Theorem ipge0

Description: The inner product in a subcomplex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Hypotheses	reipcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	reipcl.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
Assertion	ipge0	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → 0 ≤ ( 𝐴 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		reipcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		reipcl.h	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		cphngp	⊢ ( 𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp )
4		eqid	⊢ ( norm ‘ 𝑊 ) = ( norm ‘ 𝑊 )
5	1 4	nmcl	⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ 𝐴 ) ∈ ℝ )
6	3 5	sylan	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ 𝐴 ) ∈ ℝ )
7	6	sqge0d	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → 0 ≤ ( ( ( norm ‘ 𝑊 ) ‘ 𝐴 ) ↑ 2 ) )
8	1 2 4	nmsq	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )
9	7 8	breqtrd	⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → 0 ≤ ( 𝐴 , 𝐴 ) )