Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapcl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmapcl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmapcl.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmapcl.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
hdmapcl.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
6 |
|
hdmapcl.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hdmapcl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
hdmapcl.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑉 ) |
9 |
|
eqid |
⊢ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 = 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 |
10 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
11 |
|
eqid |
⊢ ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
eqid |
⊢ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
1 9 2 3 10 4 5 11 12 6 7 8
|
hdmapval |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑇 ) = ( ℩ ℎ ∈ 𝐷 ∀ 𝑦 ∈ 𝑉 ( ¬ 𝑦 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑇 } ) ) → ℎ = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑦 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑦 〉 ) , 𝑇 〉 ) ) ) ) |
14 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
15 |
|
eqid |
⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 ) |
16 |
|
eqid |
⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
18 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
19 |
1 17 18 2 3 14 9 7
|
dvheveccl |
⊢ ( 𝜑 → 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) ) |
20 |
1 2 3 14 10 4 15 16 11 7 19
|
mapdhvmap |
⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) } ) ) |
21 |
|
eqid |
⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 ) |
22 |
1 2 3 14 4 5 21 11 7 19
|
hvmapcl2 |
⊢ ( 𝜑 → ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) ∈ ( 𝐷 ∖ { ( 0g ‘ 𝐶 ) } ) ) |
23 |
22
|
eldifad |
⊢ ( 𝜑 → ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) ∈ 𝐷 ) |
24 |
1 2 3 14 10 4 5 15 16 12 7 20 19 23 8
|
hdmap1eu |
⊢ ( 𝜑 → ∃! ℎ ∈ 𝐷 ∀ 𝑦 ∈ 𝑉 ( ¬ 𝑦 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑇 } ) ) → ℎ = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑦 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑦 〉 ) , 𝑇 〉 ) ) ) |
25 |
|
riotacl |
⊢ ( ∃! ℎ ∈ 𝐷 ∀ 𝑦 ∈ 𝑉 ( ¬ 𝑦 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑇 } ) ) → ℎ = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑦 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑦 〉 ) , 𝑇 〉 ) ) → ( ℩ ℎ ∈ 𝐷 ∀ 𝑦 ∈ 𝑉 ( ¬ 𝑦 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑇 } ) ) → ℎ = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑦 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑦 〉 ) , 𝑇 〉 ) ) ) ∈ 𝐷 ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ( ℩ ℎ ∈ 𝐷 ∀ 𝑦 ∈ 𝑉 ( ¬ 𝑦 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 } ) ∪ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑇 } ) ) → ℎ = ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 𝑦 , ( ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 , ( ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 ) , 𝑦 〉 ) , 𝑇 〉 ) ) ) ∈ 𝐷 ) |
27 |
13 26
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑇 ) ∈ 𝐷 ) |