Metamath Proof Explorer

Theorem mapdh8e

Description: Part of Part (8) in Baer p. 48. Eliminate w . (Contributed by NM, 10-May-2015)

Ref Expression
Hypotheses mapdh8a.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
mapdh8a.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
mapdh8a.s ⊢ − = ( -g ‘ 𝑈 )
mapdh8a.o ⊢ 0 = ( 0g ‘ 𝑈 )
mapdh8a.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
mapdh8a.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
mapdh8a.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
mapdh8a.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
mapdh8a.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
mapdh8a.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh8a.i ⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
mapdh8a.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
mapdh8e.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
mapdh8e.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdh8e.eg ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh8e.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8e.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8e.t ⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh8e.xy ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
mapdh8e.xt ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
mapdh8e.yt ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
mapdh8e.e ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
Assertion mapdh8e ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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 ⊢ − = ( -g ‘ 𝑈 )
5 mapdh8a.o ⊢ 0 = ( 0g ‘ 𝑈 )
6 mapdh8a.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7 mapdh8a.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 mapdh8a.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
9 mapdh8a.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
10 mapdh8a.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
11 mapdh8a.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12 mapdh8a.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13 mapdh8a.i ⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
14 mapdh8a.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 mapdh8e.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16 mapdh8e.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdh8e.eg ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18 mapdh8e.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
19 mapdh8e.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
20 mapdh8e.t ⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdh8e.xy ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
22 mapdh8e.xt ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
23 mapdh8e.yt ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
24 mapdh8e.e ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
25 18 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
26 19 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
27 1 2 3 6 14 25 26 dvh3dim ⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
28 14 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
29 15 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝐹 ∈ 𝐷 )
30 16 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
31 17 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
32 18 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
33 19 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
34 20 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
35 23 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
36 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
37 1 2 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
38 37 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑈 ∈ LMod )
39 3 36 6 37 25 26 lspprcl ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
40 39 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
41 simp2 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ 𝑉 )
42 simp3 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
43 5 36 38 40 41 42 lssneln0 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
44 1 2 14 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
45 20 eldifad ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
46 prcom ⊢ { 𝑌 , 𝑇 } = { 𝑇 , 𝑌 }
47 46 fveq2i ⊢ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) = ( 𝑁 ‘ { 𝑇 , 𝑌 } )
48 24 47 eleqtrdi ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑇 , 𝑌 } ) )
49 3 5 6 44 18 45 26 21 48 lspexch ⊢ ( 𝜑 → 𝑇 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
50 36 6 37 39 49 lspsnel5a ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
51 50 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
52 37 adantr ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → 𝑈 ∈ LMod )
53 39 adantr ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
54 simpr ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → 𝑤 ∈ 𝑉 )
55 3 36 6 52 53 54 lspsnel5 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → ( 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ↔ ( 𝑁 ‘ { 𝑤 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
56 55 biimprd ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → ( ( 𝑁 ‘ { 𝑤 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
57 56 con3d ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ) → ( ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → ¬ ( 𝑁 ‘ { 𝑤 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
58 57 3impia ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ ( 𝑁 ‘ { 𝑤 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
59 nssne2 ⊢ ( ( ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∧ ¬ ( 𝑁 ‘ { 𝑤 } ) ⊆ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑇 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
60 51 58 59 syl2anc ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑇 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
61 60 necomd ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
62 22 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
63 44 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑈 ∈ LVec )
64 25 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑋 ∈ 𝑉 )
65 26 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑌 ∈ 𝑉 )
66 3 6 63 41 64 65 42 lspindpi ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
67 66 simprd ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
68 67 necomd ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
69 21 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
70 3 5 6 63 32 65 41 69 42 lspindp2l ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) ) )
71 70 simprd ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
72 1 2 3 4 5 6 7 8 9 10 11 12 13 28 29 30 31 32 33 34 35 43 61 62 68 71 mapdh8d ⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )
73 72 rexlimdv3a ⊢ ( 𝜑 → ( ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) ) )
74 27 73 mpd ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )