Theorem mapdheq2biN

Description: Lemmma for ~? mapdh . Part (2) in Baer p. 45. The bidirectional version of mapdheq2 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdh.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	mapdh.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 ) } ) ) ) ) )
	mapdh.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh.s	⊢ − = ( -g ‘ 𝑈 )
	mapdhc.o	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdh.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdh.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdhc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdhcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdhe2.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdhe2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
	mapdh.ne3	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdh.my	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
Assertion	mapdheq2biN	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
2		mapdh.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 ) } ) ) ) ) )
3		mapdh.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		mapdh.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5		mapdh.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6		mapdh.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
7		mapdh.s	⊢ − = ( -g ‘ 𝑈 )
8		mapdhc.o	⊢ 0 = ( 0g ‘ 𝑈 )
9		mapdh.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10		mapdh.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11		mapdh.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
12		mapdh.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
13		mapdh.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
14		mapdh.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdhc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdhcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		mapdhe2.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdhe2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
20		mapdh.ne3	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
21		mapdh.my	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
22	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	mapdheq2	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )
23	20	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 21 18 17 15 23	mapdheq2	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ) )
25	22 24	impbid	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )