Theorem isnzr 14300

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 5661	. . . 4 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
2		isnzr.o	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	eqtr4di 2283	. . 3 |- ( r = R -> ( 1r ` r ) = .1. )
4		fveq2 5661	. . . 4 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
5		isnzr.z	. . . 4 |- .0. = ( 0g ` R )
6	4, 5	eqtr4di 2283	. . 3 |- ( r = R -> ( 0g ` r ) = .0. )
7	3, 6	neeq12d 2432	. 2 |- ( r = R -> ( ( 1r ` r ) =/= ( 0g ` r ) <-> .1. =/= .0. ) )
8		df-nzr 14299	. 2 |- NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
9	7, 8	elrab2 2975	1 |- ( R e. NzRing <-> ( R e. Ring /\ .1. =/= .0. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   ` cfv 5343   0gc0g 13443   1rcur 14077   Ringcrg 14114  NzRingcnzr 14298

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-rex 2526  df-rab 2529  df-v 2814  df-un 3214  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-br 4103  df-iota 5303  df-fv 5351  df-nzr 14299

This theorem is referenced by:  nzrnz  14301  isnzr2  14303  opprnzrbg  14304  ringelnzr  14306  subrgnzr  14361  zringnzr  14722