Metamath Proof Explorer

Theorem mapdh6dN

Description: Lemmma for mapdh6N . (Contributed by NM, 1-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
mapdh.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 ) } ) ) ) ) )
mapdh.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
mapdh.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
mapdh.s ⊢ − = ( -g ‘ 𝑈 )
mapdhc.o ⊢ 0 = ( 0g ‘ 𝑈 )
mapdh.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
mapdh.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
mapdh.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
mapdh.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
mapdh.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
mapdhc.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
mapdh.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdhcl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh.p ⊢ + = ( +g ‘ 𝑈 )
mapdh.a ⊢ ✚ = ( +g ‘ 𝐶 )
mapdh6d.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh6d.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
mapdh6d.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.wn ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion mapdh6dN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )

Proof

Step Hyp Ref Expression
1 mapdh.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
2 mapdh.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 ) } ) ) ) ) )
3 mapdh.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 mapdh.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5 mapdh.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 mapdh.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
7 mapdh.s ⊢ − = ( -g ‘ 𝑈 )
8 mapdhc.o ⊢ 0 = ( 0g ‘ 𝑈 )
9 mapdh.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10 mapdh.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 mapdh.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
12 mapdh.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
13 mapdh.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
14 mapdh.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 mapdhc.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16 mapdh.mn ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdhcl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 mapdh.p ⊢ + = ( +g ‘ 𝑈 )
19 mapdh.a ⊢ ✚ = ( +g ‘ 𝐶 )
20 mapdh6d.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21 mapdh6d.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22 mapdh6d.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23 mapdh6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24 mapdh6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25 mapdh6d.wn ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26 3 10 14 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
27 24 eldifad ⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
28 3 5 14 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
29 17 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
30 22 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
31 6 9 28 27 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 27 33 mapdhcl ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 )
35 11 19 1 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 8 17 15 mapdhval0 ⊢ ( 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 3 5 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
45 6 18 8 lmod0vrid ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ) → ( 𝑤 + 0 ) = 𝑤 )
46 44 27 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 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
52 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝐹 ∈ 𝐷 )
53 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
54 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
55 24 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
56 23 eldifad ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
57 6 18 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 6 9 28 29 30 56 20 lspindpi ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
63 62 simpld ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
64 6 18 8 9 28 17 22 23 24 21 63 25 mapdindp1 ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
65 6 18 8 9 28 17 22 23 24 21 63 25 mapdindp2 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , ( 𝑌 + 𝑍 ) } ) )
66 6 8 9 28 17 58 27 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 6 8 9 28 24 30 69 lspsnne1 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑌 } ) )
71 eqid ⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
72 6 9 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 6 74 9 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 6 18 9 44 30 56 27 82 lspindp4 ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑌 , ( 𝑌 + 𝑍 ) } ) )
84 6 9 28 27 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 51 52 53 54 18 19 55 61 68 86 87 88 mapdh6aN ⊢ ( ( 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 ) ⟩ ) ) )