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Theorem domnrrg 16041
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 16040 . . . 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 3749 . . 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 3181 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400  NzRingcnzr 16009  RLRegcrlreg 16020  Domncdomn 16021
This theorem is referenced by:  deg1ldgdomn  19480
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-rlreg 16024  df-domn 16025
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