Step |
Hyp |
Ref |
Expression |
1 |
|
lineset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lineset.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
lineset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
lineset.n |
⊢ 𝑁 = ( Lines ‘ 𝐾 ) |
5 |
|
elex |
⊢ ( 𝐾 ∈ 𝐵 → 𝐾 ∈ V ) |
6 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) ) |
7 |
6 3
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 ) |
8 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) ) |
9 |
8 1
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ ) |
10 |
9
|
breqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) ↔ 𝑝 ≤ ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ( join ‘ 𝐾 ) ) |
12 |
11 2
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ∨ ) |
13 |
12
|
oveqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) = ( 𝑞 ∨ 𝑟 ) ) |
14 |
13
|
breq2d |
⊢ ( 𝑘 = 𝐾 → ( 𝑝 ≤ ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) ↔ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ) ) |
15 |
10 14
|
bitrd |
⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) ↔ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) ) ) |
16 |
7 15
|
rabeqbidv |
⊢ ( 𝑘 = 𝐾 → { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) |
17 |
16
|
eqeq2d |
⊢ ( 𝑘 = 𝐾 → ( 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ↔ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑘 = 𝐾 → ( ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ) ↔ ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) ) |
19 |
7 18
|
rexeqbidv |
⊢ ( 𝑘 = 𝐾 → ( ∃ 𝑟 ∈ ( Atoms ‘ 𝑘 ) ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ) ↔ ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) ) |
20 |
7 19
|
rexeqbidv |
⊢ ( 𝑘 = 𝐾 → ( ∃ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∃ 𝑟 ∈ ( Atoms ‘ 𝑘 ) ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ) ↔ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) ) ) |
21 |
20
|
abbidv |
⊢ ( 𝑘 = 𝐾 → { 𝑠 ∣ ∃ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∃ 𝑟 ∈ ( Atoms ‘ 𝑘 ) ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ) } = { 𝑠 ∣ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ) |
22 |
|
df-lines |
⊢ Lines = ( 𝑘 ∈ V ↦ { 𝑠 ∣ ∃ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∃ 𝑟 ∈ ( Atoms ‘ 𝑘 ) ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑝 ( le ‘ 𝑘 ) ( 𝑞 ( join ‘ 𝑘 ) 𝑟 ) } ) } ) |
23 |
3
|
fvexi |
⊢ 𝐴 ∈ V |
24 |
|
df-sn |
⊢ { { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } } = { 𝑠 ∣ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } } |
25 |
|
snex |
⊢ { { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } } ∈ V |
26 |
24 25
|
eqeltrri |
⊢ { 𝑠 ∣ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } } ∈ V |
27 |
|
simpr |
⊢ ( ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) → 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) |
28 |
27
|
ss2abi |
⊢ { 𝑠 ∣ ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ⊆ { 𝑠 ∣ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } } |
29 |
26 28
|
ssexi |
⊢ { 𝑠 ∣ ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ∈ V |
30 |
23 23 29
|
ab2rexex2 |
⊢ { 𝑠 ∣ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ∈ V |
31 |
21 22 30
|
fvmpt |
⊢ ( 𝐾 ∈ V → ( Lines ‘ 𝐾 ) = { 𝑠 ∣ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ) |
32 |
4 31
|
syl5eq |
⊢ ( 𝐾 ∈ V → 𝑁 = { 𝑠 ∣ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ) |
33 |
5 32
|
syl |
⊢ ( 𝐾 ∈ 𝐵 → 𝑁 = { 𝑠 ∣ ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑞 ≠ 𝑟 ∧ 𝑠 = { 𝑝 ∈ 𝐴 ∣ 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) } ) } ) |