# Metamath Proof Explorer

## Theorem mapdh8ab

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 ) )
mapdh8ab.f ( 𝜑𝐹𝐷 )
mapdh8ab.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdh8ab.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh8ab.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
mapdh8ab.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ab.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ab.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ab.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8ab.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
mapdh8ab.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh8ab.yn ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
Assertion mapdh8ab ( 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 mapdh8ab.f ( 𝜑𝐹𝐷 )
16 mapdh8ab.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdh8ab.eg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18 mapdh8ab.ee ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19 mapdh8ab.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20 mapdh8ab.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdh8ab.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdh8ab.t ( 𝜑𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
23 mapdh8ab.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
24 mapdh8ab.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
25 mapdh8ab.yn ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
26 1 2 14 dvhlvec ( 𝜑𝑈 ∈ LVec )
27 19 eldifad ( 𝜑𝑋𝑉 )
28 20 eldifad ( 𝜑𝑌𝑉 )
29 21 eldifad ( 𝜑𝑍𝑉 )
30 3 6 26 27 28 29 24 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
31 30 simprd ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
32 31 necomd ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
33 32 25 neeqtrd ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
34 25 sseq1d ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
35 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
36 1 2 14 dvhlmod ( 𝜑𝑈 ∈ LMod )
37 3 35 6 36 28 29 lspprcl ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38 3 35 6 36 37 27 lspsnel5 ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
39 22 eldifad ( 𝜑𝑇𝑉 )
40 3 35 6 36 37 39 lspsnel5 ( 𝜑 → ( 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
41 34 38 40 3bitr4d ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
42 24 41 mtbid ( 𝜑 → ¬ 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
43 26 adantr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑈 ∈ LVec )
44 20 adantr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
45 39 adantr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑇𝑉 )
46 29 adantr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑍𝑉 )
47 23 adantr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
48 simpr ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
49 prcom { 𝑍 , 𝑇 } = { 𝑇 , 𝑍 }
50 49 fveq2i ( 𝑁 ‘ { 𝑍 , 𝑇 } ) = ( 𝑁 ‘ { 𝑇 , 𝑍 } )
51 48 50 eleqtrdi ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑍 } ) )
52 3 5 6 43 44 45 46 47 51 lspexch ( ( 𝜑𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
53 42 52 mtand ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 33 22 53 24 mapdh8aa ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )