Step |
Hyp |
Ref |
Expression |
1 |
|
4thatlem.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
2 |
|
4thatlem0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
4thatlem0.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
4thatlem0.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
4thatlem0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
4thatlem0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
4thatlem0.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
4thatlem0.v |
⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
9 |
|
4thatlem0.c |
⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) |
10 |
|
4thatlem0.d |
⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) |
11 |
1 2 3 5 7
|
4atexlemswapqr |
⊢ ( 𝜑 → ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ) |
12 |
1 2 3 4 5 6 7 8 9 10
|
4atexlemcnd |
⊢ ( 𝜑 → 𝐶 ≠ 𝐷 ) |
13 |
|
pm13.18 |
⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝑆 ≠ 𝐷 ) |
14 |
13
|
necomd |
⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝐷 ≠ 𝑆 ) |
15 |
14
|
expcom |
⊢ ( 𝐶 ≠ 𝐷 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) ) |
16 |
12 15
|
syl |
⊢ ( 𝜑 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) ) |
17 |
|
biid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ) |
18 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) |
19 |
17 2 3 4 5 6 18 8 10
|
4atexlemex2 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ∧ 𝐷 ≠ 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) |
20 |
11 16 19
|
syl6an |
⊢ ( 𝜑 → ( 𝐶 = 𝑆 → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) |
21 |
20
|
imp |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) |