Theorem hdmap1l6j

Description: Lemmma for hdmap1l6 . Eliminate ( N { Y } ) = ( N { Z } ) hypothesis. (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 } ) )
	hdmap1l6i.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	hdmap1l6i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1l6i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hdmap1l6j	⊢ ( 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		hdmap1l6i.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		hdmap1l6i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
22		hdmap1l6i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
23	16	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24	17	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝐹 ∈ 𝐷 )
25	18	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
26	19	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
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 14 15 23 24 25 26 27 28 29 30	hdmap1l6i	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
32	16	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33	17	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝐹 ∈ 𝐷 )
34	18	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
35	19	adantr	⊢ ( ( 𝜑 ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
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 14 15 32 33 34 35 36 37 38 39 40 41	hdmap1l6a	⊢ ( ( 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 ⟩ ) ) )