Metamath Proof Explorer

Theorem hdmap1eulem

Description: Lemma for hdmap1eu . TODO: combine with hdmap1eu or at least share some hypotheses. (Contributed by NM, 15-May-2015)

Ref Expression
Hypotheses hdmap1eulem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmap1eulem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eulem.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmap1eulem.s ⊢ − = ( -g ‘ 𝑈 )
hdmap1eulem.o ⊢ 0 = ( 0g ‘ 𝑈 )
hdmap1eulem.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1eulem.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eulem.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmap1eulem.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
hdmap1eulem.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
hdmap1eulem.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
hdmap1eulem.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eulem.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1eulem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmap1eulem.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
hdmap1eulem.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1eulem.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
hdmap1eulem.y ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
hdmap1eulem.l ⊢ 𝐿 = ( 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 ) } ) ) ) ) )
Assertion hdmap1eulem ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1eulem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap1eulem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap1eulem.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmap1eulem.s ⊢ − = ( -g ‘ 𝑈 )
5 hdmap1eulem.o ⊢ 0 = ( 0g ‘ 𝑈 )
6 hdmap1eulem.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7 hdmap1eulem.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 hdmap1eulem.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
9 hdmap1eulem.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
10 hdmap1eulem.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
11 hdmap1eulem.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12 hdmap1eulem.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13 hdmap1eulem.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
14 hdmap1eulem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 hdmap1eulem.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
16 hdmap1eulem.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
17 hdmap1eulem.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
18 hdmap1eulem.y ⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
19 hdmap1eulem.l ⊢ 𝐿 = ( 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 ) } ) ) ) ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 19 14 17 15 16 18 mapdh9a ⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
21 14 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 16 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
23 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → 𝐹 ∈ 𝐷 )
24 simplr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → 𝑧 ∈ 𝑉 )
25 1 2 3 4 5 6 7 8 9 10 11 12 13 21 22 23 24 19 hdmap1valc ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) )
26 25 oteq2d ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ )
27 26 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
28 elun1 ⊢ ( 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) → 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) )
29 28 con3i ⊢ ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) )
30 14 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
32 1 2 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
33 32 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑈 ∈ LMod )
34 16 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
35 34 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑋 ∈ 𝑉 )
36 3 31 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
37 33 35 36 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38 simplr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑧 ∈ 𝑉 )
39 simpr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) )
40 5 31 33 37 38 39 lssneln0 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑧 ∈ ( 𝑉 ∖ { 0 } ) )
41 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝐹 ∈ 𝐷 )
42 15 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
43 16 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
44 3 6 33 38 35 39 lspsnne2 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
45 44 necomd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑧 } ) )
46 10 19 1 12 2 3 4 5 6 7 8 9 11 30 41 42 43 38 45 mapdhcl ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ∈ 𝐷 )
47 18 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → 𝑇 ∈ 𝑉 )
48 1 2 3 4 5 6 7 8 9 10 11 12 13 30 40 46 47 19 hdmap1valc ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
49 29 48 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
50 27 49 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
51 50 eqeq2d ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) ) → ( 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ↔ 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
52 51 pm5.74da ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) → ( ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
53 52 ralbidva ⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
54 53 reubidv ⊢ ( 𝜑 → ( ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
55 20 54 mpbird ⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑇 } ) ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )