Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap1eu.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap1eu.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap1eu.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap1eu.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
5 |
|
hdmap1eu.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
6 |
|
hdmap1eu.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hdmap1eu.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
8 |
|
hdmap1eu.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
9 |
|
hdmap1eu.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
hdmap1eu.i |
⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
hdmap1eu.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
hdmap1eu.mn |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) ) |
13 |
|
hdmap1eu.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
14 |
|
hdmap1eu.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
15 |
|
hdmap1eu.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑉 ) |
16 |
|
eqid |
⊢ ( -g ‘ 𝑈 ) = ( -g ‘ 𝑈 ) |
17 |
|
eqid |
⊢ ( -g ‘ 𝐶 ) = ( -g ‘ 𝐶 ) |
18 |
|
eqid |
⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 ) |
19 |
|
eqid |
⊢ ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) |
20 |
19
|
hdmap1cbv |
⊢ ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) = ( 𝑤 ∈ V ↦ if ( ( 2nd ‘ 𝑤 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ 𝑔 ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑤 ) } ) ) = ( 𝐿 ‘ { 𝑔 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑤 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑤 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑤 ) ) ( -g ‘ 𝐶 ) 𝑔 ) } ) ) ) ) ) |
21 |
1 2 3 16 4 5 6 7 17 18 8 9 10 11 12 13 14 15 20
|
hdmap1eulemOLDN |
⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ 〈 𝑧 , ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑧 〉 ) , 𝑇 〉 ) ) ) |