Metamath Proof Explorer

Theorem ipf

Description: Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	ipcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
		ipcl.7	⊢ 𝑃 = ( ·_𝑖OLD ‘ 𝑈 )
	Assertion	ipf	⊢ ( 𝑈 ∈ NrmCVec → 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ )

Proof

Step	Hyp	Ref	Expression
1		ipcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		ipcl.7	⊢ 𝑃 = ( ·_𝑖OLD ‘ 𝑈 )
3		eqid	⊢ ( +_𝑣 ‘ 𝑈 ) = ( +_𝑣 ‘ 𝑈 )
4		eqid	⊢ ( ·_𝑠OLD ‘ 𝑈 ) = ( ·_𝑠OLD ‘ 𝑈 )
5		eqid	⊢ ( norm_CV ‘ 𝑈 ) = ( norm_CV ‘ 𝑈 )
6	1 3 4 5 2	ipval	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝑃 𝑦 ) = ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) )
7	1 2	dipcl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝑃 𝑦 ) ∈ ℂ )
8	6 7	eqeltrrd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ∈ ℂ )
9	8	3expib	⊢ ( 𝑈 ∈ NrmCVec → ( ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ∈ ℂ ) )
10	9	ralrimivv	⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ∈ ℂ )
11		eqid	⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) )
12	11	fmpo	⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ∈ ℂ ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ) : ( 𝑋 × 𝑋 ) ⟶ ℂ )
13	10 12	sylib	⊢ ( 𝑈 ∈ NrmCVec → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ) : ( 𝑋 × 𝑋 ) ⟶ ℂ )
14	1 3 4 5 2	dipfval	⊢ ( 𝑈 ∈ NrmCVec → 𝑃 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ) )
15	14	feq1d	⊢ ( 𝑈 ∈ NrmCVec → ( 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( Σ 𝑘 ∈ ( 1 ... 4 ) ( ( i ↑ 𝑘 ) · ( ( ( norm_CV ‘ 𝑈 ) ‘ ( 𝑥 ( +_𝑣 ‘ 𝑈 ) ( ( i ↑ 𝑘 ) ( ·_𝑠OLD ‘ 𝑈 ) 𝑦 ) ) ) ↑ 2 ) ) / 4 ) ) : ( 𝑋 × 𝑋 ) ⟶ ℂ ) )
16	13 15	mpbird	⊢ ( 𝑈 ∈ NrmCVec → 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ )