Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrnset.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
ltrnset.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
ltrnset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
ltrnset.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
ltrnset.d |
⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
ltrnset.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
1 2 3 4 5
|
ltrnfset |
⊢ ( 𝐾 ∈ 𝐵 → ( LTrn ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ) |
9 |
8
|
fveq1d |
⊢ ( 𝐾 ∈ 𝐵 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ‘ 𝑊 ) ) |
10 |
7 9
|
syl5eq |
⊢ ( 𝐾 ∈ 𝐵 → 𝑇 = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ‘ 𝑊 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) |
12 |
11 6
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) = 𝐷 ) |
13 |
|
breq2 |
⊢ ( 𝑤 = 𝑊 → ( 𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊 ) ) |
14 |
13
|
notbid |
⊢ ( 𝑤 = 𝑊 → ( ¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊 ) ) |
15 |
|
breq2 |
⊢ ( 𝑤 = 𝑊 → ( 𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊 ) ) |
16 |
15
|
notbid |
⊢ ( 𝑤 = 𝑊 → ( ¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊 ) ) |
17 |
14 16
|
anbi12d |
⊢ ( 𝑤 = 𝑊 → ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) ↔ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ) ) |
18 |
|
oveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) ) |
19 |
|
oveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) |
20 |
18 19
|
eqeq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ↔ ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) |
21 |
17 20
|
imbi12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) ) |
22 |
21
|
2ralbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ↔ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) ) ) |
23 |
12 22
|
rabeqbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } = { 𝑓 ∈ 𝐷 ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) } ) |
24 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) |
25 |
6
|
fvexi |
⊢ 𝐷 ∈ V |
26 |
25
|
rabex |
⊢ { 𝑓 ∈ 𝐷 ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) } ∈ V |
27 |
23 24 26
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ‘ 𝑊 ) = { 𝑓 ∈ 𝐷 ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) } ) |
28 |
10 27
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻 ) → 𝑇 = { 𝑓 ∈ 𝐷 ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑊 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑊 ) ) } ) |