Metamath Proof Explorer

Theorem mapdh6kN

Description: Lemmma for mapdh6N . Eliminate nonzero vector requirement. (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 )
mapdh6k.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
mapdh6k.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
mapdh6k.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion mapdh6kN ( 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 mapdh6k.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
21 mapdh6k.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
22 mapdh6k.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 14 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24 15 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝐹 ∈ 𝐷 )
25 16 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
26 17 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
27 simpr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 )
28 21 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑍 ∈ 𝑉 )
29 22 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
30 1 2 3 4 5 6 7 8 9 10 11 12 13 23 24 25 26 18 19 27 28 29 mapdh6bN ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
31 14 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
32 15 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝐹 ∈ 𝐷 )
33 16 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
34 17 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
35 20 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝑌 ∈ 𝑉 )
36 simpr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝑍 = 0 )
37 22 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 31 32 33 34 18 19 35 36 37 mapdh6cN ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
39 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
40 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝐹 ∈ 𝐷 )
41 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
42 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
43 22 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
44 20 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑌 ∈ 𝑉 )
45 simprl ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑌 ≠ 0 )
46 eldifsn ⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) )
47 44 45 46 sylanbrc ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
48 21 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑍 ∈ 𝑉 )
49 simprr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑍 ≠ 0 )
50 eldifsn ⊢ ( 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑍 ∈ 𝑉 ∧ 𝑍 ≠ 0 ) )
51 48 49 50 sylanbrc ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
52 1 2 3 4 5 6 7 8 9 10 11 12 13 39 40 41 42 18 19 43 47 51 mapdh6jN ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
53 30 38 52 pm2.61da2ne ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )