Step | Hyp | Ref
| Expression |
1 | | riotaex 7131 |
. . . . 5
⊢
(℩𝑓
∈ 𝑇 (𝑓‘𝑃) = 𝑞) ∈ V |
2 | | cdlemm10.g |
. . . . 5
⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞)) |
3 | 1, 2 | fnmpti 6480 |
. . . 4
⊢ 𝐺 Fn 𝐶 |
4 | | fvelrnb 6730 |
. . . 4
⊢ (𝐺 Fn 𝐶 → (𝑔 ∈ ran 𝐺 ↔ ∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔)) |
5 | 3, 4 | ax-mp 5 |
. . 3
⊢ (𝑔 ∈ ran 𝐺 ↔ ∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔) |
6 | | eqeq2 2750 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑠 → ((𝑓‘𝑃) = 𝑞 ↔ (𝑓‘𝑃) = 𝑠)) |
7 | 6 | riotabidv 7129 |
. . . . . . . . . . 11
⊢ (𝑞 = 𝑠 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)) |
8 | | riotaex 7131 |
. . . . . . . . . . 11
⊢
(℩𝑓
∈ 𝑇 (𝑓‘𝑃) = 𝑠) ∈ V |
9 | 7, 2, 8 | fvmpt 6775 |
. . . . . . . . . 10
⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)) |
10 | | cdlemm10.f |
. . . . . . . . . 10
⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) |
11 | 9, 10 | eqtr4di 2791 |
. . . . . . . . 9
⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = 𝐹) |
12 | 11 | adantl 485 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐺‘𝑠) = 𝐹) |
13 | 12 | eqeq1d 2740 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → ((𝐺‘𝑠) = 𝑔 ↔ 𝐹 = 𝑔)) |
14 | 13 | rexbidva 3206 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)) |
15 | | simpl1 1192 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑔 ∈ 𝑇) |
17 | | simpl2l 1227 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑃 ∈ 𝐴) |
18 | | cdlemm10.l |
. . . . . . . . . . . 12
⊢ ≤ =
(le‘𝐾) |
19 | | cdlemm10.a |
. . . . . . . . . . . 12
⊢ 𝐴 = (Atoms‘𝐾) |
20 | | cdlemm10.h |
. . . . . . . . . . . 12
⊢ 𝐻 = (LHyp‘𝐾) |
21 | | cdlemm10.t |
. . . . . . . . . . . 12
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
22 | 18, 19, 20, 21 | ltrnat 37797 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑔‘𝑃) ∈ 𝐴) |
23 | 15, 16, 17, 22 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐴) |
24 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
(Base‘𝐾) =
(Base‘𝐾) |
25 | | simpl1l 1225 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ HL) |
26 | 25 | hllatd 37021 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ Lat) |
27 | 24, 19 | atbase 36946 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
28 | 17, 27 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑃 ∈ (Base‘𝐾)) |
29 | 24, 20, 21 | ltrncl 37782 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑔‘𝑃) ∈ (Base‘𝐾)) |
30 | 15, 16, 28, 29 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ (Base‘𝐾)) |
31 | | cdlemm10.j |
. . . . . . . . . . . . . 14
⊢ ∨ =
(join‘𝐾) |
32 | 24, 31 | latjcl 17777 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾)) |
33 | 26, 28, 30, 32 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾)) |
34 | | simpl3l 1229 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑉 ∈ 𝐴) |
35 | 24, 31, 19 | hlatjcl 37024 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
36 | 25, 17, 34, 35 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
37 | 24, 18, 31 | latlej2 17787 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃))) |
38 | 26, 28, 30, 37 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃))) |
39 | | simpl2 1193 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
40 | | cdlemm10.r |
. . . . . . . . . . . . . . 15
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
41 | 18, 31, 19, 20, 21, 40 | trljat1 37823 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃))) |
42 | 15, 16, 39, 41 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃))) |
43 | | simprr 773 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑅‘𝑔) ≤ 𝑉) |
44 | 24, 20, 21, 40 | trlcl 37821 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (𝑅‘𝑔) ∈ (Base‘𝐾)) |
45 | 15, 16, 44 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑅‘𝑔) ∈ (Base‘𝐾)) |
46 | 24, 19 | atbase 36946 |
. . . . . . . . . . . . . . . 16
⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾)) |
47 | 34, 46 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑉 ∈ (Base‘𝐾)) |
48 | 24, 18, 31 | latjlej2 17792 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝑔) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑅‘𝑔) ≤ 𝑉 → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉))) |
49 | 26, 45, 47, 28, 48 | syl13anc 1373 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ((𝑅‘𝑔) ≤ 𝑉 → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉))) |
50 | 43, 49 | mpd 15 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉)) |
51 | 42, 50 | eqbrtrrd 5054 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑔‘𝑃)) ≤ (𝑃 ∨ 𝑉)) |
52 | 24, 18, 26, 30, 33, 36, 38, 51 | lattrd 17784 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉)) |
53 | 18, 19, 20, 21 | ltrnel 37796 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑔‘𝑃) ∈ 𝐴 ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)) |
54 | 53 | simprd 499 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝑔‘𝑃) ≤ 𝑊) |
