Theorem isnzr 14194

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 5639	. . . 4 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
2		isnzr.o	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	eqtr4di 2282	. . 3 |- ( r = R -> ( 1r ` r ) = .1. )
4		fveq2 5639	. . . 4 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
5		isnzr.z	. . . 4 |- .0. = ( 0g ` R )
6	4, 5	eqtr4di 2282	. . 3 |- ( r = R -> ( 0g ` r ) = .0. )
7	3, 6	neeq12d 2422	. 2 |- ( r = R -> ( ( 1r ` r ) =/= ( 0g ` r ) <-> .1. =/= .0. ) )
8		df-nzr 14193	. 2 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
9	7, 8	elrab2 2965	1 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   =/= wne 2402   ` cfv 5326   0gc0g 13338   1rcur 13971   Ringcrg 14008  NzRingcnzr 14192

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-nzr 14193

This theorem is referenced by:  nzrnz  14195  isnzr2  14197  opprnzrbg  14198  ringelnzr  14200  subrgnzr  14255  zringnzr  14615