Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme17.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme17.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme17.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme17.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme17.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme17.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
cdleme17.f |
⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
8 |
|
cdleme17.g |
⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
9 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
10 |
1 2 3 4 5 6 7 8 9
|
cdleme17a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
11 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL ) |
12 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑊 ∈ 𝐻 ) |
13 |
|
simp21l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
14 |
|
simp21r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑃 ≤ 𝑊 ) |
15 |
|
simp22 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
16 |
|
simp23l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ∈ 𝐴 ) |
17 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
18 |
1 2 3 4 5 6 7 8 9
|
cdleme17c |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = 𝑄 ) |
19 |
11 12 13 14 15 16 17 18
|
syl223anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = 𝑄 ) |
20 |
10 19
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = 𝑄 ) |