Metamath Proof Explorer


Theorem hdmap1euOLDN

Description: Convert mapdh9aOLDN to use the HDMap1 notation. (Contributed by NM, 15-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmap1eu.h 𝐻 = ( LHyp ‘ 𝐾 )
hdmap1eu.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eu.v 𝑉 = ( Base ‘ 𝑈 )
hdmap1eu.o 0 = ( 0g𝑈 )
hdmap1eu.n 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1eu.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eu.d 𝐷 = ( Base ‘ 𝐶 )
hdmap1eu.l 𝐿 = ( LSpan ‘ 𝐶 )
hdmap1eu.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eu.i 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eu.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hdmap1eu.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
hdmap1eu.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1eu.f ( 𝜑𝐹𝐷 )
hdmap1eu.t ( 𝜑𝑇𝑉 )
Assertion hdmap1euOLDN ( 𝜑 → ∃! 𝑦𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )

Proof

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 = ( 0g𝑈 )
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 ∧ 𝑊𝐻 ) )
12 hdmap1eu.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
13 hdmap1eu.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
14 hdmap1eu.f ( 𝜑𝐹𝐷 )
15 hdmap1eu.t ( 𝜑𝑇𝑉 )
16 eqid ( -g𝑈 ) = ( -g𝑈 )
17 eqid ( -g𝐶 ) = ( -g𝐶 )
18 eqid ( 0g𝐶 ) = ( 0g𝐶 )
19 eqid ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , ( 0g𝐶 ) , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐿 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( -g𝑈 ) ( 2nd𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) ( -g𝐶 ) ) } ) ) ) ) ) = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , ( 0g𝐶 ) , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐿 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( -g𝑈 ) ( 2nd𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) ( -g𝐶 ) ) } ) ) ) ) )
20 19 hdmap1cbv ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , ( 0g𝐶 ) , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐿 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( -g𝑈 ) ( 2nd𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) ( -g𝐶 ) ) } ) ) ) ) ) = ( 𝑤 ∈ V ↦ if ( ( 2nd𝑤 ) = 0 , ( 0g𝐶 ) , ( 𝑔𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑤 ) } ) ) = ( 𝐿 ‘ { 𝑔 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑤 ) ) ( -g𝑈 ) ( 2nd𝑤 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st𝑤 ) ) ( -g𝐶 ) 𝑔 ) } ) ) ) ) )
21 1 2 3 16 4 5 6 7 17 18 8 9 10 11 12 13 14 15 20 hdmap1eulemOLDN ( 𝜑 → ∃! 𝑦𝐷𝑧𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )