Theorem hdmap1val2

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero Y . (Contributed by NM, 16-May-2015)

	Ref	Expression
Hypotheses	hdmap1val2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1val2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1val2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1val2.s	⊢ − = ( -g ‘ 𝑈 )
	hdmap1val2.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hdmap1val2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1val2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1val2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1val2.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	hdmap1val2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
	hdmap1val2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1val2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1val2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1val2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmap1val2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1val2.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hdmap1val2	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1val2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1val2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1val2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1val2.s	⊢ − = ( -g ‘ 𝑈 )
5		hdmap1val2.o	⊢ 0 = ( 0g ‘ 𝑈 )
6		hdmap1val2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		hdmap1val2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap1val2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		hdmap1val2.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
10		hdmap1val2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
11		hdmap1val2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
12		hdmap1val2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
13		hdmap1val2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14		hdmap1val2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
15		hdmap1val2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		hdmap1val2.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
17		eqid	⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
18	16	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
19	1 2 3 4 5 6 7 8 9 17 10 11 12 13 14 15 18	hdmap1val	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( 𝑌 = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) )
20		eldifsni	⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) → 𝑌 ≠ 0 )
21	20	neneqd	⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) → ¬ 𝑌 = 0 )
22		iffalse	⊢ ( ¬ 𝑌 = 0 → if ( 𝑌 = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )
23	16 21 22	3syl	⊢ ( 𝜑 → if ( 𝑌 = 0 , ( 0g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )
24	19 23	eqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )