Proof of Theorem dihord6apre
Step | Hyp | Ref
| Expression |
1 | | dihord6apre.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝐾) |
2 | | dihord6apre.h |
. . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) |
3 | | dihord6apre.t |
. . . . . . 7
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
4 | | dihord6apre.e |
. . . . . . 7
⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
5 | | dihord6apre.o |
. . . . . . 7
⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
6 | 1, 2, 3, 4, 5 | tendo1ne0 38462 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 𝑂) |
7 | 6 | 3ad2ant1 1134 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ( I ↾ 𝑇) ≠ 𝑂) |
8 | 7 | neneqd 2939 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ¬ ( I ↾ 𝑇) = 𝑂) |
9 | | dihord6apre.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
10 | | eqid 2738 |
. . . . . . 7
⊢
(join‘𝐾) =
(join‘𝐾) |
11 | | eqid 2738 |
. . . . . . 7
⊢
(meet‘𝐾) =
(meet‘𝐾) |
12 | | dihord6apre.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
13 | 1, 9, 10, 11, 12, 2 | lhpmcvr2 37658 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) |
14 | 13 | 3adant3 1133 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) |
15 | | simpl1 1192 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | | simpl2 1193 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) |
17 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) |
18 | | dihord6apre.i |
. . . . . . . . . . . 12
⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
19 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) |
20 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) |
21 | | dihord6apre.u |
. . . . . . . . . . . 12
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
22 | | dihord6apre.s |
. . . . . . . . . . . 12
⊢ ⊕ =
(LSSum‘𝑈) |
23 | 1, 9, 10, 11, 12, 2, 18, 19, 20, 21, 22 | dihvalcq 38870 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐼‘𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)))) |
24 | 15, 16, 17, 23 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐼‘𝑋) = ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)))) |
25 | | simpl3 1194 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) |
26 | 1, 9, 2, 18, 19 | dihvalb 38871 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) = (((DIsoB‘𝐾)‘𝑊)‘𝑌)) |
27 | 15, 25, 26 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐼‘𝑌) = (((DIsoB‘𝐾)‘𝑊)‘𝑌)) |
28 | 24, 27 | sseq12d 3911 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌))) |
29 | 2, 21, 15 | dvhlmod 38744 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑈 ∈ LMod) |
30 | | eqid 2738 |
. . . . . . . . . . . . . 14
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
31 | 30 | lsssssubg 19850 |
. . . . . . . . . . . . 13
⊢ (𝑈 ∈ LMod →
(LSubSp‘𝑈) ⊆
(SubGrp‘𝑈)) |
32 | 29, 31 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
33 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) |
34 | 9, 12, 2, 21, 20, 30 | diclss 38827 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ∈ (LSubSp‘𝑈)) |
35 | 15, 33, 34 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ∈ (LSubSp‘𝑈)) |
36 | 32, 35 | sseldd 3879 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ∈ (SubGrp‘𝑈)) |
37 | | simpl1l 1225 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ HL) |
38 | 37 | hllatd 36998 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ Lat) |
39 | | simpl2l 1227 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵) |
40 | | simpl1r 1226 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻) |
41 | 1, 2 | lhpbase 37632 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
42 | 40, 41 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵) |
43 | 1, 11 | latmcl 17779 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) |
44 | 38, 39, 42, 43 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) |
45 | 1, 9, 11 | latmle2 17804 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑊) ≤ 𝑊) |
46 | 38, 39, 42, 45 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋(meet‘𝐾)𝑊) ≤ 𝑊) |
47 | 1, 9, 2, 21, 19, 30 | diblss 38804 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)) ∈ (LSubSp‘𝑈)) |
48 | 15, 44, 46, 47 | syl12anc 836 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)) ∈ (LSubSp‘𝑈)) |
