Theorem mapdh8aa

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 12-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh8a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh8a.s	⊢ − = ( -g ‘ 𝑈 )
	mapdh8a.o	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdh8a.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh8a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh8a.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdh8a.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.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 ) } ) ) ) ) )
	mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdh8aa.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh8aa.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh8aa.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	mapdh8aa.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
	mapdh8aa.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8aa.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8aa.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8aa.zt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8aa.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8aa.yn	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
	mapdh8aa.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion	mapdh8aa	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdh8a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdh8a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		mapdh8a.s	⊢ − = ( -g ‘ 𝑈 )
5		mapdh8a.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		mapdh8a.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdh8a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdh8a.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		mapdh8a.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
10		mapdh8a.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
11		mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		mapdh8a.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 ) } ) ) ) ) )
14		mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdh8aa.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh8aa.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh8aa.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18		mapdh8aa.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19		mapdh8aa.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8aa.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21		mapdh8aa.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22		mapdh8aa.zt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
23		mapdh8aa.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
24		mapdh8aa.yn	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
25		mapdh8aa.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
26	20	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
27	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
28	19	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
29	21	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
30	3 6 27 28 26 29 25	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
31	30	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
32	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 26 31	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
33	17 32	eqeltrrd	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
34	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 33 31	mapdheq	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
35	17 34	mpbid	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
36	35	simpld	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
37	23	eldifad	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
38	3 6 27 26 29 37 24	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
39	38	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
40	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 25 19 20 21	mapdh75d	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 )
41	1 2 3 4 5 6 7 8 9 10 11 12 13 14 33 36 40 20 21 22 23 24	mapdh8a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) )
42	41	eqcomd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )