Theorem ipz

Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of Kreyszig p. 129. (Contributed by NM, 24-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dip0r.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	dip0r.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
	dip0r.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	ipz	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝑃 𝐴 ) = 0 ↔ 𝐴 = 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		dip0r.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		dip0r.5	⊢ 𝑍 = ( 0vec ‘ 𝑈 )
3		dip0r.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4		eqid	⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 )
5	1 4 3	ipidsq	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐴 ) = ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ↑ 2 ) )
6	5	eqeq1d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝑃 𝐴 ) = 0 ↔ ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ↑ 2 ) = 0 ) )
7	1 4	nvcl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ∈ ℝ )
8	7	recnd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ∈ ℂ )
9		sqeq0	⊢ ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ∈ ℂ → ( ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) = 0 ) )
10	8 9	syl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) = 0 ) )
11	1 2 4	nvz	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( ( normCV ‘ 𝑈 ) ‘ 𝐴 ) = 0 ↔ 𝐴 = 𝑍 ) )
12	6 10 11	3bitrd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝑃 𝐴 ) = 0 ↔ 𝐴 = 𝑍 ) )