Proof of Theorem cdlemd6
Step | Hyp | Ref
| Expression |
1 | | simp3 1140 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐹‘𝑃) = (𝐺‘𝑃)) |
2 | 1 | oveq2d 7229 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑃 ∨ (𝐺‘𝑃))) |
3 | 2 | oveq1d 7228 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
4 | | simp1l 1199 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
5 | | simp1rl 1240 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → 𝐹 ∈ 𝑇) |
6 | | simp21 1208 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
7 | | cdlemd4.l |
. . . . . . 7
⊢ ≤ =
(le‘𝐾) |
8 | | cdlemd4.j |
. . . . . . 7
⊢ ∨ =
(join‘𝐾) |
9 | | eqid 2737 |
. . . . . . 7
⊢
(meet‘𝐾) =
(meet‘𝐾) |
10 | | cdlemd4.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
11 | | cdlemd4.h |
. . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) |
12 | | cdlemd4.t |
. . . . . . 7
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
13 | | eqid 2737 |
. . . . . . 7
⊢
((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) |
14 | 7, 8, 9, 10, 11, 12, 13 | trlval2 37914 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊)) |
15 | 4, 5, 6, 14 | syl3anc 1373 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊)) |
16 | | simp1rr 1241 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → 𝐺 ∈ 𝑇) |
17 | 7, 8, 9, 10, 11, 12, 13 | trlval2 37914 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
18 | 4, 16, 6, 17 | syl3anc 1373 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
19 | 3, 15, 18 | 3eqtr4d 2787 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐹) = (((trL‘𝐾)‘𝑊)‘𝐺)) |
20 | 19 | oveq2d 7229 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐹)) = (𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐺))) |
21 | 1 | oveq1d 7228 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)) = ((𝐺‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊))) |
22 | 20, 21 | oveq12d 7231 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊))) = ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)))) |
23 | | simp22 1209 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
24 | | simp23 1210 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
25 | 7, 8, 9, 10, 11, 12, 13 | cdlemc 37948 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝐹‘𝑄) = ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)))) |
26 | 4, 5, 6, 23, 24, 25 | syl131anc 1385 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐹‘𝑄) = ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)))) |
27 | | oveq2 7221 |
. . . . . . 7
⊢ ((𝐹‘𝑃) = (𝐺‘𝑃) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑃 ∨ (𝐺‘𝑃))) |
28 | 27 | breq2d 5065 |
. . . . . 6
⊢ ((𝐹‘𝑃) = (𝐺‘𝑃) → (𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ↔ 𝑄 ≤ (𝑃 ∨ (𝐺‘𝑃)))) |
29 | 28 | notbid 321 |
. . . . 5
⊢ ((𝐹‘𝑃) = (𝐺‘𝑃) → (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ↔ ¬ 𝑄 ≤ (𝑃 ∨ (𝐺‘𝑃)))) |
30 | 29 | biimpd 232 |
. . . 4
⊢ ((𝐹‘𝑃) = (𝐺‘𝑃) → (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐺‘𝑃)))) |
31 | 1, 24, 30 | sylc 65 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐺‘𝑃))) |
32 | 7, 8, 9, 10, 11, 12, 13 | cdlemc 37948 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐺‘𝑃))) → (𝐺‘𝑄) = ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)))) |
33 | 4, 16, 6, 23, 31, 32 | syl131anc 1385 |
. 2
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐺‘𝑄) = ((𝑄 ∨ (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺‘𝑃) ∨ ((𝑃 ∨ 𝑄)(meet‘𝐾)𝑊)))) |
34 | 22, 26, 33 | 3eqtr4d 2787 |
1
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = (𝐺‘𝑃)) → (𝐹‘𝑄) = (𝐺‘𝑄)) |