Theorem nmval

Description: The value of the norm as the distance to zero. Problem 1 of Kreyszig p. 63. (Contributed by NM, 4-Dec-2006) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	nmfval.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
	nmfval.x	⊢ 𝑋 = ( Base ‘ 𝑊 )
	nmfval.z	⊢ 0 = ( 0g ‘ 𝑊 )
	nmfval.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
Assertion	nmval	⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) )

Proof

Step	Hyp	Ref	Expression
1		nmfval.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
2		nmfval.x	⊢ 𝑋 = ( Base ‘ 𝑊 )
3		nmfval.z	⊢ 0 = ( 0g ‘ 𝑊 )
4		nmfval.d	⊢ 𝐷 = ( dist ‘ 𝑊 )
5		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐷 0 ) = ( 𝐴 𝐷 0 ) )
6	1 2 3 4	nmfval	⊢ 𝑁 = ( 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝐷 0 ) )
7		ovex	⊢ ( 𝐴 𝐷 0 ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) )