# Metamath Proof Explorer

## Theorem mapdh8aa

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 12-May-2015)

Ref Expression
Hypotheses mapdh8a.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdh8a.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.v 𝑉 = ( Base ‘ 𝑈 )
mapdh8a.s = ( -g𝑈 )
mapdh8a.o 0 = ( 0g𝑈 )
mapdh8a.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdh8a.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.d 𝐷 = ( Base ‘ 𝐶 )
mapdh8a.r 𝑅 = ( -g𝐶 )
mapdh8a.q 𝑄 = ( 0g𝐶 )
mapdh8a.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdh8a.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
mapdh8a.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdh8aa.f ( 𝜑𝐹𝐷 )
mapdh8aa.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdh8aa.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh8aa.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
mapdh8aa.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8aa.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8aa.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8aa.zt ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
mapdh8aa.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8aa.yn ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
mapdh8aa.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion mapdh8aa ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )

### Proof

Step Hyp Ref Expression
1 mapdh8a.h 𝐻 = ( LHyp ‘ 𝐾 )
2 mapdh8a.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 mapdh8a.v 𝑉 = ( Base ‘ 𝑈 )
4 mapdh8a.s = ( -g𝑈 )
5 mapdh8a.o 0 = ( 0g𝑈 )
6 mapdh8a.n 𝑁 = ( LSpan ‘ 𝑈 )
7 mapdh8a.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 mapdh8a.d 𝐷 = ( Base ‘ 𝐶 )
9 mapdh8a.r 𝑅 = ( -g𝐶 )
10 mapdh8a.q 𝑄 = ( 0g𝐶 )
11 mapdh8a.j 𝐽 = ( LSpan ‘ 𝐶 )
12 mapdh8a.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13 mapdh8a.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
14 mapdh8a.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 mapdh8aa.f ( 𝜑𝐹𝐷 )
16 mapdh8aa.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdh8aa.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18 mapdh8aa.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19 mapdh8aa.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20 mapdh8aa.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdh8aa.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdh8aa.zt ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
23 mapdh8aa.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
24 mapdh8aa.yn ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
25 mapdh8aa.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
26 20 eldifad ( 𝜑𝑌𝑉 )
27 1 2 14 dvhlvec ( 𝜑𝑈 ∈ LVec )
28 19 eldifad ( 𝜑𝑋𝑉 )
29 21 eldifad ( 𝜑𝑍𝑉 )
30 3 6 27 28 26 29 25 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
31 30 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
32 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 26 31 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
33 17 32 eqeltrrd ( 𝜑𝐺𝐷 )
34 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 33 31 mapdheq ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
35 17 34 mpbid ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
36 35 simpld ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
37 23 eldifad ( 𝜑𝑇𝑉 )
38 3 6 27 26 29 37 24 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
39 38 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 25 19 20 21 mapdh75d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 33 36 40 20 21 22 23 24 mapdh8a ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) )
42 41 eqcomd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )