Metamath Proof Explorer

Theorem hdmap1l6k

Description: Lemmma for hdmap1l6 . Eliminate nonzero vector requirement. (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 } ) )
hdmap1l6k.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
hdmap1l6k.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
hdmap1l6k.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion hdmap1l6k ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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 hdmap1l6k.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
21 hdmap1l6k.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
22 hdmap1l6k.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 16 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24 17 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝐹 ∈ 𝐷 )
25 18 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
26 19 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
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 14 15 23 24 25 26 27 28 29 hdmap1l6b ⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
31 16 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
32 17 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝐹 ∈ 𝐷 )
33 18 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
34 19 adantr ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
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 14 15 31 32 33 34 35 36 37 hdmap1l6c ⊢ ( ( 𝜑 ∧ 𝑍 = 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
39 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
40 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝐹 ∈ 𝐷 )
41 18 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
42 19 adantr ⊢ ( ( 𝜑 ∧ ( 𝑌 ≠ 0 ∧ 𝑍 ≠ 0 ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
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 14 15 39 40 41 42 43 47 51 hdmap1l6j ⊢ ( ( 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 ⟩ ) ) )