Theorem mapdh8ab

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 13-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 ) )
	mapdh8ab.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh8ab.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh8ab.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	mapdh8ab.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
	mapdh8ab.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8ab.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8ab.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8ab.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8ab.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	mapdh8ab.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	mapdh8ab.yn	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
Assertion	mapdh8ab	⊢ ( 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		mapdh8ab.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh8ab.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh8ab.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18		mapdh8ab.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19		mapdh8ab.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8ab.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21		mapdh8ab.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22		mapdh8ab.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
23		mapdh8ab.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
24		mapdh8ab.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
25		mapdh8ab.yn	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑇 } ) )
26	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
27	19	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
28	20	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
29	21	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
30	3 6 26 27 28 29 24	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
31	30	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
32	31	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
33	32 25	neeqtrd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
34	25	sseq1d	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
35		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
36	1 2 14	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
37	3 35 6 36 28 29	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
38	3 35 6 36 37 27	ellspsn5b	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
39	22	eldifad	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
40	3 35 6 36 37 39	ellspsn5b	⊢ ( 𝜑 → ( 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑇 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
41	34 38 40	3bitr4d	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
42	24 41	mtbid	⊢ ( 𝜑 → ¬ 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
43	26	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑈 ∈ LVec )
44	20	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
45	39	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑇 ∈ 𝑉 )
46	29	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑍 ∈ 𝑉 )
47	23	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
48		simpr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
49		prcom	⊢ { 𝑍 , 𝑇 } = { 𝑇 , 𝑍 }
50	49	fveq2i	⊢ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) = ( 𝑁 ‘ { 𝑇 , 𝑍 } )
51	48 50	eleqtrdi	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑍 } ) )
52	3 5 6 43 44 45 46 47 51	lspexch	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) ) → 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
53	42 52	mtand	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑇 } ) )
54	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 33 22 53 24	mapdh8aa	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑇 ⟩ ) )