| Step | Hyp | Ref
| Expression |
| 1 | | idltrn.b |
. . 3
⊢ 𝐵 = (Base‘𝐾) |
| 2 | | idltrn.h |
. . 3
⊢ 𝐻 = (LHyp‘𝐾) |
| 3 | | eqid 2737 |
. . 3
⊢
((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊) |
| 4 | 1, 2, 3 | idldil 40116 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊)) |
| 5 | | simpll 767 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| 6 | | simplrr 778 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾)) |
| 7 | | simprr 773 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊) |
| 8 | | eqid 2737 |
. . . . . . 7
⊢
(le‘𝐾) =
(le‘𝐾) |
| 9 | | eqid 2737 |
. . . . . . 7
⊢
(meet‘𝐾) =
(meet‘𝐾) |
| 10 | | eqid 2737 |
. . . . . . 7
⊢
(0.‘𝐾) =
(0.‘𝐾) |
| 11 | | eqid 2737 |
. . . . . . 7
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) |
| 12 | 8, 9, 10, 11, 2 | lhpmat 40032 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾)) |
| 13 | 5, 6, 7, 12 | syl12anc 837 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾)) |
| 14 | 1, 11 | atbase 39290 |
. . . . . . . . 9
⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵) |
| 15 | | fvresi 7193 |
. . . . . . . . 9
⊢ (𝑞 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑞) = 𝑞) |
| 16 | 6, 14, 15 | 3syl 18 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑞) = 𝑞) |
| 17 | 16 | oveq2d 7447 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = (𝑞(join‘𝐾)𝑞)) |
| 18 | | simplll 775 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL) |
| 19 | | eqid 2737 |
. . . . . . . . 9
⊢
(join‘𝐾) =
(join‘𝐾) |
| 20 | 19, 11 | hlatjidm 39370 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑞(join‘𝐾)𝑞) = 𝑞) |
| 21 | 18, 6, 20 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)𝑞) = 𝑞) |
| 22 | 17, 21 | eqtrd 2777 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = 𝑞) |
| 23 | 22 | oveq1d 7446 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊) = (𝑞(meet‘𝐾)𝑊)) |
| 24 | | simplrl 777 |
. . . . . . . . . 10
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾)) |
| 25 | 1, 11 | atbase 39290 |
. . . . . . . . . 10
⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵) |
| 26 | | fvresi 7193 |
. . . . . . . . . 10
⊢ (𝑝 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝) |
| 27 | 24, 25, 26 | 3syl 18 |
. . . . . . . . 9
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑝) = 𝑝) |
| 28 | 27 | oveq2d 7447 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = (𝑝(join‘𝐾)𝑝)) |
| 29 | 19, 11 | hlatjidm 39370 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝) |
| 30 | 18, 24, 29 | syl2anc 584 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)𝑝) = 𝑝) |
| 31 | 28, 30 | eqtrd 2777 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = 𝑝) |
| 32 | 31 | oveq1d 7446 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (𝑝(meet‘𝐾)𝑊)) |
| 33 | | simprl 771 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊) |
| 34 | 8, 9, 10, 11, 2 | lhpmat 40032 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾)) |
| 35 | 5, 24, 33, 34 | syl12anc 837 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾)) |
| 36 | 32, 35 | eqtrd 2777 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (0.‘𝐾)) |
| 37 | 13, 23, 36 | 3eqtr4rd 2788 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)) |
| 38 | 37 | ex 412 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊))) |
| 39 | 38 | ralrimivva 3202 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊))) |
| 40 | | idltrn.t |
. . 3
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| 41 | 8, 19, 9, 11, 2, 3,
40 | isltrn 40121 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝐵) ∈ 𝑇 ↔ (( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊))))) |
| 42 | 4, 39, 41 | mpbir2and 713 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |