hdmap1vallem

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015)

	Ref	Expression
Hypotheses	hdmap1val.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1fval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1fval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1fval.s	⊢ − = ( -_g ‘ 𝑈 )
	hdmap1fval.o	⊢ 0 = ( 0_g ‘ 𝑈 )
	hdmap1fval.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1fval.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1fval.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1fval.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
	hdmap1fval.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
	hdmap1fval.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	hdmap1fval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1fval.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1fval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1val.t	⊢ ( 𝜑 → 𝑇 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) )
Assertion	hdmap1vallem	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑇 ) = if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1val.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1fval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1fval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1fval.s	⊢ − = ( -_g ‘ 𝑈 )
5		hdmap1fval.o	⊢ 0 = ( 0_g ‘ 𝑈 )
6		hdmap1fval.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		hdmap1fval.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap1fval.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		hdmap1fval.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
10		hdmap1fval.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
11		hdmap1fval.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		hdmap1fval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		hdmap1fval.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
14		hdmap1fval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) )
15		hdmap1val.t	⊢ ( 𝜑 → 𝑇 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) )
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14	hdmap1fval	⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) )
17	16	fveq1d	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑇 ) = ( ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ‘ 𝑇 ) )
18	10	fvexi	⊢ 𝑄 ∈ V
19		riotaex	⊢ ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ∈ V
20	18 19	ifex	⊢ if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) ∈ V
21		fveqeq2	⊢ ( 𝑥 = 𝑇 → ( ( 2^nd ‘ 𝑥 ) = 0 ↔ ( 2^nd ‘ 𝑇 ) = 0 ) )
22		fveq2	⊢ ( 𝑥 = 𝑇 → ( 2^nd ‘ 𝑥 ) = ( 2^nd ‘ 𝑇 ) )
23	22	sneqd	⊢ ( 𝑥 = 𝑇 → { ( 2^nd ‘ 𝑥 ) } = { ( 2^nd ‘ 𝑇 ) } )
24	23	fveq2d	⊢ ( 𝑥 = 𝑇 → ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) = ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) )
25	24	fveqeq2d	⊢ ( 𝑥 = 𝑇 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ) )
26		2fveq3	⊢ ( 𝑥 = 𝑇 → ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) )
27	26 22	oveq12d	⊢ ( 𝑥 = 𝑇 → ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) )
28	27	sneqd	⊢ ( 𝑥 = 𝑇 → { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } = { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } )
29	28	fveq2d	⊢ ( 𝑥 = 𝑇 → ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) = ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) )
30	29	fveq2d	⊢ ( 𝑥 = 𝑇 → ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) )
31		2fveq3	⊢ ( 𝑥 = 𝑇 → ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) = ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) )
32	31	oveq1d	⊢ ( 𝑥 = 𝑇 → ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) )
33	32	sneqd	⊢ ( 𝑥 = 𝑇 → { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } = { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } )
34	33	fveq2d	⊢ ( 𝑥 = 𝑇 → ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) )
35	30 34	eqeq12d	⊢ ( 𝑥 = 𝑇 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) )
36	25 35	anbi12d	⊢ ( 𝑥 = 𝑇 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) )
37	36	riotabidv	⊢ ( 𝑥 = 𝑇 → ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) )
38	21 37	ifbieq2d	⊢ ( 𝑥 = 𝑇 → if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) = if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) )
39		eqid	⊢ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
40	38 39	fvmptg	⊢ ( ( 𝑇 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ∧ if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) ∈ V ) → ( ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ‘ 𝑇 ) = if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) )
41	15 20 40	sylancl	⊢ ( 𝜑 → ( ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ‘ 𝑇 ) = if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) )
42	17 41	eqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑇 ) = if ( ( 2^nd ‘ 𝑇 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑇 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) − ( 2^nd ‘ 𝑇 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) 𝑅 ℎ ) } ) ) ) ) )

Metamath Proof Explorer

Theorem hdmap1vallem

Proof