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 ) ) )