49 | 32, 48 | sseldd 3879 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)) ∈ (SubGrp‘𝑈)) |
50 | 22 | lsmub1 18901 |
. . . . . . . . . . 11
⊢
(((((DIsoC‘𝐾)‘𝑊)‘𝑞) ∈ (SubGrp‘𝑈) ∧ (((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)) ∈ (SubGrp‘𝑈)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)))) |
51 | 36, 49, 50 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊)))) |
52 | | sstr 3886 |
. . . . . . . . . . 11
⊢
(((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌)) → (((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌)) |
53 | | eqidd 2739 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (( I ↾ 𝑇)‘𝐺) = (( I ↾ 𝑇)‘𝐺)) |
54 | 2, 3, 4 | tendoidcl 38403 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
55 | 15, 54 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ( I ↾ 𝑇) ∈ 𝐸) |
56 | | dihord6apre.p |
. . . . . . . . . . . . . . 15
⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
57 | | dihord6apre.g |
. . . . . . . . . . . . . . 15
⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞) |
58 | | fvex 6688 |
. . . . . . . . . . . . . . 15
⊢ (( I
↾ 𝑇)‘𝐺) ∈ V |
59 | 3 | fvexi 6689 |
. . . . . . . . . . . . . . . 16
⊢ 𝑇 ∈ V |
60 | | resiexg 7646 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ V → ( I ↾
𝑇) ∈
V) |
61 | 59, 60 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ( I
↾ 𝑇) ∈
V |
62 | 9, 12, 2, 56, 3, 4,
20, 57, 58, 61 | dicopelval2 38815 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑞) ↔ ((( I ↾ 𝑇)‘𝐺) = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸))) |
63 | 15, 33, 62 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑞) ↔ ((( I ↾ 𝑇)‘𝐺) = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸))) |
64 | 53, 55, 63 | mpbir2and 713 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑞)) |
65 | | ssel2 3873 |
. . . . . . . . . . . . 13
⊢
(((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌) ∧ 〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑞)) → 〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑌)) |
66 | | eqid 2738 |
. . . . . . . . . . . . . . . 16
⊢
((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) |
67 | 1, 9, 2, 3, 5, 66,
19 | dibopelval2 38779 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑌) ↔ ((( I ↾ 𝑇)‘𝐺) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ ( I ↾ 𝑇) = 𝑂))) |
68 | 15, 25, 67 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑌) ↔ ((( I ↾ 𝑇)‘𝐺) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ ( I ↾ 𝑇) = 𝑂))) |
69 | | simpr 488 |
. . . . . . . . . . . . . 14
⊢ (((( I
↾ 𝑇)‘𝐺) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑌) ∧ ( I ↾ 𝑇) = 𝑂) → ( I ↾ 𝑇) = 𝑂) |
70 | 68, 69 | syl6bi 256 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑌) → ( I ↾ 𝑇) = 𝑂)) |
71 | 65, 70 | syl5 34 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌) ∧ 〈(( I ↾ 𝑇)‘𝐺), ( I ↾ 𝑇)〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑞)) → ( I ↾ 𝑇) = 𝑂)) |
72 | 64, 71 | mpan2d 694 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌) → ( I ↾ 𝑇) = 𝑂)) |
73 | 52, 72 | syl5 34 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ∧ ((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌)) → ( I ↾ 𝑇) = 𝑂)) |
74 | 51, 73 | mpand 695 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((((DIsoC‘𝐾)‘𝑊)‘𝑞) ⊕
(((DIsoB‘𝐾)‘𝑊)‘(𝑋(meet‘𝐾)𝑊))) ⊆ (((DIsoB‘𝐾)‘𝑊)‘𝑌) → ( I ↾ 𝑇) = 𝑂)) |
75 | 28, 74 | sylbid 243 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → ( I ↾ 𝑇) = 𝑂)) |
76 | 75 | exp44 441 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑞 ∈ 𝐴 → (¬ 𝑞 ≤ 𝑊 → ((𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋 → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → ( I ↾ 𝑇) = 𝑂))))) |
77 | 76 | imp4a 426 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑞 ∈ 𝐴 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → ( I ↾ 𝑇) = 𝑂)))) |
78 | 77 | rexlimdv 3193 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → ( I ↾ 𝑇) = 𝑂))) |
79 | 14, 78 | mpd 15 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → ( I ↾ 𝑇) = 𝑂)) |
80 | 8, 79 | mtod 201 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ¬ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) |
81 | 80 | pm2.21d 121 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) → 𝑋 ≤ 𝑌)) |
82 | 81 | imp 410 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌) |