Proof of Theorem trljat1
| Step | Hyp | Ref
| Expression |
| 1 | | trljat.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
| 2 | | trljat.j |
. . . 4
⊢ ∨ =
(join‘𝐾) |
| 3 | | eqid 2737 |
. . . 4
⊢
(meet‘𝐾) =
(meet‘𝐾) |
| 4 | | trljat.a |
. . . 4
⊢ 𝐴 = (Atoms‘𝐾) |
| 5 | | trljat.h |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
| 6 | | trljat.t |
. . . 4
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| 7 | | trljat.r |
. . . 4
⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
| 8 | 1, 2, 3, 4, 5, 6, 7 | trlval2 40165 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊)) |
| 9 | 8 | oveq1d 7446 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘𝐹) ∨ 𝑃) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃)) |
| 10 | | simp1l 1198 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL) |
| 11 | 10 | hllatd 39365 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat) |
| 12 | | simp3l 1202 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) |
| 13 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝐾) =
(Base‘𝐾) |
| 14 | 13, 4 | atbase 39290 |
. . . 4
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
| 15 | 12, 14 | syl 17 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾)) |
| 16 | 13, 5, 6, 7 | trlcl 40166 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
| 17 | 16 | 3adant3 1133 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) ∈ (Base‘𝐾)) |
| 18 | 13, 2 | latjcom 18492 |
. . 3
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃)) |
| 19 | 11, 15, 17, 18 | syl3anc 1373 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ 𝑃)) |
| 20 | 13, 5, 6 | ltrncl 40127 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
| 21 | 15, 20 | syld3an3 1411 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾)) |
| 22 | 13, 2 | latjcl 18484 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 23 | 11, 15, 21, 22 | syl3anc 1373 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) |
| 24 | | simp1r 1199 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻) |
| 25 | 13, 5 | lhpbase 40000 |
. . . . 5
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
| 26 | 24, 25 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
| 27 | 13, 1, 2 | latlej1 18493 |
. . . . 5
⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
| 28 | 11, 15, 21, 27 | syl3anc 1373 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) |
| 29 | 13, 1, 2, 3, 4 | atmod2i1 39863 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃))) |
| 30 | 10, 12, 23, 26, 28, 29 | syl131anc 1385 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃))) |
| 31 | | eqid 2737 |
. . . . . 6
⊢
(1.‘𝐾) =
(1.‘𝐾) |
| 32 | 1, 2, 31, 4, 5 | lhpjat1 40022 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾)) |
| 33 | 32 | 3adant2 1132 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑊 ∨ 𝑃) = (1.‘𝐾)) |
| 34 | 33 | oveq2d 7447 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(𝑊 ∨ 𝑃)) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾))) |
| 35 | | hlol 39362 |
. . . . 5
⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
| 36 | 10, 35 | syl 17 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ OL) |
| 37 | 13, 3, 31 | olm11 39228 |
. . . 4
⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
| 38 | 36, 23, 37 | syl2anc 584 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)(1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃))) |
| 39 | 30, 34, 38 | 3eqtrrd 2782 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) = (((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) ∨ 𝑃)) |
| 40 | 9, 19, 39 | 3eqtr4d 2787 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃))) |