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 |
|
elex |
⊢ ( 𝐾 ∈ 𝐶 → 𝐾 ∈ V ) |
7 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) ) |
8 |
7 5
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 ) |
9 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( LDil ‘ 𝑘 ) = ( LDil ‘ 𝐾 ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝑘 = 𝐾 → ( ( LDil ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) ) |
12 |
11 4
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 ) |
13 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) ) |
14 |
13 1
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ ) |
15 |
14
|
breqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( le ‘ 𝑘 ) 𝑤 ↔ 𝑝 ≤ 𝑤 ) ) |
16 |
15
|
notbid |
⊢ ( 𝑘 = 𝐾 → ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ↔ ¬ 𝑝 ≤ 𝑤 ) ) |
17 |
14
|
breqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( le ‘ 𝑘 ) 𝑤 ↔ 𝑞 ≤ 𝑤 ) ) |
18 |
17
|
notbid |
⊢ ( 𝑘 = 𝐾 → ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ↔ ¬ 𝑞 ≤ 𝑤 ) ) |
19 |
16 18
|
anbi12d |
⊢ ( 𝑘 = 𝐾 → ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) ↔ ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( meet ‘ 𝑘 ) = ( meet ‘ 𝐾 ) ) |
21 |
20 3
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( meet ‘ 𝑘 ) = ∧ ) |
22 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ( join ‘ 𝐾 ) ) |
23 |
22 2
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ∨ ) |
24 |
23
|
oveqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) = ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ) |
25 |
|
eqidd |
⊢ ( 𝑘 = 𝐾 → 𝑤 = 𝑤 ) |
26 |
21 24 25
|
oveq123d |
⊢ ( 𝑘 = 𝐾 → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) ) |
27 |
23
|
oveqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) = ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ) |
28 |
21 27 25
|
oveq123d |
⊢ ( 𝑘 = 𝐾 → ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) |
29 |
26 28
|
eqeq12d |
⊢ ( 𝑘 = 𝐾 → ( ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ↔ ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ) |
30 |
19 29
|
imbi12d |
⊢ ( 𝑘 = 𝐾 → ( ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ) ) |
31 |
12 30
|
raleqbidv |
⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) ↔ ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ) ) |
32 |
12 31
|
raleqbidv |
⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) ↔ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) ) ) |
33 |
10 32
|
rabeqbidv |
⊢ ( 𝑘 = 𝐾 → { 𝑓 ∈ ( ( LDil ‘ 𝑘 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) } = { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) |
34 |
8 33
|
mpteq12dv |
⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑓 ∈ ( ( LDil ‘ 𝑘 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ) |
35 |
|
df-ltrn |
⊢ LTrn = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑓 ∈ ( ( LDil ‘ 𝑘 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑤 ∧ ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ) → ( ( 𝑝 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝑘 ) 𝑤 ) = ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑓 ‘ 𝑞 ) ) ( meet ‘ 𝑘 ) 𝑤 ) ) } ) ) |
36 |
34 35 5
|
mptfvmpt |
⊢ ( 𝐾 ∈ V → ( LTrn ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ) |
37 |
6 36
|
syl |
⊢ ( 𝐾 ∈ 𝐶 → ( LTrn ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤 ) → ( ( 𝑝 ∨ ( 𝑓 ‘ 𝑝 ) ) ∧ 𝑤 ) = ( ( 𝑞 ∨ ( 𝑓 ‘ 𝑞 ) ) ∧ 𝑤 ) ) } ) ) |