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Theorem isdomn2 16056
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 2296 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 isdomn2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdomn 16051 . 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 3183 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  E  <->  A. x  e.  ( B  \  {  .0.  } ) x  e.  E )
6 isdomn2.t . . . . . . . . 9  |-  E  =  (RLReg `  R )
76, 1, 2, 3isrrg 16045 . . . . . . . 8  |-  ( x  e.  E  <->  ( x  e.  B  /\  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
87baib 871 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  E  <->  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
98imbi2d 307 . . . . . 6  |-  ( x  e.  B  ->  (
( x  =/=  .0.  ->  x  e.  E )  <-> 
( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) ) )
109ralbiia 2588 . . . . 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 3762 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
1211imbi1i 315 . . . . . . 7  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E )
)
13 impexp 433 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E
)  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1412, 13bitri 240 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1514ralbii2 2584 . . . . 5  |-  ( A. x  e.  ( B  \  {  .0.  } ) x  e.  E  <->  A. x  e.  B  ( x  =/=  .0.  ->  x  e.  E ) )
16 con34b 283 . . . . . . . . 9  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( -.  (
x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r `  R ) y )  =  .0.  ) )
17 impexp 433 . . . . . . . . . 10  |-  ( ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( -.  x  =  .0. 
->  ( -.  y  =  .0.  ->  -.  (
x ( .r `  R ) y )  =  .0.  ) ) )
18 ioran 476 . . . . . . . . . . 11  |-  ( -.  ( x  =  .0. 
\/  y  =  .0.  )  <->  ( -.  x  =  .0.  /\  -.  y  =  .0.  ) )
1918imbi1i 315 . . . . . . . . . 10  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
20 df-ne 2461 . . . . . . . . . . 11  |-  ( x  =/=  .0.  <->  -.  x  =  .0.  )
21 con34b 283 . . . . . . . . . . 11  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  )  <->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
2220, 21imbi12i 316 . . . . . . . . . 10  |-  ( ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) )  <-> 
( -.  x  =  .0.  ->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
) )
2317, 19, 223bitr4i 268 . . . . . . . . 9  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
2416, 23bitri 240 . . . . . . . 8  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( x  =/= 
.0.  ->  ( ( x ( .r `  R
) y )  =  .0.  ->  y  =  .0.  ) ) )
2524ralbii 2580 . . . . . . 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 2643 . . . . . . 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 240 . . . . . 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 2580 . . . . 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 268 . . . 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 241 . . 3  |-  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( B  \  {  .0.  } )  C_  E )
3130anbi2i 675 . 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 240 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416  NzRingcnzr 16025  RLRegcrlreg 16036  Domncdomn 16037
This theorem is referenced by:  domnrrg  16057  drngdomn  16060
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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