Theorem hdmap1l6f

Description: Lemmma for hdmap1l6 . Part (6) in Baer p. 47 line 38. (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 } ) )
	hdmap1l6d.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	hdmap1l6d.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
	hdmap1l6d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1l6d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1l6d.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1l6d.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion	hdmap1l6f	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) )

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		hdmap1l6d.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		hdmap1l6d.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22		hdmap1l6d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23		hdmap1l6d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24		hdmap1l6d.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25		hdmap1l6d.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26	1 2 16	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
27	22	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
28	24	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
29	18	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
30	23	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
31	3 7 26 29 27 30 20	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
32	31	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
33	3 6 7 26 18 27 28 32 25	lspindp1	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) ) )
34	33	simprd	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑤 , 𝑌 } ) )
35	3 7 26 28 29 27 25	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
36	35	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
37		eqidd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) )
38		eqidd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
39	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 24 22 34 36 37 38	hdmap1l6a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) )