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 β€˜ π‘Š )
reipcl.h ⊒ , = ( ·𝑖 β€˜ π‘Š )
Assertion ipge0 ( ( π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ) β†’ 0 ≀ ( 𝐴 , 𝐴 ) )

Proof

Step Hyp Ref Expression
1 reipcl.v ⊒ 𝑉 = ( Base β€˜ π‘Š )
2 reipcl.h ⊒ , = ( ·𝑖 β€˜ π‘Š )
3 cphngp ⊒ ( π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmGrp )
4 eqid ⊒ ( norm β€˜ π‘Š ) = ( norm β€˜ π‘Š )
5 1 4 nmcl ⊒ ( ( π‘Š ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ) β†’ ( ( norm β€˜ π‘Š ) β€˜ 𝐴 ) ∈ ℝ )
6 3 5 sylan ⊒ ( ( π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ) β†’ ( ( norm β€˜ π‘Š ) β€˜ 𝐴 ) ∈ ℝ )
7 6 sqge0d ⊒ ( ( π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ) β†’ 0 ≀ ( ( ( norm β€˜ π‘Š ) β€˜ 𝐴 ) ↑ 2 ) )
8 1 2 4 nmsq ⊒ ( ( π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ) β†’ ( ( ( norm β€˜ π‘Š ) β€˜ 𝐴 ) ↑ 2 ) = ( 𝐴 , 𝐴 ) )
9 7 8 breqtrd ⊒ ( ( π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ) β†’ 0 ≀ ( 𝐴 , 𝐴 ) )