Step |
Hyp |
Ref |
Expression |
1 |
|
dochval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dochval.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
3 |
|
dochval.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
4 |
|
dochval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
dochval.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dochval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dochval.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
8 |
|
dochval.n |
⊢ 𝑁 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
1 2 3 4
|
dochffval |
⊢ ( 𝐾 ∈ 𝑋 → ( ocH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝐾 ∈ 𝑋 → ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) ‘ 𝑊 ) ) |
11 |
8 10
|
syl5eq |
⊢ ( 𝐾 ∈ 𝑋 → 𝑁 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) ‘ 𝑊 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
13 |
12 6
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = 𝑈 ) |
14 |
13
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( Base ‘ 𝑈 ) ) |
15 |
14 7
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑉 ) |
16 |
15
|
pweqd |
⊢ ( 𝑤 = 𝑊 → 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝒫 𝑉 ) |
17 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
18 |
17 5
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) = 𝐼 ) |
19 |
18
|
fveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) = ( 𝐼 ‘ 𝑦 ) ) |
20 |
19
|
sseq2d |
⊢ ( 𝑤 = 𝑊 → ( 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) ↔ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) ) ) |
21 |
20
|
rabbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } = { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) |
22 |
21
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) = ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) |
23 |
22
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) = ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) |
24 |
18 23
|
fveq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) = ( 𝐼 ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) |
25 |
16 24
|
mpteq12dv |
⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) = ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ) |
26 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) |
27 |
7
|
fvexi |
⊢ 𝑉 ∈ V |
28 |
27
|
pwex |
⊢ 𝒫 𝑉 ∈ V |
29 |
28
|
mptex |
⊢ ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ∈ V |
30 |
25 26 29
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝒫 ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↦ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑦 ) } ) ) ) ) ) ‘ 𝑊 ) = ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ) |
31 |
11 30
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝑁 = ( 𝑥 ∈ 𝒫 𝑉 ↦ ( 𝐼 ‘ ( ⊥ ‘ ( 𝐺 ‘ { 𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ ( 𝐼 ‘ 𝑦 ) } ) ) ) ) ) |