Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem5.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem5.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem5.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihmeetlem5.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihmeetlem5.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
7 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
8 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
9 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 ) |
10 |
|
simprr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ≤ 𝑋 ) |
11 |
1 2 3 4 5
|
atmod2i1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑄 ≤ 𝑋 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) = ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ) |
12 |
6 7 8 9 10 11
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) = ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ) |
13 |
12
|
eqcomd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |