mapdh8c

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 = ( 0_g ‘ 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 = ( 0_g ‘ 𝐶 )
	mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
	mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdh8c.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh8c.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh8c.a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
	mapdh8c.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8c.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8c.xt	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8c.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8c.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8c.wt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8c.ut	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8c.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
	mapdh8c.e	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
	mapdh8c.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
Assertion	mapdh8c	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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 = ( 0_g ‘ 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 = ( 0_g ‘ 𝐶 )
11		mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		mapdh8a.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
14		mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdh8c.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh8c.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh8c.a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
18		mapdh8c.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdh8c.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8c.xt	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
21		mapdh8c.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
22		mapdh8c.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
23		mapdh8c.wt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
24		mapdh8c.ut	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
25		mapdh8c.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
26		mapdh8c.e	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
27		mapdh8c.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
28	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
29	18	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
30	19	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
31	22	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
32	3 6 28 29 30 31 27	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) ) )
33	32	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
34	20	eldifad	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
35	3 5 6 28 18 30 34 24 26	lspexch	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
36	28	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑈 ∈ LVec )
37	19	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
38	29	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑋 ∈ 𝑉 )
39	31	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑤 ∈ 𝑉 )
40	25	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
41		simpr	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) )
42	3 5 6 36 37 38 39 40 41	lspexch	⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑤 } ) )
43	27 42	mtand	⊢ ( 𝜑 → ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑤 } ) )
44	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 22 23 20 33 35 43	mapdh8b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )

Metamath Proof Explorer

Theorem mapdh8c

Proof