hdmap1val

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval .) TODO: change I = ( x e. _V |-> ... to ( ph -> ( I<. X , F , Y > ) = ... in e.g. mapdh8 to shorten proofs with no $d on x . (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.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmap1val.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1val.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	hdmap1val	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( 𝑌 = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 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.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
16		hdmap1val.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
17		hdmap1val.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
18		df-ot	⊢ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ = ⟨ ⟨ 𝑋 , 𝐹 ⟩ , 𝑌 ⟩
19		opelxp	⊢ ( ⟨ 𝑋 , 𝐹 ⟩ ∈ ( 𝑉 × 𝐷 ) ↔ ( 𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) )
20	15 16 19	sylanbrc	⊢ ( 𝜑 → ⟨ 𝑋 , 𝐹 ⟩ ∈ ( 𝑉 × 𝐷 ) )
21		opelxp	⊢ ( ⟨ ⟨ 𝑋 , 𝐹 ⟩ , 𝑌 ⟩ ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ↔ ( ⟨ 𝑋 , 𝐹 ⟩ ∈ ( 𝑉 × 𝐷 ) ∧ 𝑌 ∈ 𝑉 ) )
22	20 17 21	sylanbrc	⊢ ( 𝜑 → ⟨ ⟨ 𝑋 , 𝐹 ⟩ , 𝑌 ⟩ ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) )
23	18 22	eqeltrid	⊢ ( 𝜑 → ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ∈ ( ( 𝑉 × 𝐷 ) × 𝑉 ) )
24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 23	hdmap1vallem	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) ) ) ) )
25		ot3rdg	⊢ ( 𝑌 ∈ 𝑉 → ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝑌 )
26	17 25	syl	⊢ ( 𝜑 → ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝑌 )
27	26	eqeq1d	⊢ ( 𝜑 → ( ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 0 ↔ 𝑌 = 0 ) )
28	26	sneqd	⊢ ( 𝜑 → { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } = { 𝑌 } )
29	28	fveq2d	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) = ( 𝑁 ‘ { 𝑌 } ) )
30	29	fveqeq2d	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) ) = ( 𝐽 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ) )
31		ot1stg	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉 ) → ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) = 𝑋 )
32	15 16 17 31	syl3anc	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) = 𝑋 )
33	32 26	oveq12d	⊢ ( 𝜑 → ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) = ( 𝑋 − 𝑌 ) )
34	33	sneqd	⊢ ( 𝜑 → { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } = { ( 𝑋 − 𝑌 ) } )
35	34	fveq2d	⊢ ( 𝜑 → ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) = ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) )
36	35	fveq2d	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) )
37		ot2ndg	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉 ) → ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) = 𝐹 )
38	15 16 17 37	syl3anc	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) = 𝐹 )
39	38	oveq1d	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) = ( 𝐹 𝑅 ℎ ) )
40	39	sneqd	⊢ ( 𝜑 → { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } = { ( 𝐹 𝑅 ℎ ) } )
41	40	fveq2d	⊢ ( 𝜑 → ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) )
42	36 41	eqeq12d	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) )
43	30 42	anbi12d	⊢ ( 𝜑 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )
44	43	riotabidv	⊢ ( 𝜑 → ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )
45	27 44	ifbieq2d	⊢ ( 𝜑 → if ( ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) − ( 2^nd ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) 𝑅 ℎ ) } ) ) ) ) = if ( 𝑌 = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) )
46	24 45	eqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = if ( 𝑌 = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) )

Metamath Proof Explorer

Theorem hdmap1val

Proof