Theorem isnzr 13325

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 5516	. . . 4 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
2		isnzr.o	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	eqtr4di 2228	. . 3 |- ( r = R -> ( 1r ` r ) = .1. )
4		fveq2 5516	. . . 4 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
5		isnzr.z	. . . 4 |- .0. = ( 0g ` R )
6	4, 5	eqtr4di 2228	. . 3 |- ( r = R -> ( 0g ` r ) = .0. )
7	3, 6	neeq12d 2367	. 2 |- ( r = R -> ( ( 1r ` r ) =/= ( 0g ` r ) <-> .1. =/= .0. ) )
8		df-nzr 13324	. 2 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
9	7, 8	elrab2 2897	1 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   =/= wne 2347   ` cfv 5217   0gc0g 12705   1rcur 13142   Ringcrg 13179  NzRingcnzr 13323

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-nzr 13324

This theorem is referenced by:  nzrnz  13326  ringelnzr  13328  subrgnzr  13363  zringnzr  13495