Theorem norm-i-i

Description: Theorem 3.3(i) of Beran p. 97. (Contributed by NM, 5-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	normcl.1	⊢ 𝐴 ∈ ℋ
Assertion	norm-i-i	⊢ ( ( normℎ ‘ 𝐴 ) = 0 ↔ 𝐴 = 0ℎ )

Proof

Step	Hyp	Ref	Expression
1		normcl.1	⊢ 𝐴 ∈ ℋ
2		norm-i	⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) = 0 ↔ 𝐴 = 0ℎ ) )
3	1 2	ax-mp	⊢ ( ( normℎ ‘ 𝐴 ) = 0 ↔ 𝐴 = 0ℎ )