Theorem isnrg 22464

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.)

Hypotheses

Ref	Expression
isnrg.1	|- N = ( norm ` R )
isnrg.2	|- A = (AbsVal ` R )

Assertion

Ref	Expression
isnrg	|- ( R e. NrmRing <-> ( R e. NrmGrp /\ N e. A ) )

Proof of Theorem isnrg

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( r = R -> ( norm ` r ) = ( norm ` R ) )
2		isnrg.1	. . . 4 |- N = ( norm ` R )
3	1, 2	syl6eqr 2674	. . 3 |- ( r = R -> ( norm ` r ) = N )
4		fveq2 6191	. . . 4 |- ( r = R -> (AbsVal ` r ) = (AbsVal ` R ) )
5		isnrg.2	. . . 4 |- A = (AbsVal ` R )
6	4, 5	syl6eqr 2674	. . 3 |- ( r = R -> (AbsVal ` r ) = A )
7	3, 6	eleq12d 2695	. 2 |- ( r = R -> ( ( norm ` r ) e. (AbsVal ` r ) <-> N e. A ) )
8		df-nrg 22390	. 2 |- NrmRing = { r e. NrmGrp | ( norm ` r ) e. (AbsVal ` r ) }
9	7, 8	elrab2 3366	1 |- ( R e. NrmRing <-> ( R e. NrmGrp /\ N e. A ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  AbsValcabv 18816   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-nrg 22390

This theorem is referenced by:  nrgabv  22465  nrgngp  22466  subrgnrg  22477  tngnrg  22478  cnnrg  22584  zhmnrg  30011