Theorem isnzr 13399

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 5527	. . . 4 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
2		isnzr.o	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	eqtr4di 2238	. . 3 |- ( r = R -> ( 1r ` r ) = .1. )
4		fveq2 5527	. . . 4 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
5		isnzr.z	. . . 4 |- .0. = ( 0g ` R )
6	4, 5	eqtr4di 2238	. . 3 |- ( r = R -> ( 0g ` r ) = .0. )
7	3, 6	neeq12d 2377	. 2 |- ( r = R -> ( ( 1r ` r ) =/= ( 0g ` r ) <-> .1. =/= .0. ) )
8		df-nzr 13398	. 2 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
9	7, 8	elrab2 2908	1 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   =/= wne 2357   ` cfv 5228   0gc0g 12722   1rcur 13206   Ringcrg 13243  NzRingcnzr 13397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-nzr 13398

This theorem is referenced by:  nzrnz  13400  ringelnzr  13402  subrgnzr  13457  zringnzr  13749