Theorem mapdh75e

Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). X , Y , Z are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN .) (Contributed by NM, 2-May-2015)

	Ref	Expression
Hypotheses	mapdh75.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh75.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh75.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh75.s	⊢ − = ( -g ‘ 𝑈 )
	mapdh75.o	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdh75.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh75.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh75.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh75.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdh75.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	mapdh75.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh75.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh75.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 ) } ) ) ) ) )
	mapdh75.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdh75.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh75.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh75b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
	mapdh75e.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	mapdh75e.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh75e.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	mapdh75e	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑋 ⟩ ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		mapdh75.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdh75.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdh75.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		mapdh75.s	⊢ − = ( -g ‘ 𝑈 )
5		mapdh75.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		mapdh75.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdh75.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdh75.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		mapdh75.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
10		mapdh75.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
11		mapdh75.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdh75.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		mapdh75.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		mapdh75.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdh75.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh75.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh75b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
18		mapdh75e.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
19		mapdh75e.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh75e.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
21	20	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
22	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 21 18	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
23	17 22	eqeltrrd	⊢ ( 𝜑 → 𝐸 ∈ 𝐷 )
24	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 23 18	mapdheq2	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑋 ⟩ ) = 𝐹 ) )
25	17 24	mpd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑋 ⟩ ) = 𝐹 )