Theorem normge0

Description: The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	normge0	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( normℎ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		hiidrcl	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )
2		hiidge0	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( 𝐴 ·ih 𝐴 ) )
3	1 2	sqrtge0d	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )
4		normval	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )
5	3 4	breqtrrd	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( normℎ ‘ 𝐴 ) )