Theorem reipcl

Description: An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015)

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

Proof

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