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Theorem domnrrg 16352
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 16351 . . . 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 3919 . . 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 3341 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 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   {csn 3806   ` cfv 5446   Basecbs 13461   0gc0g 13715  NzRingcnzr 16320  RLRegcrlreg 16331  Domncdomn 16332
This theorem is referenced by:  deg1ldgdomn  20009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-rlreg 16335  df-domn 16336
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