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Metamath Proof Explorer

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 ) )