| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝐾 ∈ HL) |
| 2 | | eqid 2737 |
. . . . 5
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) |
| 3 | | paddidm.s |
. . . . 5
⊢ 𝑆 = (PSubSp‘𝐾) |
| 4 | 2, 3 | psubssat 39756 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
| 5 | 4 | 3adant3 1133 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾)) |
| 6 | 2, 3 | psubssat 39756 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
| 7 | 6 | 3adant2 1132 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾)) |
| 8 | | paddidm.p |
. . . 4
⊢ + =
(+𝑃‘𝐾) |
| 9 | 2, 8 | paddssat 39816 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) |
| 10 | 1, 5, 7, 9 | syl3anc 1373 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) |
| 11 | | olc 869 |
. . . . 5
⊢ ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))) |
| 12 | | eqid 2737 |
. . . . . . . 8
⊢
(le‘𝐾) =
(le‘𝐾) |
| 13 | | eqid 2737 |
. . . . . . . 8
⊢
(join‘𝐾) =
(join‘𝐾) |
| 14 | 12, 13, 2, 8 | elpadd 39801 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))) |
| 15 | 1, 10, 10, 14 | syl3anc 1373 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))) |
| 16 | 2, 8 | padd4N 39842 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌))) |
| 17 | 1, 5, 7, 5, 7, 16 | syl122anc 1381 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌))) |
| 18 | 3, 8 | paddidm 39843 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋) |
| 19 | 18 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋) |
| 20 | 3, 8 | paddidm 39843 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌) |
| 21 | 20 | 3adant2 1132 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑌 + 𝑌) = 𝑌) |
| 22 | 19, 21 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = (𝑋 + 𝑌)) |
| 23 | 17, 22 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = (𝑋 + 𝑌)) |
| 24 | 23 | eleq2d 2827 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ 𝑝 ∈ (𝑋 + 𝑌))) |
| 25 | 15, 24 | bitr3d 281 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) ↔ 𝑝 ∈ (𝑋 + 𝑌))) |
| 26 | 11, 25 | imbitrid 244 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ (𝑋 + 𝑌))) |
| 27 | 26 | expd 415 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))) |
| 28 | 27 | ralrimiv 3145 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))) |
| 29 | 12, 13, 2, 3 | ispsubsp2 39748 |
. . 3
⊢ (𝐾 ∈ HL → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))))) |
| 30 | 29 | 3ad2ant1 1134 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))))) |
| 31 | 10, 28, 30 | mpbir2and 713 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |