Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapval2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmapval2.e |
⊢ 𝐸 = 〈 ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) 〉 |
3 |
|
hdmapval2.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
hdmapval2.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
hdmapval2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
6 |
|
hdmapval2.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hdmapval2.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
8 |
|
hdmapval2.j |
⊢ 𝐽 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
hdmapval2.i |
⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
hdmapval2.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
hdmapval2.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
hdmapval2.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑉 ) |
13 |
|
hdmapval2.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
14 |
1 2 3 4 5 6 7 8 9 10 11 12
|
hdmapval |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑇 ) = ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑇 ) = 𝐹 ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) = 𝐹 ) ) |
16 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
17 |
|
eqid |
⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 ) |
18 |
|
eqid |
⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
20 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
21 |
1 19 20 3 4 16 2 11
|
dvheveccl |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) ) |
22 |
1 3 4 16 5 6 17 18 8 11 21
|
mapdhvmap |
⊢ ( 𝜑 → ( ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑁 ‘ { 𝐸 } ) ) = ( ( LSpan ‘ 𝐶 ) ‘ { ( 𝐽 ‘ 𝐸 ) } ) ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 ) |
24 |
1 3 4 16 6 7 23 8 11 21
|
hvmapcl2 |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝐸 ) ∈ ( 𝐷 ∖ { ( 0g ‘ 𝐶 ) } ) ) |
25 |
24
|
eldifad |
⊢ ( 𝜑 → ( 𝐽 ‘ 𝐸 ) ∈ 𝐷 ) |
26 |
1 3 4 16 5 6 7 17 18 9 11 22 21 25 12
|
hdmap1eu |
⊢ ( 𝜑 → ∃! ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) |
27 |
|
nfv |
⊢ Ⅎ ℎ 𝜑 |
28 |
|
nfcvd |
⊢ ( 𝜑 → Ⅎ ℎ 𝐹 ) |
29 |
|
nfvd |
⊢ ( 𝜑 → Ⅎ ℎ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) |
30 |
|
eqeq1 |
⊢ ( ℎ = 𝐹 → ( ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ↔ 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) |
31 |
30
|
imbi2d |
⊢ ( ℎ = 𝐹 → ( ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ↔ ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) ) |
32 |
31
|
ralbidv |
⊢ ( ℎ = 𝐹 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝜑 ∧ ℎ = 𝐹 ) → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) ) |
34 |
27 28 29 13 33
|
riota2df |
⊢ ( ( 𝜑 ∧ ∃! ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) = 𝐹 ) ) |
35 |
26 34
|
mpdan |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ↔ ( ℩ ℎ ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ℎ = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) = 𝐹 ) ) |
36 |
15 35
|
bitr4d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑇 ) = 𝐹 ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝐸 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝐹 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝐸 , ( 𝐽 ‘ 𝐸 ) , 𝑧 〉 ) , 𝑇 〉 ) ) ) ) |