Theorem mapdh75fN

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

	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 } ) )
	mapdh75a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	mapdh75d.b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
	mapdh75d.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	mapdh75d.un	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	mapdh75d.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh75d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh75d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	mapdh75fN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑌 ⟩ ) = 𝐺 )

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		mapdh75a	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18		mapdh75d.b	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
19		mapdh75d.vw	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
20		mapdh75d.un	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		mapdh75d.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
22		mapdh75d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23		mapdh75d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24	22	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
25	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
26	21	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
27	23	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
28	3 6 25 26 24 27 20	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
29	28	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
30	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 21 24 29	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
31	17 30	eqeltrrd	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
32	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 21 22 31 29	mapdheq	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
33	17 32	mpbid	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
34	33	simpld	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
35	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23	mapdh75d	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 )
36	1 2 3 4 5 6 7 8 9 10 11 12 13 14 31 34 35 19 22 23	mapdh75e	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , 𝐸 , 𝑌 ⟩ ) = 𝐺 )