Database
BASIC TOPOLOGY
Metric spaces
Normed algebraic structures
nmval
Metamath Proof Explorer
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 ) )