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Theorem isdomn2 16287
Description: A ring is a domain iff all nonzero elements are non-zero-divisors. (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
isdomn2  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)

Proof of Theorem isdomn2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdomn2.b . . 3  |-  B  =  ( Base `  R
)
2 eqid 2388 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 isdomn2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdomn 16282 . 2  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
5 dfss3 3282 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  E  <->  A. x  e.  ( B  \  {  .0.  } ) x  e.  E )
6 isdomn2.t . . . . . . . . 9  |-  E  =  (RLReg `  R )
76, 1, 2, 3isrrg 16276 . . . . . . . 8  |-  ( x  e.  E  <->  ( x  e.  B  /\  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
87baib 872 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  E  <->  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
98imbi2d 308 . . . . . 6  |-  ( x  e.  B  ->  (
( x  =/=  .0.  ->  x  e.  E )  <-> 
( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) ) )
109ralbiia 2682 . . . . 5  |-  ( A. x  e.  B  (
x  =/=  .0.  ->  x  e.  E )  <->  A. x  e.  B  ( x  =/=  .0.  ->  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
11 eldifsn 3871 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
1211imbi1i 316 . . . . . . 7  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E )
)
13 impexp 434 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E
)  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1412, 13bitri 241 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1514ralbii2 2678 . . . . 5  |-  ( A. x  e.  ( B  \  {  .0.  } ) x  e.  E  <->  A. x  e.  B  ( x  =/=  .0.  ->  x  e.  E ) )
16 con34b 284 . . . . . . . . 9  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( -.  (
x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r `  R ) y )  =  .0.  ) )
17 impexp 434 . . . . . . . . . 10  |-  ( ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( -.  x  =  .0. 
->  ( -.  y  =  .0.  ->  -.  (
x ( .r `  R ) y )  =  .0.  ) ) )
18 ioran 477 . . . . . . . . . . 11  |-  ( -.  ( x  =  .0. 
\/  y  =  .0.  )  <->  ( -.  x  =  .0.  /\  -.  y  =  .0.  ) )
1918imbi1i 316 . . . . . . . . . 10  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
20 df-ne 2553 . . . . . . . . . . 11  |-  ( x  =/=  .0.  <->  -.  x  =  .0.  )
21 con34b 284 . . . . . . . . . . 11  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  )  <->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
2220, 21imbi12i 317 . . . . . . . . . 10  |-  ( ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) )  <-> 
( -.  x  =  .0.  ->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
) )
2317, 19, 223bitr4i 269 . . . . . . . . 9  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
2416, 23bitri 241 . . . . . . . 8  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( x  =/= 
.0.  ->  ( ( x ( .r `  R
) y )  =  .0.  ->  y  =  .0.  ) ) )
2524ralbii 2674 . . . . . . 7  |-  ( A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  A. y  e.  B  ( x  =/=  .0.  ->  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) )
26 r19.21v 2737 . . . . . . 7  |-  ( A. y  e.  B  (
x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) )  <-> 
( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) )
2725, 26bitri 241 . . . . . 6  |-  ( A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( x  =/= 
.0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) )
2827ralbii 2674 . . . . 5  |-  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  A. x  e.  B  ( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) )
2910, 15, 283bitr4i 269 . . . 4  |-  ( A. x  e.  ( B  \  {  .0.  } ) x  e.  E  <->  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) )
305, 29bitr2i 242 . . 3  |-  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( B  \  {  .0.  } )  C_  E )
3130anbi2i 676 . 2  |-  ( ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) )  <->  ( R  e. NzRing  /\  ( B  \  {  .0.  } )  C_  E ) )
324, 31bitri 241 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650    \ cdif 3261    C_ wss 3264   {csn 3758   ` cfv 5395  (class class class)co 6021   Basecbs 13397   .rcmulr 13458   0gc0g 13651  NzRingcnzr 16256  RLRegcrlreg 16267  Domncdomn 16268
This theorem is referenced by:  domnrrg  16288  drngdomn  16291
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-rlreg 16271  df-domn 16272
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