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	$$⊢A\in \mathscr{H}$$
Assertion	norm-i-i	$$⊢{norm}_{ℎ}\left(A\right)=0\leftrightarrow A=0_{ℎ}$$

Proof

Step	Hyp	Ref	Expression
1		normcl.1	$$⊢A\in \mathscr{H}$$
2		norm-i	$$⊢A\in \mathscr{H}\to \left({norm}_{ℎ}\left(A\right)=0\leftrightarrow A=0_{ℎ}\right)$$
3	1 2	ax-mp	$$⊢{norm}_{ℎ}\left(A\right)=0\leftrightarrow A=0_{ℎ}$$