Theorem mapdh8b

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 6-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 ) )
	mapdh8b.f	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
	mapdh8b.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
	mapdh8b.a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑤 ⟩ ) = 𝐸 )
	mapdh8b.x	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8b.y	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8b.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8b.xt	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8b.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
	mapdh8b.e	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
	mapdh8b.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
Assertion	mapdh8b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 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		mapdh8b.f	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
16		mapdh8b.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
17		mapdh8b.a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑤 ⟩ ) = 𝐸 )
18		mapdh8b.x	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdh8b.y	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8b.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
21		mapdh8b.xt	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
22		mapdh8b.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
23		mapdh8b.e	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
24		mapdh8b.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
25	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
26	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
27	19	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
28	21	eldifad	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
29	3 6 25 26 27 28 23 24	lspindp5	⊢ ( 𝜑 → ¬ 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
30		prcom	⊢ { 𝑤 , 𝑇 } = { 𝑇 , 𝑤 }
31	30	fveq2i	⊢ ( 𝑁 ‘ { 𝑤 , 𝑇 } ) = ( 𝑁 ‘ { 𝑇 , 𝑤 } )
32	31	eleq2i	⊢ ( 𝑌 ∈ ( 𝑁 ‘ { 𝑤 , 𝑇 } ) ↔ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) )
33	25	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑈 ∈ LVec )
34	18	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
35	28	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑇 ∈ 𝑉 )
36	27	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑤 ∈ 𝑉 )
37	22	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
38		simpr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) )
39	3 5 6 33 34 35 36 37 38	lspexch	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) ) → 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
40	39	ex	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑁 ‘ { 𝑇 , 𝑤 } ) → 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) ) )
41	32 40	syl5bi	⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑁 ‘ { 𝑤 , 𝑇 } ) → 𝑇 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) ) )
42	29 41	mtod	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑤 , 𝑇 } ) )
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 42	mapdh8a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) )