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Metamath Proof Explorer

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 )