Proof of Theorem cdlemg6c
Step | Hyp | Ref
| Expression |
1 | | simpl1 1192 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simprl 771 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊)) |
3 | | simpl22 1253 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
4 | | simpl23 1254 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐹 ∈ 𝑇) |
5 | | simpl31 1255 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐺 ∈ 𝑇) |
6 | | simprr 773 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑟 ≤ (𝑃 ∨ 𝑉)) |
7 | | simpl1l 1225 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐾 ∈ HL) |
8 | | simp22l 1293 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑄 ∈ 𝐴) |
9 | 8 | adantr 484 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ∈ 𝐴) |
10 | | simprll 779 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑟 ∈ 𝐴) |
11 | | cdlemg4b.v |
. . . . . . 7
⊢ 𝑉 = (𝑅‘𝐺) |
12 | | eqid 2738 |
. . . . . . . . 9
⊢
(Base‘𝐾) =
(Base‘𝐾) |
13 | | cdlemg4.h |
. . . . . . . . 9
⊢ 𝐻 = (LHyp‘𝐾) |
14 | | cdlemg4.t |
. . . . . . . . 9
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
15 | | cdlemg4.r |
. . . . . . . . 9
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
16 | 12, 13, 14, 15 | trlcl 37821 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
17 | 1, 5, 16 | syl2anc 587 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
18 | 11, 17 | eqeltrid 2837 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ∈ (Base‘𝐾)) |
19 | | simp22r 1294 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ¬ 𝑄 ≤ 𝑊) |
20 | 19 | adantr 484 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ 𝑊) |
21 | | cdlemg4.l |
. . . . . . . . . . 11
⊢ ≤ =
(le‘𝐾) |
22 | 21, 13, 14, 15 | trlle 37841 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊) |
23 | 1, 5, 22 | syl2anc 587 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑅‘𝐺) ≤ 𝑊) |
24 | 11, 23 | eqbrtrid 5065 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ≤ 𝑊) |
25 | | simp1l 1198 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐾 ∈ HL) |
26 | 25 | hllatd 37021 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐾 ∈ Lat) |
27 | 26 | adantr 484 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐾 ∈ Lat) |
28 | | cdlemg4.a |
. . . . . . . . . . . 12
⊢ 𝐴 = (Atoms‘𝐾) |
29 | 12, 28 | atbase 36946 |
. . . . . . . . . . 11
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
30 | 8, 29 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑄 ∈ (Base‘𝐾)) |
31 | 30 | adantr 484 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ∈ (Base‘𝐾)) |
32 | | simp1r 1199 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑊 ∈ 𝐻) |
33 | 12, 13 | lhpbase 37655 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
34 | 32, 33 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑊 ∈ (Base‘𝐾)) |
35 | 34 | adantr 484 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑊 ∈ (Base‘𝐾)) |
36 | 12, 21 | lattr 17782 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊) → 𝑄 ≤ 𝑊)) |
37 | 27, 31, 18, 35, 36 | syl13anc 1373 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊) → 𝑄 ≤ 𝑊)) |
38 | 24, 37 | mpan2d 694 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ 𝑉 → 𝑄 ≤ 𝑊)) |
39 | 20, 38 | mtod 201 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ 𝑉) |
40 | | cdlemg4.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
41 | 12, 21, 40, 28 | hlexch2 37040 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑉 ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ 𝑉) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑄 ∨ 𝑉))) |
42 | 7, 9, 10, 18, 39, 41 | syl131anc 1384 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑄 ∨ 𝑉))) |
43 | | simpl32 1256 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ≤ (𝑃 ∨ 𝑉)) |
44 | | simp21l 1291 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑃 ∈ 𝐴) |
45 | 44 | adantr 484 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑃 ∈ 𝐴) |
46 | 12, 28 | atbase 36946 |
. . . . . . . . 9
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
47 | 45, 46 | syl 17 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑃 ∈ (Base‘𝐾)) |
48 | 12, 21, 40 | latlej2 17787 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉)) |
49 | 27, 47, 18, 48 | syl3anc 1372 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ≤ (𝑃 ∨ 𝑉)) |
50 | 12, 40 | latjcl 17777 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
51 | 27, 47, 18, 50 | syl3anc 1372 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
52 | 12, 21, 40 | latjle12 17788 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉))) |
53 | 27, 31, 18, 51, 52 | syl13anc 1373 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉))) |
54 | 43, 49, 53 | mpbi2and 712 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) |
55 | 12, 28 | atbase 36946 |
. . . . . . . 8
⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾)) |
56 | 10, 55 | syl 17 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑟 ∈ (Base‘𝐾)) |
57 | 12, 40 | latjcl 17777 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾)) |
58 | 27, 31, 18, 57 | syl3anc 1372 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾)) |
59 | 12, 21 | lattr 17782 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑟 ≤ (𝑄 ∨ 𝑉) ∧ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) → 𝑟 ≤ (𝑃 ∨ 𝑉))) |
60 | 27, 56, 58, 51, 59 | syl13anc 1373 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑟 ≤ (𝑄 ∨ 𝑉) ∧ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) → 𝑟 ≤ (𝑃 ∨ 𝑉))) |
61 | 54, 60 | mpan2d 694 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑟 ≤ (𝑄 ∨ 𝑉) → 𝑟 ≤ (𝑃 ∨ 𝑉))) |
62 | 42, 61 | syld 47 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑃 ∨ 𝑉))) |
63 | 6, 62 | mtod 201 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ (𝑟 ∨ 𝑉)) |
64 | | simpl21 1252 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
65 | | simpl33 1257 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑃)) = 𝑃) |
66 | 21, 28, 13, 14, 15, 40, 11 | cdlemg6a 38275 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑟)) = 𝑟) |
67 | 1, 64, 2, 4, 5, 6, 65, 66 | syl133anc 1394 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑟)) = 𝑟) |
68 | 21, 28, 13, 14, 15, 40, 11 | cdlemg6b 38276 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑟 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑟)) = 𝑟)) → (𝐹‘(𝐺‘𝑄)) = 𝑄) |
69 | 1, 2, 3, 4, 5, 63,
67, 68 | syl133anc 1394 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑄)) = 𝑄) |
70 | 69 | ex 416 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉)) → (𝐹‘(𝐺‘𝑄)) = 𝑄)) |