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Metamath Proof Explorer

Theorem normval

Description: The value of the norm of a vector in Hilbert space. Definition of norm in Beran p. 96. In the literature, the norm of A is usually written as "|| A ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

Ref Expression
Assertion normval ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 oveq12 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐴 ) → ( 𝑥 ·ih 𝑥 ) = ( 𝐴 ·ih 𝐴 ) )
2 1 anidms ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·ih 𝑥 ) = ( 𝐴 ·ih 𝐴 ) )
3 2 fveq2d ⊢ ( 𝑥 = 𝐴 → ( √ ‘ ( 𝑥 ·ih 𝑥 ) ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )
4 dfhnorm2 ⊢ normℎ = ( 𝑥 ∈ ℋ ↦ ( √ ‘ ( 𝑥 ·ih 𝑥 ) ) )
5 fvex ⊢ ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) ∈ V
6 3 4 5 fvmpt ⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )