Step | Hyp | Ref
| Expression |
1 | | cdlemf.l |
. . 3
⊢ ≤ =
(le‘𝐾) |
2 | | eqid 2738 |
. . 3
⊢
(join‘𝐾) =
(join‘𝐾) |
3 | | cdlemf.a |
. . 3
⊢ 𝐴 = (Atoms‘𝐾) |
4 | | cdlemf.h |
. . 3
⊢ 𝐻 = (LHyp‘𝐾) |
5 | | eqid 2738 |
. . 3
⊢
(meet‘𝐾) =
(meet‘𝐾) |
6 | 1, 2, 3, 4, 5 | cdlemf2 38576 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) |
7 | | simp1l 1196 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
8 | | simp2l 1198 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑝 ∈ 𝐴) |
9 | | simp3ll 1243 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑝 ≤ 𝑊) |
10 | | simp2r 1199 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑞 ∈ 𝐴) |
11 | | simp3lr 1244 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑞 ≤ 𝑊) |
12 | | cdlemf.t |
. . . . . . 7
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
13 | 1, 3, 4, 12 | cdleme50ex 38573 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) |
14 | 7, 8, 9, 10, 11, 13 | syl122anc 1378 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) |
15 | | simp3r 1201 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑓‘𝑝) = 𝑞) |
16 | 15 | oveq2d 7291 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑝(join‘𝐾)(𝑓‘𝑝)) = (𝑝(join‘𝐾)𝑞)) |
17 | 16 | oveq1d 7290 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) |
18 | | simp11 1202 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
19 | | simp3l 1200 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑓 ∈ 𝑇) |
20 | | simp13l 1287 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑝 ∈ 𝐴) |
21 | | simp2ll 1239 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → ¬ 𝑝 ≤ 𝑊) |
22 | | cdlemf.r |
. . . . . . . . . . . . 13
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
23 | 1, 2, 5, 3, 4, 12,
22 | trlval2 38177 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑅‘𝑓) = ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊)) |
24 | 18, 19, 20, 21, 23 | syl112anc 1373 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑅‘𝑓) = ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊)) |
25 | | simp2r 1199 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) |
26 | 17, 24, 25 | 3eqtr4d 2788 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑅‘𝑓) = 𝑈) |
27 | 26 | 3exp 1118 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈))) |
28 | 27 | 3expia 1120 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈)))) |
29 | 28 | 3imp 1110 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈)) |
30 | 29 | expd 416 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝑓 ∈ 𝑇 → ((𝑓‘𝑝) = 𝑞 → (𝑅‘𝑓) = 𝑈))) |
31 | 30 | reximdvai 3200 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞 → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)) |
32 | 14, 31 | mpd 15 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈) |
33 | 32 | 3exp 1118 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈))) |
34 | 33 | rexlimdvv 3222 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)) |
35 | 6, 34 | mpd 15 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈) |