Step |
Hyp |
Ref |
Expression |
1 |
|
oldmm1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
oldmm1.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
oldmm1.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
oldmm1.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
5 |
|
olop |
⊢ ( 𝐾 ∈ OL → 𝐾 ∈ OP ) |
6 |
1 4
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ) |
8 |
7
|
3adant3 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ) |
9 |
1 2 3 4
|
oldmm1 |
⊢ ( ( 𝐾 ∈ OL ∧ ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ 𝑌 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∨ ( ⊥ ‘ 𝑌 ) ) ) |
10 |
8 9
|
syld3an2 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ 𝑌 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∨ ( ⊥ ‘ 𝑌 ) ) ) |
11 |
1 4
|
opococ |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) |
12 |
5 11
|
sylan |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) |
13 |
12
|
3adant3 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∨ ( ⊥ ‘ 𝑌 ) ) = ( 𝑋 ∨ ( ⊥ ‘ 𝑌 ) ) ) |
15 |
10 14
|
eqtrd |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ 𝑌 ) ) = ( 𝑋 ∨ ( ⊥ ‘ 𝑌 ) ) ) |