Step |
Hyp |
Ref |
Expression |
1 |
|
diaval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
diaval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
diaval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
diaval.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
diaval.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
diaval.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
1 2 3
|
diaffval |
⊢ ( 𝐾 ∈ 𝑉 → ( DIsoA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ) |
8 |
7
|
fveq1d |
⊢ ( 𝐾 ∈ 𝑉 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) ) |
9 |
6 8
|
syl5eq |
⊢ ( 𝐾 ∈ 𝑉 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) ) |
10 |
|
breq2 |
⊢ ( 𝑤 = 𝑊 → ( 𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑊 ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } = { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ) |
12 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
13 |
12 4
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 ) |
14 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ) |
15 |
14 5
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = 𝑅 ) |
16 |
15
|
fveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) = ( 𝑅 ‘ 𝑓 ) ) |
17 |
16
|
breq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 ↔ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 ) ) |
18 |
13 17
|
rabeqbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) |
19 |
11 18
|
mpteq12dv |
⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ) |
20 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) |
21 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
22 |
21
|
mptrabex |
⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ∈ V |
23 |
19 20 22
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ) |
24 |
9 23
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ) |