Proof of Theorem pl42lem3N
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl1 1191 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → 𝐾 ∈ HL) | 
| 2 |  | simpl2 1192 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | 
| 3 |  | pl42lem.b | . . . . . 6
⊢ 𝐵 = (Base‘𝐾) | 
| 4 |  | eqid 2736 | . . . . . 6
⊢
(Atoms‘𝐾) =
(Atoms‘𝐾) | 
| 5 |  | pl42lem.f | . . . . . 6
⊢ 𝐹 = (pmap‘𝐾) | 
| 6 | 3, 4, 5 | pmapssat 39762 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) | 
| 7 | 1, 2, 6 | syl2anc 584 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (𝐹‘𝑋) ⊆ (Atoms‘𝐾)) | 
| 8 |  | simpl3 1193 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | 
| 9 | 3, 4, 5 | pmapssat 39762 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) | 
| 10 | 1, 8, 9 | syl2anc 584 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) | 
| 11 |  | pl42lem.p | . . . . 5
⊢  + =
(+𝑃‘𝐾) | 
| 12 | 4, 11 | paddssat 39817 | . . . 4
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) | 
| 13 | 1, 7, 10, 12 | syl3anc 1372 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) | 
| 14 |  | simpr2 1195 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → 𝑊 ∈ 𝐵) | 
| 15 | 3, 4, 5 | pmapssat 39762 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵) → (𝐹‘𝑊) ⊆ (Atoms‘𝐾)) | 
| 16 | 1, 14, 15 | syl2anc 584 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (𝐹‘𝑊) ⊆ (Atoms‘𝐾)) | 
| 17 |  | inss1 4236 | . . . 4
⊢ (((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) ⊆ ((𝐹‘𝑋) + (𝐹‘𝑌)) | 
| 18 | 4, 11 | paddss1 39820 | . . . 4
⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑊) ⊆ (Atoms‘𝐾)) → ((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) ⊆ ((𝐹‘𝑋) + (𝐹‘𝑌)) → ((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)))) | 
| 19 | 17, 18 | mpi 20 | . . 3
⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹‘𝑊) ⊆ (Atoms‘𝐾)) → ((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊))) | 
| 20 | 1, 13, 16, 19 | syl3anc 1372 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊))) | 
| 21 |  | simpr3 1196 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → 𝑉 ∈ 𝐵) | 
| 22 | 3, 4, 5 | pmapssat 39762 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐵) → (𝐹‘𝑉) ⊆ (Atoms‘𝐾)) | 
| 23 | 1, 21, 22 | syl2anc 584 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (𝐹‘𝑉) ⊆ (Atoms‘𝐾)) | 
| 24 | 4, 11 | sspadd2 39819 | . . 3
⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹‘𝑋) + (𝐹‘𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹‘𝑉) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉))) | 
| 25 | 1, 23, 13, 24 | syl3anc 1372 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (𝐹‘𝑉) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉))) | 
| 26 |  | ss2in 4244 | . 2
⊢
((((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)) ∧ (𝐹‘𝑉) ⊆ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉))) → (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉)) ⊆ ((((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)) ∩ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉)))) | 
| 27 | 20, 25, 26 | syl2anc 584 | 1
⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉)) ⊆ ((((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)) ∩ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉)))) |