Metamath Proof Explorer


Theorem mapdh8ac

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 13-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 ) )
mapdh8ac.f ( 𝜑𝐹𝐷 )
mapdh8ac.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdh8ac.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh8ac.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
mapdh8ac.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ac.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ac.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ac.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ac.yn ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
mapdh8ac.ew ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = 𝐵 )
mapdh8ac.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ac.yw ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
mapdh8ac.xy ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
mapdh8ac.wz ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
mapdh8ac.xz ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑍 } ) )
Assertion mapdh8ac ( 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 mapdh8ac.f ( 𝜑𝐹𝐷 )
16 mapdh8ac.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdh8ac.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18 mapdh8ac.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19 mapdh8ac.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20 mapdh8ac.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdh8ac.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdh8ac.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
23 mapdh8ac.yn ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
24 mapdh8ac.ew ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = 𝐵 )
25 mapdh8ac.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
26 mapdh8ac.yw ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
27 mapdh8ac.xy ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
28 mapdh8ac.wz ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
29 mapdh8ac.xz ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑍 } ) )
30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 24 19 20 25 22 26 27 23 mapdh8ab ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , 𝐵 , 𝑇 ⟩ ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 18 19 25 21 22 28 29 23 mapdh8ab ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐵 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )
32 30 31 eqtrd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )