Theorem isnrg

Description: A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	isnrg.1	|- N = ( norm ` R )
	isnrg.2	|- A = ( AbsVal ` R )
Assertion	isnrg	|- ( R e. NrmRing <-> ( R e. NrmGrp /\ N e. A ) )

Proof

Step	Hyp	Ref	Expression
1		isnrg.1	|- N = ( norm ` R )
2		isnrg.2	|- A = ( AbsVal ` R )
3		fveq2	|- ( r = R -> ( norm ` r ) = ( norm ` R ) )
4	3 1	eqtr4di	|- ( r = R -> ( norm ` r ) = N )
5		fveq2	|- ( r = R -> ( AbsVal ` r ) = ( AbsVal ` R ) )
6	5 2	eqtr4di	|- ( r = R -> ( AbsVal ` r ) = A )
7	4 6	eleq12d	|- ( r = R -> ( ( norm ` r ) e. ( AbsVal ` r ) <-> N e. A ) )
8		df-nrg	|- NrmRing = { r e. NrmGrp | ( norm ` r ) e. ( AbsVal ` r ) }
9	7 8	elrab2	|- ( R e. NrmRing <-> ( R e. NrmGrp /\ N e. A ) )