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Theorem isnzr 16330
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 5728 . . . 4  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
2 isnzr.o . . . 4  |-  .1.  =  ( 1r `  R )
31, 2syl6eqr 2486 . . 3  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
4 fveq2 5728 . . . 4  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
5 isnzr.z . . . 4  |-  .0.  =  ( 0g `  R )
64, 5syl6eqr 2486 . . 3  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
73, 6neeq12d 2616 . 2  |-  ( r  =  R  ->  (
( 1r `  r
)  =/=  ( 0g
`  r )  <->  .1.  =/=  .0.  ) )
8 df-nzr 16329 . 2  |- NzRing  =  {
r  e.  Ring  |  ( 1r `  r )  =/=  ( 0g `  r ) }
97, 8elrab2 3094 1  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454   0gc0g 13723   Ringcrg 15660   1rcur 15662  NzRingcnzr 16328
This theorem is referenced by:  nzrnz  16331  nzrrng  16332  drngnzr  16333  isnzr2  16334  rngelnzr  16336  subrgnzr  16338  chrnzr  16811  nrginvrcn  18727  ply1nzb  20045  zrhnm  24353  isdomn3  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-nzr 16329
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