Theorem isnzr 19259

Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
isnzr.o	|- .1. = ( 1r ` R )
isnzr.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
isnzr	|- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Proof of Theorem isnzr

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
2		isnzr.o	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	syl6eqr 2674	. . 3 |- ( r = R -> ( 1r ` r ) = .1. )
4		fveq2 6191	. . . 4 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
5		isnzr.z	. . . 4 |- .0. = ( 0g ` R )
6	4, 5	syl6eqr 2674	. . 3 |- ( r = R -> ( 0g ` r ) = .0. )
7	3, 6	neeq12d 2855	. 2 |- ( r = R -> ( ( 1r ` r ) =/= ( 0g ` r ) <-> .1. =/= .0. ) )
8		df-nzr 19258	. 2 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
9	7, 8	elrab2 3366	1 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   0gc0g 16100   1rcur 18501   Ringcrg 18547  NzRingcnzr 19257

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-ne 2795  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-nzr 19258

This theorem is referenced by:  nzrnz  19260  nzrring  19261  drngnzr  19262  isnzr2  19263  isnzr2hash  19264  ringelnzr  19266  subrgnzr  19268  zringnzr  19830  chrnzr  19878  nrginvrcn  22496  ply1nzb  23882  zrhnm  30013  isdomn3  37782