Theorem mapdh7dN

Description: Part (7) of Baer p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdh7.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh7.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh7.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh7.s	⊢ − = ( -g ‘ 𝑈 )
	mapdh7.o	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdh7.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh7.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh7.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh7.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdh7.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	mapdh7.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh7.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh7.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 ) } ) ) ) ) )
	mapdh7.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdh7.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh7.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh7.x	⊢ ( 𝜑 → 𝑢 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh7.y	⊢ ( 𝜑 → 𝑣 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh7.z	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh7.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑢 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) )
	mapdh7.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑢 , 𝑣 } ) )
	mapdh7a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑣 ⟩ ) = 𝐺 )
	mapdh7.b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
Assertion	mapdh7dN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑣 , 𝐺 , 𝑤 ⟩ ) = 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		mapdh7.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdh7.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdh7.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		mapdh7.s	⊢ − = ( -g ‘ 𝑈 )
5		mapdh7.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		mapdh7.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdh7.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdh7.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		mapdh7.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
10		mapdh7.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
11		mapdh7.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdh7.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		mapdh7.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		mapdh7.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdh7.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh7.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑢 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh7.x	⊢ ( 𝜑 → 𝑢 ∈ ( 𝑉 ∖ { 0 } ) )
18		mapdh7.y	⊢ ( 𝜑 → 𝑣 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdh7.z	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh7.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑢 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) )
21		mapdh7.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑢 , 𝑣 } ) )
22		mapdh7a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑣 ⟩ ) = 𝐺 )
23		mapdh7.b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
24	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
25	18	eldifad	⊢ ( 𝜑 → 𝑣 ∈ 𝑉 )
26	19	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
27	3 5 6 24 17 25 26 20 21	lspindp1	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) ∧ ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑤 , 𝑣 } ) ) )
28	27	simprd	⊢ ( 𝜑 → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑤 , 𝑣 } ) )
29		prcom	⊢ { 𝑣 , 𝑤 } = { 𝑤 , 𝑣 }
30	29	fveq2i	⊢ ( 𝑁 ‘ { 𝑣 , 𝑤 } ) = ( 𝑁 ‘ { 𝑤 , 𝑣 } )
31	30	eleq2i	⊢ ( 𝑢 ∈ ( 𝑁 ‘ { 𝑣 , 𝑤 } ) ↔ 𝑢 ∈ ( 𝑁 ‘ { 𝑤 , 𝑣 } ) )
32	28 31	sylnibr	⊢ ( 𝜑 → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑣 , 𝑤 } ) )
33	17	eldifad	⊢ ( 𝜑 → 𝑢 ∈ 𝑉 )
34	3 6 24 26 33 25 21	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑢 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) ) )
35	34	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) )
36	35	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑣 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
37	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 18 19 32 36 22 23	mapdheq4	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑣 , 𝐺 , 𝑤 ⟩ ) = 𝐸 )