55 | 15, 16, 39, 54 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ¬ (𝑔‘𝑃) ≤ 𝑊) |
56 | 52, 55 | jca 515 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)) |
57 | | breq1 5033 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉))) |
58 | | breq1 5033 |
. . . . . . . . . . . . 13
⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ 𝑊 ↔ (𝑔‘𝑃) ≤ 𝑊)) |
59 | 58 | notbid 321 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑔‘𝑃) → (¬ 𝑟 ≤ 𝑊 ↔ ¬ (𝑔‘𝑃) ≤ 𝑊)) |
60 | 57, 59 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑔‘𝑃) → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))) |
61 | | cdlemm10.c |
. . . . . . . . . . 11
⊢ 𝐶 = {𝑟 ∈ 𝐴 ∣ (𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊)} |
62 | 60, 61 | elrab2 3591 |
. . . . . . . . . 10
⊢ ((𝑔‘𝑃) ∈ 𝐶 ↔ ((𝑔‘𝑃) ∈ 𝐴 ∧ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))) |
63 | 23, 56, 62 | sylanbrc 586 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐶) |
64 | 18, 19, 20, 21 | cdlemeiota 38242 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑔 ∈ 𝑇) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃))) |
65 | 15, 39, 16, 64 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃))) |
66 | 65 | eqcomd 2744 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔) |
67 | | eqeq2 2750 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑔‘𝑃) → ((𝑓‘𝑃) = 𝑠 ↔ (𝑓‘𝑃) = (𝑔‘𝑃))) |
68 | 67 | riotabidv 7129 |
. . . . . . . . . . . 12
⊢ (𝑠 = (𝑔‘𝑃) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃))) |
69 | 10, 68 | syl5eq 2785 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑔‘𝑃) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃))) |
70 | 69 | eqeq1d 2740 |
. . . . . . . . . 10
⊢ (𝑠 = (𝑔‘𝑃) → (𝐹 = 𝑔 ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔)) |
71 | 70 | rspcev 3526 |
. . . . . . . . 9
⊢ (((𝑔‘𝑃) ∈ 𝐶 ∧ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔) |
72 | 63, 66, 71 | syl2anc 587 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔) |
73 | 72 | ex 416 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)) |
74 | | breq1 5033 |
. . . . . . . . . . . . 13
⊢ (𝑟 = 𝑠 → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ 𝑠 ≤ (𝑃 ∨ 𝑉))) |
75 | | breq1 5033 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = 𝑠 → (𝑟 ≤ 𝑊 ↔ 𝑠 ≤ 𝑊)) |
76 | 75 | notbid 321 |
. . . . . . . . . . . . 13
⊢ (𝑟 = 𝑠 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑠 ≤ 𝑊)) |
77 | 74, 76 | anbi12d 634 |
. . . . . . . . . . . 12
⊢ (𝑟 = 𝑠 → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) |
78 | 77, 61 | elrab2 3591 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝐶 ↔ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) |
79 | | simpl1 1192 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
80 | | simpl2l 1227 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ∈ 𝐴) |
81 | | simpl2r 1228 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ¬ 𝑃 ≤ 𝑊) |
82 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ 𝐴) |
83 | | simprrr 782 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ¬ 𝑠 ≤ 𝑊) |
84 | 18, 19, 20, 21, 10 | ltrniotacl 38236 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
85 | 18, 19, 20, 21, 10 | ltrniotaval 38238 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑠) |
86 | 84, 85 | jca 515 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) |
87 | 79, 80, 81, 82, 83, 86 | syl122anc 1380 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) |
88 | | simp3l 1202 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → 𝐹 ∈ 𝑇) |
89 | | simp11 1204 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
90 | | simp12 1205 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
91 | | eqid 2738 |
. . . . . . . . . . . . . . . . 17
⊢
(meet‘𝐾) =
(meet‘𝐾) |
92 | 18, 31, 91, 19, 20, 21, 40 | trlval2 37820 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊)) |
93 | 89, 88, 90, 92 | syl3anc 1372 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊)) |
94 | | simp3r 1203 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐹‘𝑃) = 𝑠) |
95 | 94 | oveq2d 7186 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑠)) |
96 | 95 | oveq1d 7185 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) = ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊)) |
97 | 93, 96 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) = ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊)) |
98 | | simpl1l 1225 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝐾 ∈ HL) |
99 | | simpl3l 1229 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑉 ∈ 𝐴) |
100 | 18, 31, 19 | hlatlej1 37032 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑉)) |
101 | 98, 80, 99, 100 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ≤ (𝑃 ∨ 𝑉)) |
102 | | simprrl 781 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ≤ (𝑃 ∨ 𝑉)) |
103 | 98 | hllatd 37021 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝐾 ∈ Lat) |
