Theorem hdmap1cl

Description: Convert closure theorem mapdhcl to use HDMap1 function. (Contributed by NM, 15-May-2015)

	Ref	Expression
Hypotheses	hdmap1eq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1eq2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1eq2.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hdmap1eq2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1eq2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1eq2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
	hdmap1eq2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1eq2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1eq2.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
	hdmap1cl.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	hdmap1cl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1cl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	hdmap1cl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		hdmap1eq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1eq2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1eq2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1eq2.o	⊢ 0 = ( 0g ‘ 𝑈 )
5		hdmap1eq2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6		hdmap1eq2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
7		hdmap1eq2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
8		hdmap1eq2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
9		hdmap1eq2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
10		hdmap1eq2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
11		hdmap1eq2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		hdmap1eq2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
13		hdmap1eq2.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
14		hdmap1cl.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
15		hdmap1cl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
16		hdmap1cl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
17		eqid	⊢ ( -g ‘ 𝑈 ) = ( -g ‘ 𝑈 )
18		eqid	⊢ ( -g ‘ 𝐶 ) = ( -g ‘ 𝐶 )
19		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
20		eqid	⊢ ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) )
21	1 2 3 17 4 5 6 7 18 19 8 9 10 11 15 12 16 20	hdmap1valc	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
22	19 20 1 9 2 3 17 4 5 6 7 18 8 11 12 13 15 16 14	mapdhcl	⊢ ( 𝜑 → ( ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑈 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
23	21 22	eqeltrd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )