Proof of Theorem pmod2iN
| Step | Hyp | Ref
| Expression |
| 1 | | incom 4209 |
. . . . . 6
⊢ (𝑋 ∩ 𝑌) = (𝑌 ∩ 𝑋) |
| 2 | 1 | oveq1i 7441 |
. . . . 5
⊢ ((𝑋 ∩ 𝑌) + 𝑍) = ((𝑌 ∩ 𝑋) + 𝑍) |
| 3 | | hllat 39364 |
. . . . . . 7
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
| 4 | 3 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → 𝐾 ∈ Lat) |
| 5 | | simp22 1208 |
. . . . . . 7
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → 𝑌 ⊆ 𝐴) |
| 6 | | ssinss1 4246 |
. . . . . . 7
⊢ (𝑌 ⊆ 𝐴 → (𝑌 ∩ 𝑋) ⊆ 𝐴) |
| 7 | 5, 6 | syl 17 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → (𝑌 ∩ 𝑋) ⊆ 𝐴) |
| 8 | | simp23 1209 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝐴) |
| 9 | | pmod.a |
. . . . . . 7
⊢ 𝐴 = (Atoms‘𝐾) |
| 10 | | pmod.p |
. . . . . . 7
⊢ + =
(+𝑃‘𝐾) |
| 11 | 9, 10 | paddcom 39815 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ (𝑌 ∩ 𝑋) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑌 ∩ 𝑋) + 𝑍) = (𝑍 + (𝑌 ∩ 𝑋))) |
| 12 | 4, 7, 8, 11 | syl3anc 1373 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑌 ∩ 𝑋) + 𝑍) = (𝑍 + (𝑌 ∩ 𝑋))) |
| 13 | 2, 12 | eqtrid 2789 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑋 ∩ 𝑌) + 𝑍) = (𝑍 + (𝑌 ∩ 𝑋))) |
| 14 | | simp21 1207 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ 𝑆) |
| 15 | 8, 5, 14 | 3jca 1129 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → (𝑍 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆)) |
| 16 | | pmod.s |
. . . . . . 7
⊢ 𝑆 = (PSubSp‘𝐾) |
| 17 | 9, 16, 10 | pmod1i 39850 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ (𝑍 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆)) → (𝑍 ⊆ 𝑋 → ((𝑍 + 𝑌) ∩ 𝑋) = (𝑍 + (𝑌 ∩ 𝑋)))) |
| 18 | 17 | 3impia 1118 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑍 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ 𝑍 ⊆ 𝑋) → ((𝑍 + 𝑌) ∩ 𝑋) = (𝑍 + (𝑌 ∩ 𝑋))) |
| 19 | 15, 18 | syld3an2 1413 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑍 + 𝑌) ∩ 𝑋) = (𝑍 + (𝑌 ∩ 𝑋))) |
| 20 | 9, 10 | paddcom 39815 |
. . . . . 6
⊢ ((𝐾 ∈ Lat ∧ 𝑍 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑍 + 𝑌) = (𝑌 + 𝑍)) |
| 21 | 4, 8, 5, 20 | syl3anc 1373 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → (𝑍 + 𝑌) = (𝑌 + 𝑍)) |
| 22 | 21 | ineq1d 4219 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑍 + 𝑌) ∩ 𝑋) = ((𝑌 + 𝑍) ∩ 𝑋)) |
| 23 | 13, 19, 22 | 3eqtr2d 2783 |
. . 3
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑋 ∩ 𝑌) + 𝑍) = ((𝑌 + 𝑍) ∩ 𝑋)) |
| 24 | | incom 4209 |
. . 3
⊢ ((𝑌 + 𝑍) ∩ 𝑋) = (𝑋 ∩ (𝑌 + 𝑍)) |
| 25 | 23, 24 | eqtrdi 2793 |
. 2
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ 𝑍 ⊆ 𝑋) → ((𝑋 ∩ 𝑌) + 𝑍) = (𝑋 ∩ (𝑌 + 𝑍))) |
| 26 | 25 | 3expia 1122 |
1
⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑍 ⊆ 𝑋 → ((𝑋 ∩ 𝑌) + 𝑍) = (𝑋 ∩ (𝑌 + 𝑍)))) |