Theorem mapdh7eN

Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (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 } ) )
	mapdh7b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
Assertion	mapdh7eN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑢 ⟩ ) = 𝐹 )

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		mapdh7b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) = 𝐸 )
23	19	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
24	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
25	17	eldifad	⊢ ( 𝜑 → 𝑢 ∈ 𝑉 )
26	18	eldifad	⊢ ( 𝜑 → 𝑣 ∈ 𝑉 )
27	3 6 24 23 25 26 21	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑢 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑣 } ) ) )
28	27	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑢 } ) )
29	28	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑢 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
30	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 23 29	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 )
31	22 30	eqeltrrd	⊢ ( 𝜑 → 𝐸 ∈ 𝐷 )
32	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 19 31 29	mapdheq2	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑢 , 𝐹 , 𝑤 ⟩ ) = 𝐸 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑢 ⟩ ) = 𝐹 ) )
33	22 32	mpd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑤 , 𝐸 , 𝑢 ⟩ ) = 𝐹 )