Theorem obsip

Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015)

	Ref	Expression
Hypotheses	isobs.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	isobs.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
	isobs.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	isobs.u	⊢ 1 = ( 1r ‘ 𝐹 )
	isobs.z	⊢ 0 = ( 0g ‘ 𝐹 )
Assertion	obsip	⊢ ( ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 , 𝑄 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		isobs.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		isobs.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		isobs.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
4		isobs.u	⊢ 1 = ( 1r ‘ 𝐹 )
5		isobs.z	⊢ 0 = ( 0g ‘ 𝐹 )
6		eqid	⊢ ( ocv ‘ 𝑊 ) = ( ocv ‘ 𝑊 )
7		eqid	⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
8	1 2 3 4 5 6 7	isobs	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) ↔ ( 𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 , 𝑦 ) = if ( 𝑥 = 𝑦 , 1 , 0 ) ∧ ( ( ocv ‘ 𝑊 ) ‘ 𝐵 ) = { ( 0g ‘ 𝑊 ) } ) ) )
9	8	simp3bi	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 , 𝑦 ) = if ( 𝑥 = 𝑦 , 1 , 0 ) ∧ ( ( ocv ‘ 𝑊 ) ‘ 𝐵 ) = { ( 0g ‘ 𝑊 ) } ) )
10	9	simpld	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 , 𝑦 ) = if ( 𝑥 = 𝑦 , 1 , 0 ) )
11		oveq1	⊢ ( 𝑥 = 𝑃 → ( 𝑥 , 𝑦 ) = ( 𝑃 , 𝑦 ) )
12		eqeq1	⊢ ( 𝑥 = 𝑃 → ( 𝑥 = 𝑦 ↔ 𝑃 = 𝑦 ) )
13	12	ifbid	⊢ ( 𝑥 = 𝑃 → if ( 𝑥 = 𝑦 , 1 , 0 ) = if ( 𝑃 = 𝑦 , 1 , 0 ) )
14	11 13	eqeq12d	⊢ ( 𝑥 = 𝑃 → ( ( 𝑥 , 𝑦 ) = if ( 𝑥 = 𝑦 , 1 , 0 ) ↔ ( 𝑃 , 𝑦 ) = if ( 𝑃 = 𝑦 , 1 , 0 ) ) )
15		oveq2	⊢ ( 𝑦 = 𝑄 → ( 𝑃 , 𝑦 ) = ( 𝑃 , 𝑄 ) )
16		eqeq2	⊢ ( 𝑦 = 𝑄 → ( 𝑃 = 𝑦 ↔ 𝑃 = 𝑄 ) )
17	16	ifbid	⊢ ( 𝑦 = 𝑄 → if ( 𝑃 = 𝑦 , 1 , 0 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) )
18	15 17	eqeq12d	⊢ ( 𝑦 = 𝑄 → ( ( 𝑃 , 𝑦 ) = if ( 𝑃 = 𝑦 , 1 , 0 ) ↔ ( 𝑃 , 𝑄 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) ) )
19	14 18	rspc2v	⊢ ( ( 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 , 𝑦 ) = if ( 𝑥 = 𝑦 , 1 , 0 ) → ( 𝑃 , 𝑄 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) ) )
20	10 19	syl5com	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ( 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 , 𝑄 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) ) )
21	20	3impib	⊢ ( ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 , 𝑄 ) = if ( 𝑃 = 𝑄 , 1 , 0 ) )