Proof of Theorem cdlemg6c
Step | Hyp | Ref
| Expression |
1 | | simpl1 1064 |
. . 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 794 |
. . 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 1140 |
. . 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 1141 |
. . 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 1142 |
. . 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 796 |
. . . 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 1112 |
. . . . . 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 1180 |
. . . . . . 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 481 |
. . . . . 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 802 |
. . . . . 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 2622 |
. . . . . . . . 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 35451 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾)) |
17 | 1, 5, 16 | syl2anc 693 |
. . . . . . 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 | syl5eqel 2705 |
. . . . . 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 1181 |
. . . . . . . 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 481 |
. . . . . . 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 35471 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊) |
23 | 1, 5, 22 | syl2anc 693 |
. . . . . . . . 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 | syl5eqbr 4688 |
. . . . . . . 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 1085 |
. . . . . . . . . . 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 | | hllat 34650 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
27 | 25, 26 | 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)) → 𝐾 ∈ Lat) |
28 | 27 | adantr 481 |
. . . . . . . . 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) |
29 | | cdlemg4.a |
. . . . . . . . . . . 12
⊢ 𝐴 = (Atoms‘𝐾) |
30 | 12, 29 | atbase 34576 |
. . . . . . . . . . 11
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
31 | 8, 30 | 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)) |
32 | 31 | adantr 481 |
. . . . . . . . 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)) |
33 | | simp1r 1086 |
. . . . . . . . . . 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) |
34 | 12, 13 | lhpbase 35284 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
35 | 33, 34 | 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)) |
36 | 35 | adantr 481 |
. . . . . . . . 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)) |
37 | 12, 21 | lattr 17056 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊) → 𝑄 ≤ 𝑊)) |
38 | 28, 32, 18, 36, 37 | syl13anc 1328 |
. . . . . . . 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)) |
39 | 24, 38 | mpan2d 710 |
. . . . . . 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)) |
40 | 20, 39 | mtod 189 |
. . . . . 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) |
41 | | cdlemg4.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
42 | 12, 21, 41, 29 | hlexch2 34669 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑉 ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ 𝑉) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑄 ∨ 𝑉))) |
43 | 7, 9, 10, 18, 40, 42 | syl131anc 1339 |
. . . . 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))) |
44 | | simpl32 1143 |
. . . . . . 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)) |
45 | | simp21l 1178 |
. . . . . . . . . 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) |
46 | 45 | adantr 481 |
. . . . . . . . 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) |
47 | 12, 29 | atbase 34576 |
. . . . . . . . 9
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
48 | 46, 47 | 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)) |
49 | 12, 21, 41 | latlej2 17061 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉)) |
50 | 28, 48, 18, 49 | syl3anc 1326 |
. . . . . . 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)) |
51 | 12, 41 | latjcl 17051 |
. . . . . . . . 9
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾)) |
52 | 28, 48, 18, 51 | syl3anc 1326 |
. . . . . . . 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)) |
53 | 12, 21, 41 | latjle12 17062 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉))) |
54 | 28, 32, 18, 52, 53 | syl13anc 1328 |
. . . . . . 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))) |
55 | 44, 50, 54 | mpbi2and 956 |
. . . . . 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)) |
56 | 12, 29 | atbase 34576 |
. . . . . . . 8
⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾)) |
57 | 10, 56 | 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)) |
58 | 12, 41 | latjcl 17051 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾)) |
59 | 28, 32, 18, 58 | syl3anc 1326 |
. . . . . . 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)) |
60 | 12, 21 | lattr 17056 |
. . . . . . 7
⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑟 ≤ (𝑄 ∨ 𝑉) ∧ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) → 𝑟 ≤ (𝑃 ∨ 𝑉))) |
61 | 28, 57, 59, 52, 60 | syl13anc 1328 |
. . . . . 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))) |
62 | 55, 61 | mpan2d 710 |
. . . . 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))) |
63 | 43, 62 | 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))) |
64 | 6, 63 | mtod 189 |
. . 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)) |
65 | | simpl21 1139 |
. . . 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)) |
66 | | simpl33 1144 |
. . . 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) |
67 | 21, 29, 13, 14, 15, 41, 11 | cdlemg6a 35906 |
. . . 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) |
68 | 1, 65, 2, 4, 5, 6, 66, 67 | syl133anc 1349 |
. . 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) |
69 | 21, 29, 13, 14, 15, 41, 11 | cdlemg6b 35907 |
. . 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) |
70 | 1, 2, 3, 4, 5, 64,
68, 69 | syl133anc 1349 |
. 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) |
71 | 70 | ex 450 |
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)) |