Step |
Hyp |
Ref |
Expression |
1 |
|
isline.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
isline.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
isline.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
isline.n |
⊢ 𝑁 = ( Lines ‘ 𝐾 ) |
5 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → 𝑄 ∈ 𝐴 ) |
6 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → 𝑅 ∈ 𝐴 ) |
7 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) |
8 |
|
neeq1 |
⊢ ( 𝑞 = 𝑄 → ( 𝑞 ≠ 𝑟 ↔ 𝑄 ≠ 𝑟 ) ) |
9 |
|
oveq1 |
⊢ ( 𝑞 = 𝑄 → ( 𝑞 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) |
10 |
9
|
breq2d |
⊢ ( 𝑞 = 𝑄 → ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ↔ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑞 = 𝑄 → { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } ) |
12 |
11
|
eqeq2d |
⊢ ( 𝑞 = 𝑄 → ( 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ↔ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } ) ) |
13 |
8 12
|
anbi12d |
⊢ ( 𝑞 = 𝑄 → ( ( 𝑞 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ↔ ( 𝑄 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } ) ) ) |
14 |
|
neeq2 |
⊢ ( 𝑟 = 𝑅 → ( 𝑄 ≠ 𝑟 ↔ 𝑄 ≠ 𝑅 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑟 = 𝑅 → ( 𝑄 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑅 ) ) |
16 |
15
|
breq2d |
⊢ ( 𝑟 = 𝑅 → ( 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) ↔ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) ) ) |
17 |
16
|
rabbidv |
⊢ ( 𝑟 = 𝑅 → { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) |
18 |
17
|
eqeq2d |
⊢ ( 𝑟 = 𝑅 → ( 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } ↔ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) |
19 |
14 18
|
anbi12d |
⊢ ( 𝑟 = 𝑅 → ( ( 𝑄 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑟 ) } ) ↔ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) ) |
20 |
13 19
|
rspc2ev |
⊢ ( ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) |
21 |
5 6 7 20
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) |
22 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → 𝐾 ∈ 𝐷 ) |
23 |
1 2 3 4
|
isline |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) ) |
24 |
22 23
|
syl |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → ( 𝑋 ∈ 𝑁 ↔ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) ) |
25 |
21 24
|
mpbird |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑋 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑄 ∨ 𝑅 ) } ) ) → 𝑋 ∈ 𝑁 ) |