MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnzr Unicode version

Theorem isnzr 16027
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 5541 . . . 4  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
2 isnzr.o . . . 4  |-  .1.  =  ( 1r `  R )
31, 2syl6eqr 2346 . . 3  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
4 fveq2 5541 . . . 4  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
5 isnzr.z . . . 4  |-  .0.  =  ( 0g `  R )
64, 5syl6eqr 2346 . . 3  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
73, 6neeq12d 2474 . 2  |-  ( r  =  R  ->  (
( 1r `  r
)  =/=  ( 0g
`  r )  <->  .1.  =/=  .0.  ) )
8 df-nzr 16026 . 2  |- NzRing  =  {
r  e.  Ring  |  ( 1r `  r )  =/=  ( 0g `  r ) }
97, 8elrab2 2938 1  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271   0gc0g 13416   Ringcrg 15353   1rcur 15355  NzRingcnzr 16025
This theorem is referenced by:  nzrnz  16028  nzrrng  16029  drngnzr  16030  isnzr2  16031  rngelnzr  16033  subrgnzr  16035  chrnzr  16500  nrginvrcn  18218  ply1nzb  19524  isdomn3  27626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-nzr 16026
  Copyright terms: Public domain W3C validator