Step |
Hyp |
Ref |
Expression |
1 |
|
lcdval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcdval.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcdval.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lcdval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
lcdval.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
6 |
|
lcdval.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
7 |
|
lcdval.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
8 |
|
lcdval.k |
⊢ ( 𝜑 → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
1
|
lcdfval |
⊢ ( 𝐾 ∈ 𝑋 → ( LCDual ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝐾 ∈ 𝑋 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) ) |
11 |
3 10
|
eqtrid |
⊢ ( 𝐾 ∈ 𝑋 → 𝐶 = ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
13 |
12 4
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = 𝑈 ) |
14 |
13
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LDual ‘ 𝑈 ) ) |
15 |
14 7
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐷 ) |
16 |
13
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LFnl ‘ 𝑈 ) ) |
17 |
16 5
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐹 ) |
18 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ) |
19 |
18 2
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) = ⊥ ) |
20 |
13
|
fveq2d |
⊢ ( 𝑤 = 𝑊 → ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LKer ‘ 𝑈 ) ) |
21 |
20 6
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐿 ) |
22 |
21
|
fveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) = ( 𝐿 ‘ 𝑓 ) ) |
23 |
19 22
|
fveq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) |
24 |
19 23
|
fveq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) ) |
25 |
24 22
|
eqeq12d |
⊢ ( 𝑤 = 𝑊 → ( ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) ) |
26 |
17 25
|
rabeqbidv |
⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) |
27 |
15 26
|
oveq12d |
⊢ ( 𝑤 = 𝑊 → ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ) |
28 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) |
29 |
|
ovex |
⊢ ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ∈ V |
30 |
27 28 29
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ) |
31 |
11 30
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ) |
32 |
8 31
|
syl |
⊢ ( 𝜑 → 𝐶 = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ) |