Step |
Hyp |
Ref |
Expression |
1 |
|
llni2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
llni2.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
llni2.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
4 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 ) |
5 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 ) |
6 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ≠ 𝑄 ) |
7 |
|
eqidd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑄 ) ) |
8 |
|
neeq1 |
⊢ ( 𝑟 = 𝑃 → ( 𝑟 ≠ 𝑠 ↔ 𝑃 ≠ 𝑠 ) ) |
9 |
|
oveq1 |
⊢ ( 𝑟 = 𝑃 → ( 𝑟 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑠 ) ) |
10 |
9
|
eqeq2d |
⊢ ( 𝑟 = 𝑃 → ( ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ↔ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑠 ) ) ) |
11 |
8 10
|
anbi12d |
⊢ ( 𝑟 = 𝑃 → ( ( 𝑟 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ) ↔ ( 𝑃 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑠 ) ) ) ) |
12 |
|
neeq2 |
⊢ ( 𝑠 = 𝑄 → ( 𝑃 ≠ 𝑠 ↔ 𝑃 ≠ 𝑄 ) ) |
13 |
|
oveq2 |
⊢ ( 𝑠 = 𝑄 → ( 𝑃 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑄 ) ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑠 = 𝑄 → ( ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑠 ) ↔ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑄 ) ) ) |
15 |
12 14
|
anbi12d |
⊢ ( 𝑠 = 𝑄 → ( ( 𝑃 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑠 ) ) ↔ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑄 ) ) ) ) |
16 |
11 15
|
rspc2ev |
⊢ ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( 𝑟 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ) ) |
17 |
4 5 6 7 16
|
syl112anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( 𝑟 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ) ) |
18 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ HL ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
20 |
19 1 2
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
19 1 2 3
|
islln3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∈ 𝑁 ↔ ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( 𝑟 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ) ) ) |
23 |
18 21 22
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑄 ) ∈ 𝑁 ↔ ∃ 𝑟 ∈ 𝐴 ∃ 𝑠 ∈ 𝐴 ( 𝑟 ≠ 𝑠 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑟 ∨ 𝑠 ) ) ) ) |
24 |
17 23
|
mpbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝑁 ) |