Metamath Proof Explorer

Theorem mapdh6aN

Description: Lemma for mapdh6N . Part (6) in Baer p. 47, case 1. (Contributed by NM, 23-Apr-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 )
mapdhe6.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe6.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe6.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh6.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
mapdh6.fg ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh6.fe ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
Assertion mapdh6aN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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 mapdhe6.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdhe6.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdhe6.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 mapdh6.yz ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
24 mapdh6.fg ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
25 mapdh6.fe ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 mapdh6lem2N ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 ✚ 𝐸 ) } ) )
27 24 25 oveq12d ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( 𝐺 ✚ 𝐸 ) )
28 27 sneqd ⊢ ( 𝜑 → { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } = { ( 𝐺 ✚ 𝐸 ) } )
29 28 fveq2d ⊢ ( 𝜑 → ( 𝐽 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) = ( 𝐽 ‘ { ( 𝐺 ✚ 𝐸 ) } ) )
30 26 29 eqtr4d ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 mapdh6lem1N ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) )
32 27 oveq2d ⊢ ( 𝜑 → ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) = ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) )
33 32 sneqd ⊢ ( 𝜑 → { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } = { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } )
34 33 fveq2d ⊢ ( 𝜑 → ( 𝐽 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) )
35 31 34 eqtr4d ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) )
36 3 5 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
37 20 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
38 21 eldifad ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
39 6 18 lmodvacl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
40 36 37 38 39 syl3anc ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
41 6 18 8 9 36 37 38 23 lmodindp1 ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ≠ 0 )
42 eldifsn ⊢ ( ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝑌 + 𝑍 ) ∈ 𝑉 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) )
43 40 41 42 sylanbrc ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) )
44 3 10 14 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
45 3 5 14 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
46 17 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
47 6 8 9 45 37 21 46 23 22 lspindp2 ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
48 47 simpld ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 37 48 mapdhcl ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
50 6 8 9 45 20 38 46 23 22 lspindp1 ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) )
51 50 simpld ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 38 51 mapdhcl ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
53 11 19 lmodvacl ⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
54 44 49 52 53 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
55 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
56 6 55 9 36 37 38 lspprcl ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
57 6 18 9 36 37 38 lspprvacl ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
58 55 9 36 56 57 lspsnel5a ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
59 6 55 9 36 56 46 lspsnel5 ⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
60 22 59 mtbid ⊢ ( 𝜑 → ¬ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
61 nssne2 ⊢ ( ( ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∧ ¬ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
62 58 60 61 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
63 62 necomd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 43 54 63 mapdheq ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) ) ) )
65 30 35 64 mpbir2and ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )