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Theorem isnzr 16011
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.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
2 isnzr.o . . . 4  |-  .1.  =  ( 1r `  R )
31, 2syl6eqr 2333 . . 3  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
4 fveq2 5525 . . . 4  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
5 isnzr.z . . . 4  |-  .0.  =  ( 0g `  R )
64, 5syl6eqr 2333 . . 3  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
73, 6neeq12d 2461 . 2  |-  ( r  =  R  ->  (
( 1r `  r
)  =/=  ( 0g
`  r )  <->  .1.  =/=  .0.  ) )
8 df-nzr 16010 . 2  |- NzRing  =  {
r  e.  Ring  |  ( 1r `  r )  =/=  ( 0g `  r ) }
97, 8elrab2 2925 1  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   0gc0g 13400   Ringcrg 15337   1rcur 15339  NzRingcnzr 16009
This theorem is referenced by:  nzrnz  16012  nzrrng  16013  drngnzr  16014  isnzr2  16015  rngelnzr  16017  subrgnzr  16019  chrnzr  16484  nrginvrcn  18202  ply1nzb  19508  isdomn3  27523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-nzr 16010
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