Metamath Proof Explorer

Theorem mapdh9aOLDN

Description: Lemma for part (9) in Baer p. 48. (Contributed by NM, 14-May-2015) (New usage is discouraged.)

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 ) )
mapdh8h.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
mapdh8h.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdh9a.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh9a.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
Assertion mapdh9aOLDN ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 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 mapdh8h.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16 mapdh8h.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdh9a.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 mapdh9a.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
19 14 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
20 15 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝐹 ∈ 𝐷 )
21 16 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
22 17 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
23 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
24 1 2 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
25 24 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑈 ∈ LMod )
26 17 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
27 3 23 6 24 26 18 lspprcl ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∈ ( LSubSp ‘ 𝑈 ) )
28 27 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∈ ( LSubSp ‘ 𝑈 ) )
29 simp2l ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑧 ∈ 𝑉 )
30 simp3l ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
31 5 23 25 28 29 30 lssneln0 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑧 ∈ ( 𝑉 ∖ { 0 } ) )
32 simp2r ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑤 ∈ 𝑉 )
33 simp3r ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
34 5 23 25 28 32 33 lssneln0 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
35 1 2 14 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
36 35 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑈 ∈ LVec )
37 26 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑋 ∈ 𝑉 )
38 18 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → 𝑇 ∈ 𝑉 )
39 3 6 36 29 37 38 30 lspindpi ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
40 39 simpld ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
41 40 necomd ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑧 } ) )
42 3 6 36 32 37 38 33 lspindpi ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
43 42 simpld ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
44 43 necomd ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
45 39 simprd ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
46 42 simprd ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
47 1 2 3 4 5 6 7 8 9 10 11 12 13 19 20 21 22 31 34 41 44 45 46 38 mapdh8 ⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) ∧ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) )
48 47 3exp ⊢ ( 𝜑 → ( ( 𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉 ) → ( ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) ) ) )
49 48 ralrimivv ⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) ) )
50 1 2 3 6 14 26 18 dvh3dim ⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
51 14 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
52 15 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝐹 ∈ 𝐷 )
53 16 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
54 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
55 simplr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑧 ∈ 𝑉 )
56 35 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑈 ∈ LVec )
57 26 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑋 ∈ 𝑉 )
58 18 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑇 ∈ 𝑉 )
59 simpr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
60 3 6 56 55 57 58 59 lspindpi ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
61 60 simpld ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
62 61 necomd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑧 } ) )
63 10 13 1 12 2 3 4 5 6 7 8 9 11 51 52 53 54 55 62 mapdhcl ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ∈ 𝐷 )
64 eqidd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) )
65 24 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑈 ∈ LMod )
66 27 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∈ ( LSubSp ‘ 𝑈 ) )
67 5 23 65 66 55 59 lssneln0 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑧 ∈ ( 𝑉 ∖ { 0 } ) )
68 10 13 1 12 2 3 4 5 6 7 8 9 11 51 52 53 54 67 63 62 mapdheq ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑧 } ) ) = ( 𝐽 ‘ { ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑧 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ) } ) ) ) )
69 64 68 mpbid ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑧 } ) ) = ( 𝐽 ‘ { ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑧 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ) } ) ) )
70 69 simpld ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑧 } ) ) = ( 𝐽 ‘ { ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) } ) )
71 60 simprd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
72 10 13 1 12 2 3 4 5 6 7 8 9 11 51 63 70 67 58 71 mapdhcl ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 )
73 72 ex ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) → ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 ) )
74 73 ancld ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) → ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 ) ) )
75 74 reximdva ⊢ ( 𝜑 → ( ∃ 𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → ∃ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 ) ) )
76 50 75 mpd ⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 ) )
77 eleq1w ⊢ ( 𝑧 = 𝑤 → ( 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ↔ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) )
78 77 notbid ⊢ ( 𝑧 = 𝑤 → ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ↔ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) )
79 oteq1 ⊢ ( 𝑧 = 𝑤 → ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ )
80 oteq3 ⊢ ( 𝑧 = 𝑤 → ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ = ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ )
81 80 fveq2d ⊢ ( 𝑧 = 𝑤 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
82 81 oteq2d ⊢ ( 𝑧 = 𝑤 → ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ )
83 79 82 eqtrd ⊢ ( 𝑧 = 𝑤 → ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ )
84 83 fveq2d ⊢ ( 𝑧 = 𝑤 → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) )
85 78 84 reusv3 ⊢ ( ∃ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ∈ 𝐷 ) → ( ∀ 𝑧 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃ 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
86 76 85 syl ⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑉 ∀ 𝑤 ∈ 𝑉 ( ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑤 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃ 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
87 49 86 mpbid ⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
88 reusv1 ⊢ ( ∃ 𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → ( ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃ 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
89 50 88 syl ⊢ ( 𝜑 → ( ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃ 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
90 87 89 mpbird ⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )