Step |
Hyp |
Ref |
Expression |
1 |
|
atcvrne.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
atcvrne.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
3 |
|
atcvrne.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝐾 ∈ AtLat ) |
6 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ∈ 𝐴 ) |
7 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
8 |
7 3
|
atn0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) ) |
9 |
5 6 8
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) ) |
10 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝐾 ∈ HL ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
11 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
13 |
6 12
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
14 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ∈ 𝐴 ) |
15 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑅 ∈ 𝐴 ) |
16 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) |
17 |
11 1 7 2 3
|
atcvrj0 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 = ( 0. ‘ 𝐾 ) ↔ 𝑄 = 𝑅 ) ) |
18 |
10 13 14 15 16 17
|
syl131anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 = ( 0. ‘ 𝐾 ) ↔ 𝑄 = 𝑅 ) ) |
19 |
18
|
necon3bid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 ≠ ( 0. ‘ 𝐾 ) ↔ 𝑄 ≠ 𝑅 ) ) |
20 |
9 19
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ≠ 𝑅 ) |