104 | 80, 27 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾)) |
105 | 24, 19 | atbase 36946 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾)) |
106 | 105 | ad2antrl 728 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ (Base‘𝐾)) |
107 | 98, 80, 99, 35 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
108 | 24, 18, 31 | latjle12 17788 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑉) ∧ 𝑠 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉))) |
109 | 103, 104,
106, 107, 108 | syl13anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ≤ (𝑃 ∨ 𝑉) ∧ 𝑠 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉))) |
110 | 101, 102,
109 | mpbi2and 712 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉)) |
111 | 24, 31, 19 | hlatjcl 37024 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾)) |
112 | 98, 80, 82, 111 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾)) |
113 | | simpl1r 1226 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑊 ∈ 𝐻) |
114 | 24, 20 | lhpbase 37655 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
115 | 113, 114 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾)) |
116 | 24, 18, 91 | latmlem1 17807 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑠) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊))) |
117 | 103, 112,
107, 115, 116 | syl13anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊))) |
118 | 110, 117 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊)) |
119 | 18, 31, 91, 19, 20 | lhpat4N 37701 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉) |
120 | 119 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉) |
121 | 118, 120 | breqtrd 5056 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ 𝑉) |
122 | 121 | 3adant3 1133 |
. . . . . . . . . . . . . 14
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ 𝑉) |
123 | 97, 122 | eqbrtrd 5052 |
. . . . . . . . . . . . 13
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) ≤ 𝑉) |
124 | 88, 123 | jca 515 |
. . . . . . . . . . . 12
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)) |
125 | 87, 124 | mpd3an3 1463 |
. . . . . . . . . . 11
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)) |
126 | 78, 125 | sylan2b 597 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)) |
127 | 126 | ex 416 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))) |
128 | | eleq1 2820 |
. . . . . . . . . . 11
⊢ (𝐹 = 𝑔 → (𝐹 ∈ 𝑇 ↔ 𝑔 ∈ 𝑇)) |
129 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝐹 = 𝑔 → (𝑅‘𝐹) = (𝑅‘𝑔)) |
130 | 129 | breq1d 5040 |
. . . . . . . . . . 11
⊢ (𝐹 = 𝑔 → ((𝑅‘𝐹) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉)) |
131 | 128, 130 | anbi12d 634 |
. . . . . . . . . 10
⊢ (𝐹 = 𝑔 → ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))) |
132 | 131 | biimpcd 252 |
. . . . . . . . 9
⊢ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉) → (𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))) |
133 | 127, 132 | syl6 35 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))) |
134 | 133 | rexlimdv 3193 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))) |
135 | 73, 134 | impbid 215 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) ↔ ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)) |
136 | 14, 135 | bitr4d 285 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))) |
137 | | fveq2 6674 |
. . . . . . 7
⊢ (𝑓 = 𝑔 → (𝑅‘𝑓) = (𝑅‘𝑔)) |
138 | 137 | breq1d 5040 |
. . . . . 6
⊢ (𝑓 = 𝑔 → ((𝑅‘𝑓) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉)) |
139 | 138 | elrab 3588 |
. . . . 5
⊢ (𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉} ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) |
140 | 136, 139 | bitr4di 292 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ 𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})) |
141 | | simp1l 1198 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ HL) |
142 | | simp1r 1199 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑊 ∈ 𝐻) |
143 | | simp3l 1202 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ 𝐴) |
144 | 143, 46 | syl 17 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ (Base‘𝐾)) |
145 | | simp3r 1203 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ≤ 𝑊) |
146 | | cdlemm10.i |
. . . . . . 7
⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
147 | 24, 18, 20, 21, 40, 146 | diaval 38689 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉}) |
148 | 141, 142,
144, 145, 147 | syl22anc 838 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉}) |
149 | 148 | eleq2d 2818 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ (𝐼‘𝑉) ↔ 𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})) |
150 | 140, 149 | bitr4d 285 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ 𝑔 ∈ (𝐼‘𝑉))) |
151 | 5, 150 | syl5bb 286 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ ran 𝐺 ↔ 𝑔 ∈ (𝐼‘𝑉))) |
152 | 151 | eqrdv 2736 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉)) |