Theorem nmmul

Description: The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	nmmul.x	|- X = ( Base ` R )
	nmmul.n	|- N = ( norm ` R )
	nmmul.t	|- .x. = ( .r ` R )
Assertion	nmmul	|- ( ( R e. NrmRing /\ A e. X /\ B e. X ) -> ( N ` ( A .x. B ) ) = ( ( N ` A ) x. ( N ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		nmmul.x	|- X = ( Base ` R )
2		nmmul.n	|- N = ( norm ` R )
3		nmmul.t	|- .x. = ( .r ` R )
4		eqid	|- ( AbsVal ` R ) = ( AbsVal ` R )
5	2 4	nrgabv	|- ( R e. NrmRing -> N e. ( AbsVal ` R ) )
6	4 1 3	abvmul	|- ( ( N e. ( AbsVal ` R ) /\ A e. X /\ B e. X ) -> ( N ` ( A .x. B ) ) = ( ( N ` A ) x. ( N ` B ) ) )
7	5 6	syl3an1	|- ( ( R e. NrmRing /\ A e. X /\ B e. X ) -> ( N ` ( A .x. B ) ) = ( ( N ` A ) x. ( N ` B ) ) )