Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg4.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
cdlemg4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
cdlemg4.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
cdlemg4.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
cdlemg4.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
7 |
|
cdlemg4b.v |
⊢ 𝑉 = ( 𝑅 ‘ 𝐺 ) |
8 |
|
cdlemg4.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
9 |
1 2 3 4 5 6 7 8
|
cdlemg4f |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = ( ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) |
10 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ HL ) |
11 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ 𝐻 ) |
12 |
|
simp21 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
13 |
|
simp22l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑄 ∈ 𝐴 ) |
14 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
15 |
1 6 8 2 3 14
|
cdleme0cp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
16 |
10 11 12 13 15
|
syl22anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
17 |
16
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) = ( ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) |
18 |
9 17
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = ( ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) |