Proof of Theorem cdlemk19u
Step | Hyp | Ref
| Expression |
1 | | simp1l 1198 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simp1 1137 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) |
3 | | simp2l 1200 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
4 | | simp2r 1201 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑁 ∈ 𝑇) |
5 | | simp3 1139 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
6 | | cdlemk5.b |
. . . 4
⊢ 𝐵 = (Base‘𝐾) |
7 | | cdlemk5.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
8 | | cdlemk5.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
9 | | cdlemk5.m |
. . . 4
⊢ ∧ =
(meet‘𝐾) |
10 | | cdlemk5.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
11 | | cdlemk5.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
12 | | cdlemk5.t |
. . . 4
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
13 | | cdlemk5.r |
. . . 4
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
14 | | cdlemk5.z |
. . . 4
⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) |
15 | | cdlemk5.y |
. . . 4
⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) |
16 | | cdlemk5.x |
. . . 4
⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) |
17 | | cdlemk5.u |
. . . 4
⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) |
18 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | cdlemk35u 38590 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) ∈ 𝑇) |
19 | 2, 3, 4, 3, 5, 18 | syl131anc 1384 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) ∈ 𝑇) |
20 | | simpr 488 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → 𝐹 = 𝑁) |
21 | | simpl2l 1227 |
. . . . . 6
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → 𝐹 ∈ 𝑇) |
22 | 16, 17 | cdlemk40t 38544 |
. . . . . 6
⊢ ((𝐹 = 𝑁 ∧ 𝐹 ∈ 𝑇) → (𝑈‘𝐹) = 𝐹) |
23 | 20, 21, 22 | syl2anc 587 |
. . . . 5
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → (𝑈‘𝐹) = 𝐹) |
24 | 23 | fveq1d 6670 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝐹‘𝑃)) |
25 | | fveq1 6667 |
. . . . 5
⊢ (𝐹 = 𝑁 → (𝐹‘𝑃) = (𝑁‘𝑃)) |
26 | 25 | adantl 485 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → (𝐹‘𝑃) = (𝑁‘𝑃)) |
27 | 24, 26 | eqtrd 2773 |
. . 3
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) |
28 | | simpl1 1192 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) |
29 | | simpl2l 1227 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝐹 ∈ 𝑇) |
30 | | simpr 488 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝐹 ≠ 𝑁) |
31 | | simpl2r 1228 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝑁 ∈ 𝑇) |
32 | | simpl3 1194 |
. . . 4
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
33 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | cdlemk19u1 38595 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ 𝑁 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) |
34 | 28, 29, 30, 31, 32, 33 | syl131anc 1384 |
. . 3
⊢
(((((𝐾 ∈ HL
∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) |
35 | 27, 34 | pm2.61dane 3021 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) |
36 | 7, 10, 11, 12 | cdlemd 37833 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈‘𝐹) ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) → (𝑈‘𝐹) = 𝑁) |
37 | 1, 19, 4, 5, 35, 36 | syl311anc 1385 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) = 𝑁) |