Proof of Theorem dihjustlem
Step | Hyp | Ref
| Expression |
1 | | simp1l 1078 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝐾 ∈ HL) |
2 | | hllat 33668 |
. . . . . 6
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
3 | 1, 2 | syl 17 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝐾 ∈ Lat) |
4 | | simp21l 1171 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑄 ∈ 𝐴) |
5 | | dihjust.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐾) |
6 | | dihjust.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
7 | 5, 6 | atbase 33594 |
. . . . . 6
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
8 | 4, 7 | syl 17 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑄 ∈ 𝐵) |
9 | | simp23 1089 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑋 ∈ 𝐵) |
10 | | simp1r 1079 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐻) |
11 | | dihjust.h |
. . . . . . . 8
⊢ 𝐻 = (LHyp‘𝐾) |
12 | 5, 11 | lhpbase 34302 |
. . . . . . 7
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
13 | 10, 12 | syl 17 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐵) |
14 | | dihjust.m |
. . . . . . 7
⊢ ∧ =
(meet‘𝐾) |
15 | 5, 14 | latmcl 16875 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵) |
16 | 3, 9, 13, 15 | syl3anc 1318 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑋 ∧ 𝑊) ∈ 𝐵) |
17 | | dihjust.l |
. . . . . 6
⊢ ≤ =
(le‘𝐾) |
18 | | dihjust.j |
. . . . . 6
⊢ ∨ =
(join‘𝐾) |
19 | 5, 17, 18 | latlej1 16883 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → 𝑄 ≤ (𝑄 ∨ (𝑋 ∧ 𝑊))) |
20 | 3, 8, 16, 19 | syl3anc 1318 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑄 ≤ (𝑄 ∨ (𝑋 ∧ 𝑊))) |
21 | | simp3 1056 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) |
22 | 20, 21 | breqtrd 4609 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑄 ≤ (𝑅 ∨ (𝑋 ∧ 𝑊))) |
23 | | simp1 1054 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
24 | | simp22 1088 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) |
25 | | simp21 1087 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
26 | 5, 17, 14 | latmle2 16900 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊) |
27 | 3, 9, 13, 26 | syl3anc 1318 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑋 ∧ 𝑊) ≤ 𝑊) |
28 | 16, 27 | jca 553 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) |
29 | | dihjust.i |
. . . . 5
⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
30 | | dihjust.J |
. . . . 5
⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) |
31 | | dihjust.u |
. . . . 5
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
32 | | dihjust.s |
. . . . 5
⊢ ⊕ =
(LSSum‘𝑈) |
33 | 5, 17, 18, 6, 11, 29, 30, 31, 32 | cdlemn 35519 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊))) → (𝑄 ≤ (𝑅 ∨ (𝑋 ∧ 𝑊)) ↔ (𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))))) |
34 | 23, 24, 25, 28, 33 | syl13anc 1320 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝑄 ≤ (𝑅 ∨ (𝑋 ∧ 𝑊)) ↔ (𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))))) |
35 | 22, 34 | mpbid 221 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) |
36 | 11, 31, 23 | dvhlmod 35417 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → 𝑈 ∈ LMod) |
37 | | eqid 2610 |
. . . . . 6
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
38 | 37 | lsssssubg 18779 |
. . . . 5
⊢ (𝑈 ∈ LMod →
(LSubSp‘𝑈) ⊆
(SubGrp‘𝑈)) |
39 | 36, 38 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
40 | 17, 6, 11, 31, 30, 37 | diclss 35500 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐽‘𝑅) ∈ (LSubSp‘𝑈)) |
41 | 23, 24, 40 | syl2anc 691 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐽‘𝑅) ∈ (LSubSp‘𝑈)) |
42 | 39, 41 | sseldd 3569 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐽‘𝑅) ∈ (SubGrp‘𝑈)) |
43 | 5, 17, 11, 31, 29, 37 | diblss 35477 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (LSubSp‘𝑈)) |
44 | 23, 16, 27, 43 | syl12anc 1316 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (LSubSp‘𝑈)) |
45 | 39, 44 | sseldd 3569 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑊)) ∈ (SubGrp‘𝑈)) |
46 | 32 | lsmub2 17895 |
. . 3
⊢ (((𝐽‘𝑅) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ∈ (SubGrp‘𝑈)) → (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) |
47 | 42, 45, 46 | syl2anc 691 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) |
48 | 17, 6, 11, 31, 30, 37 | diclss 35500 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐽‘𝑄) ∈ (LSubSp‘𝑈)) |
49 | 23, 25, 48 | syl2anc 691 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐽‘𝑄) ∈ (LSubSp‘𝑈)) |
50 | 39, 49 | sseldd 3569 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (𝐽‘𝑄) ∈ (SubGrp‘𝑈)) |
51 | 37, 32 | lsmcl 18904 |
. . . . 5
⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑅) ∈ (LSubSp‘𝑈) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ∈ (LSubSp‘𝑈)) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∈ (LSubSp‘𝑈)) |
52 | 36, 41, 44, 51 | syl3anc 1318 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∈ (LSubSp‘𝑈)) |
53 | 39, 52 | sseldd 3569 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∈ (SubGrp‘𝑈)) |
54 | 32 | lsmlub 17901 |
. . 3
⊢ (((𝐽‘𝑄) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ∈ (SubGrp‘𝑈) ∧ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∈ (SubGrp‘𝑈)) → (((𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) ↔ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))))) |
55 | 50, 45, 53, 54 | syl3anc 1318 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → (((𝐽‘𝑄) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ∧ (𝐼‘(𝑋 ∧ 𝑊)) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) ↔ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊))))) |
56 | 35, 47, 55 | mpbi2and 958 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = (𝑅 ∨ (𝑋 ∧ 𝑊))) → ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑅) ⊕ (𝐼‘(𝑋 ∧ 𝑊)))) |