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

Theorem nzrnz 16362
Description: One and zero are different in 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
nzrnz  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )

Proof of Theorem nzrnz
StepHypRef Expression
1 isnzr.o . . 3  |-  .1.  =  ( 1r `  R )
2 isnzr.z . . 3  |-  .0.  =  ( 0g `  R )
31, 2isnzr 16361 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
43simprbi 452 1  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    =/= wne 2605   ` cfv 5483   0gc0g 13754   Ringcrg 15691   1rcur 15693  NzRingcnzr 16359
This theorem is referenced by:  nzrunit  16368  subrgnzr  16369  fidomndrng  16398  nm1  18734  deg1pw  20074  ply1nz  20075  ply1nzb  20076  lgsqrlem4  21159  zrhnm  24384  uvcf1  27256  lindfind2  27303  mon1pid  27539  deg1mhm  27541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-nzr 16360
  Copyright terms: Public domain W3C validator