) |
27 | 17, 19, 21, 25, 26 | syl13anc 1328 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊))) |
28 | 11, 13, 27 | mpbi2and 956 |
. . . . . . . . 9
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ (𝑌(meet‘𝐾)𝑊)) |
29 | | simp1l1 1154 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
30 | | simp1l2 1155 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) |
31 | 3, 6 | latmcl 17052 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵) |
32 | 17, 21, 25, 31 | syl3anc 1326 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵) |
33 | 3, 4, 6 | latmle2 17077 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌(meet‘𝐾)𝑊) ≤ 𝑊) |
34 | 17, 21, 25, 33 | syl3anc 1326 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) ≤ 𝑊) |
35 | | eqid 2622 |
. . . . . . . . . . 11
⊢
((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) |
36 | 3, 4, 8, 35 | dibord 36448 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊))) |
37 | 29, 30, 32, 34, 36 | syl112anc 1330 |
. . . . . . . . 9
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊))) |
38 | 28, 37 | mpbird 247 |
. . . . . . . 8
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))) |
39 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) |
40 | 8, 39, 29 | dvhlmod 36399 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((DVecH‘𝐾)‘𝑊) ∈ LMod) |
41 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) |
42 | 41 | lsssssubg 18958 |
. . . . . . . . . . 11
⊢
(((DVecH‘𝐾)‘𝑊) ∈ LMod →
(LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
43 | 40, 42 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
44 | | simp2 1062 |
. . . . . . . . . . 11
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊)) |
45 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) |
46 | 4, 7, 8, 39, 45, 41 | diclss 36482 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
47 | 29, 44, 46 | syl2anc 693 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
48 | 43, 47 | sseldd 3604 |
. . . . . . . . 9
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
49 | 3, 4, 8, 39, 35, 41 | diblss 36459 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
50 | 29, 32, 34, 49 | syl12anc 1324 |
. . . . . . . . . 10
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
51 | 43, 50 | sseldd 3604 |
. . . . . . . . 9
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) |
52 | | eqid 2622 |
. . . . . . . . . 10
⊢
(LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊)) |
53 | 52 | lsmub2 18072 |
. . . . . . . . 9
⊢
(((((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)) ∧ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) |
54 | 48, 51, 53 | syl2anc 693 |
. . . . . . . 8
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) |
55 | 38, 54 | sstrd 3613 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) |
56 | | dihord3.i |
. . . . . . . . 9
⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
57 | 3, 4, 8, 56, 35 | dihvalb 36526 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋)) |
58 | 29, 30, 57 | syl2anc 693 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋)) |
59 | | simp1l3 1156 |
. . . . . . . 8
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) |
60 | | simp3 1063 |
. . . . . . . 8
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) |
61 | 3, 4, 5, 6, 7, 8, 56, 35, 45, 39, 52 | dihvalcq 36525 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) |
62 | 29, 59, 44, 60, 61 | syl112anc 1330 |
. . . . . . 7
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))) |
63 | 55, 58, 62 | 3sstr4d 3648 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) |
64 | 63 | 3exp 1264 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)))) |
65 | 64 | expd 452 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → (¬ 𝑟 ≤ 𝑊 → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))))) |
66 | 65 | imp4a 614 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)))) |
67 | 66 | rexlimdv 3030 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))) |
68 | 10, 67 | mpd 15 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) |