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Metamath Proof Explorer

Theorem nmsq

Description: The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of Ponnusamy p. 362. (Contributed by NM, 1-Feb-2007) (Revised by Mario Carneiro, 7-Oct-2015)

Ref Expression
Hypotheses nmsq.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
nmsq.h ⊢ , = ( ·𝑖 ‘ 𝑊 )
nmsq.n ⊢ 𝑁 = ( norm ‘ 𝑊 )
Assertion nmsq ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )

Proof

Step Hyp Ref Expression
1 nmsq.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 nmsq.h ⊢ , = ( ·𝑖 ‘ 𝑊 )
3 nmsq.n ⊢ 𝑁 = ( norm ‘ 𝑊 )
4 1 2 3 cphnm ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) = ( √ ‘ ( 𝐴 , 𝐴 ) ) )
5 4 oveq1d ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ ( 𝐴 , 𝐴 ) ) ↑ 2 ) )
6 1 2 cphipcl ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 𝐴 ) ∈ ℂ )
7 6 3anidm23 ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 , 𝐴 ) ∈ ℂ )
8 7 sqsqrtd ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( √ ‘ ( 𝐴 , 𝐴 ) ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )
9 5 8 eqtrd ⊢ ( ( 𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )