Theorem hdmap1valc

Description: Connect the value of the preliminary map from vectors to functionals I to the hypothesis L used by earlier theorems. Note: the X e. ( V \ { .0. } ) hypothesis could be the more general X e. V but the former will be easier to use. TODO: use the I function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015)

	Ref	Expression
Hypotheses	hdmap1valc.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1valc.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1valc.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1valc.s	⊢ − = ( -g ‘ 𝑈 )
	hdmap1valc.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hdmap1valc.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1valc.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1valc.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1valc.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	hdmap1valc.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	hdmap1valc.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	hdmap1valc.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1valc.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1valc.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1valc.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1valc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1valc.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	hdmap1valc.l	⊢ 𝐿 = ( 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 ) } ) ) ) ) )
Assertion	hdmap1valc	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1valc.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1valc.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1valc.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1valc.s	⊢ − = ( -g ‘ 𝑈 )
5		hdmap1valc.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		hdmap1valc.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		hdmap1valc.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap1valc.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		hdmap1valc.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
10		hdmap1valc.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
11		hdmap1valc.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		hdmap1valc.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		hdmap1valc.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
14		hdmap1valc.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		hdmap1valc.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
16		hdmap1valc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
17		hdmap1valc.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
18		hdmap1valc.l	⊢ 𝐿 = ( 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 ) } ) ) ) ) )
19	15	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
20	1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 16 17	hdmap1val	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( 𝑌 = 0 , 𝑄 , ( ℩ 𝑔 ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑔 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝑔 ) } ) ) ) ) )
21	18	hdmap1cbv	⊢ 𝐿 = ( 𝑤 ∈ V ↦ if ( ( 2nd ‘ 𝑤 ) = 0 , 𝑄 , ( ℩ 𝑔 ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑤 ) } ) ) = ( 𝐽 ‘ { 𝑔 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑤 ) ) − ( 2nd ‘ 𝑤 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑤 ) ) 𝑅 𝑔 ) } ) ) ) ) )
22	10 21 19 16 17	mapdhval	⊢ ( 𝜑 → ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( 𝑌 = 0 , 𝑄 , ( ℩ 𝑔 ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑔 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝑔 ) } ) ) ) ) )
23	20 22	eqtr4d	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )