Metamath Proof Explorer

Theorem hdmap1l6d

Description: Lemmma for hdmap1l6 . (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses hdmap1l6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmap1l6.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmap1l6.p ⊢ + = ( +g ‘ 𝑈 )
hdmap1l6.s ⊢ − = ( -g ‘ 𝑈 )
hdmap1l6c.o ⊢ 0 = ( 0g ‘ 𝑈 )
hdmap1l6.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1l6.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmap1l6.a ⊢ ✚ = ( +g ‘ 𝐶 )
hdmap1l6.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
hdmap1l6.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
hdmap1l6.l ⊢ 𝐿 = ( LSpan ‘ 𝐶 )
hdmap1l6.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmap1l6.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
hdmap1l6cl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
hdmap1l6d.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
hdmap1l6d.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
hdmap1l6d.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.wn ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion hdmap1l6d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1l6.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap1l6.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap1l6.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmap1l6.p ⊢ + = ( +g ‘ 𝑈 )
5 hdmap1l6.s ⊢ − = ( -g ‘ 𝑈 )
6 hdmap1l6c.o ⊢ 0 = ( 0g ‘ 𝑈 )
7 hdmap1l6.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 hdmap1l6.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9 hdmap1l6.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
10 hdmap1l6.a ⊢ ✚ = ( +g ‘ 𝐶 )
11 hdmap1l6.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
12 hdmap1l6.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
13 hdmap1l6.l ⊢ 𝐿 = ( LSpan ‘ 𝐶 )
14 hdmap1l6.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
15 hdmap1l6.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
16 hdmap1l6.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17 hdmap1l6.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
18 hdmap1l6cl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
19 hdmap1l6.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
20 hdmap1l6d.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21 hdmap1l6d.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22 hdmap1l6d.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23 hdmap1l6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24 hdmap1l6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25 hdmap1l6d.wn ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26 1 8 16 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
27 1 2 16 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
28 24 eldifad ⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
29 18 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
30 22 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
31 3 7 27 28 29 30 25 lspindpi ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
32 31 simpld ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
33 32 necomd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
34 1 2 3 6 7 8 9 13 14 15 16 17 19 33 18 28 hdmap1cl ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 )
35 9 10 12 lmod0vrid ⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ 𝑄 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
36 26 34 35 syl2anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ 𝑄 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
37 36 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ 𝑄 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
38 oteq3 ⊢ ( ( 𝑌 + 𝑍 ) = 0 → ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ = ⟨ 𝑋 , 𝐹 , 0 ⟩ )
39 38 fveq2d ⊢ ( ( 𝑌 + 𝑍 ) = 0 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) )
40 1 2 3 6 8 9 12 15 16 17 29 hdmap1val0 ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) = 𝑄 )
41 39 40 sylan9eqr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = 𝑄 )
42 41 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ 𝑄 ) )
43 oveq2 ⊢ ( ( 𝑌 + 𝑍 ) = 0 → ( 𝑤 + ( 𝑌 + 𝑍 ) ) = ( 𝑤 + 0 ) )
44 1 2 16 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
45 3 4 6 lmod0vrid ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ) → ( 𝑤 + 0 ) = 𝑤 )
46 44 28 45 syl2anc ⊢ ( 𝜑 → ( 𝑤 + 0 ) = 𝑤 )
47 43 46 sylan9eqr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( 𝑤 + ( 𝑌 + 𝑍 ) ) = 𝑤 )
48 47 oteq3d ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ = ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ )
49 48 fveq2d ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
50 37 42 49 3eqtr4rd ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )
51 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
52 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝐹 ∈ 𝐷 )
53 18 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
54 19 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
55 24 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
56 23 eldifad ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
57 3 4 lmodvacl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
58 44 30 56 57 syl3anc ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
59 58 anim1i ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( ( 𝑌 + 𝑍 ) ∈ 𝑉 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) )
60 eldifsn ⊢ ( ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝑌 + 𝑍 ) ∈ 𝑉 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) )
61 59 60 sylibr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) )
62 3 7 27 29 30 56 20 lspindpi ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
63 62 simpld ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
64 3 4 6 7 27 18 22 23 24 21 63 25 mapdindp1 ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
65 3 4 6 7 27 18 22 23 24 21 63 25 mapdindp2 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , ( 𝑌 + 𝑍 ) } ) )
66 3 6 7 27 18 58 28 64 65 lspindp1 ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , ( 𝑌 + 𝑍 ) } ) ) )
67 66 simprd ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , ( 𝑌 + 𝑍 ) } ) )
68 67 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , ( 𝑌 + 𝑍 ) } ) )
69 31 simprd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
70 3 6 7 27 24 30 69 lspsnne1 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑌 } ) )
71 eqid ⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
72 3 7 71 44 30 56 lsmpr ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) = ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) )
73 21 oveq2d ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) = ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) )
74 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
75 3 74 7 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
76 44 30 75 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
77 74 lsssubg ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) )
78 44 76 77 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) )
79 71 lsmidm ⊢ ( ( 𝑁 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝑈 ) → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝑁 ‘ { 𝑌 } ) )
80 78 79 syl ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝑁 ‘ { 𝑌 } ) )
81 72 73 80 3eqtr2d ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) = ( 𝑁 ‘ { 𝑌 } ) )
82 70 81 neleqtrrd ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
83 3 4 7 44 30 56 28 82 lspindp4 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑌 , ( 𝑌 + 𝑍 ) } ) )
84 3 7 27 28 30 58 83 lspindpi ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) )
85 84 simprd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
86 85 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
87 eqidd ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
88 eqidd ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) )
89 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 51 52 53 54 55 61 68 86 87 88 hdmap1l6a ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )
90 50 89 pm2.61dane ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )