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

Proof

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