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Theorem domnrrg 16281
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 16280 . . . 4  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)
54simprbi 451 . . 3  |-  ( R  e. Domn  ->  ( B  \  {  .0.  } )  C_  E )
653ad2ant1 978 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  ( B  \  {  .0.  }
)  C_  E )
7 simp2 958 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  B )
8 simp3 959 . . 3  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  =/=  .0.  )
9 eldifsn 3864 . . 3  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
107, 8, 9sylanbrc 646 . 2  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  ( B  \  {  .0.  } ) )
116, 10sseldd 3286 1  |-  ( ( R  e. Domn  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  X  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2544    \ cdif 3254    C_ wss 3257   {csn 3751   ` cfv 5388   Basecbs 13390   0gc0g 13644  NzRingcnzr 16249  RLRegcrlreg 16260  Domncdomn 16261
This theorem is referenced by:  deg1ldgdomn  19878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-iota 5352  df-fun 5390  df-fv 5396  df-ov 6017  df-rlreg 16264  df-domn 16265
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