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Theorem domnrrg 16057
Description: In a domain, any nonzero element is a non-zero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
isdomn2.b  |-  B  =  ( Base `  R
)
isdomn2.t  |-  E  =  (RLReg `  R )
isdomn2.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
domnrrg  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  E )

Proof of Theorem domnrrg
StepHypRef Expression
1 isdomn2.b . . . . 5  |-  B  =  ( Base `  R
)
2 isdomn2.t . . . . 5  |-  E  =  (RLReg `  R )
3 isdomn2.z . . . . 5  |-  .0.  =  ( 0g `  R )
41, 2, 3isdomn2 16056 . . . 4  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)
54simprbi 450 . . 3  |-  ( R  e. Domn  ->  ( B  \  {  .0.  } )  C_  E )
653ad2ant1 976 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  ( B  \  {  .0.  }
)  C_  E )
7 simp2 956 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  B )
8 simp3 957 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  =/=  .0.  )
9 eldifsn 3762 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
107, 8, 9sylanbrc 645 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  ( B  \  {  .0.  } ) )
116, 10sseldd 3194 1  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416  NzRingcnzr 16025  RLRegcrlreg 16036  Domncdomn 16037
This theorem is referenced by:  deg1ldgdomn  19496
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-rlreg 16040  df-domn 16041
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