Metamath Proof Explorer

Theorem mapdh6jN

Description: Lemmma for mapdh6N . Eliminate ( N{ Y } ) = ( N{ Z } ) hypothesis. (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 )
mapdh6i.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh6i.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6i.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion mapdh6jN ( 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 mapdh6i.xn ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21 mapdh6i.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdh6i.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
23 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝐹 ∈ 𝐷 )
25 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
26 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
27 20 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
28 21 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
29 22 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
30 simpr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 23 24 25 26 18 19 27 28 29 30 mapdh6iN ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
32 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33 15 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝐹 ∈ 𝐷 )
34 16 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
35 17 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
36 21 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
37 22 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
38 20 adantr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
39 simpr ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
40 eqidd ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
41 eqidd ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
42 1 2 3 4 5 6 7 8 9 10 11 12 13 32 33 34 35 18 19 36 37 38 39 40 41 mapdh6aN ⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
43 31 42 pm2.61dane ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )