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Theorem isdomn2 16040
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 2283 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 isdomn2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdomn 16035 . 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 3170 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  E  <->  A. x  e.  ( B  \  {  .0.  } ) x  e.  E )
6 isdomn2.t . . . . . . . . 9  |-  E  =  (RLReg `  R )
76, 1, 2, 3isrrg 16029 . . . . . . . 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 2575 . . . . 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 3749 . . . . . . . 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 2571 . . . . 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 2448 . . . . . . . . . . 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 2567 . . . . . . 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 2630 . . . . . . 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 2567 . . . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  NzRingcnzr 16009  RLRegcrlreg 16020  Domncdomn 16021
This theorem is referenced by:  domnrrg  16041  drngdomn  16044
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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