Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme35.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme35.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme35.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme35.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme35.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme35.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
cdleme35.f |
⊢ 𝐹 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) ) |
8 |
|
cdleme35.g |
⊢ 𝐺 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
9 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑅 ≠ 𝑆 ) |
10 |
|
simpl1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) |
11 |
|
simpl2 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) |
12 |
|
simpl31 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) |
13 |
|
simpl32 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
14 |
|
simpr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → 𝐹 = 𝐺 ) |
15 |
1 2 3 4 5 6 7 8
|
cdleme35h |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ) → 𝑅 = 𝑆 ) |
16 |
10 11 12 13 14 15
|
syl113anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝐺 ) → 𝑅 = 𝑆 ) |
17 |
16
|
ex |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( 𝐹 = 𝐺 → 𝑅 = 𝑆 ) ) |
18 |
17
|
necon3d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( 𝑅 ≠ 𝑆 → 𝐹 ≠ 𝐺 ) ) |
19 |
9 18
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝐹 ≠ 𝐺 ) |