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Theorem isdomn2 18143
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 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 isdomn2.z . . 3  |-  .0.  =  ( 0g `  R )
41, 2, 3isdomn 18138 . 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 3479 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  E  <->  A. x  e.  ( B  \  {  .0.  } ) x  e.  E )
6 isdomn2.t . . . . . . . . 9  |-  E  =  (RLReg `  R )
76, 1, 2, 3isrrg 18131 . . . . . . . 8  |-  ( x  e.  E  <->  ( x  e.  B  /\  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
87baib 901 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  E  <->  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
98imbi2d 314 . . . . . 6  |-  ( x  e.  B  ->  (
( x  =/=  .0.  ->  x  e.  E )  <-> 
( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) ) )
109ralbiia 2884 . . . . 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 4141 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
1211imbi1i 323 . . . . . . 7  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E )
)
13 impexp 444 . . . . . . 7  |-  ( ( ( x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E
)  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1412, 13bitri 249 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( x  e.  B  ->  ( x  =/=  .0.  ->  x  e.  E ) ) )
1514ralbii2 2883 . . . . 5  |-  ( A. x  e.  ( B  \  {  .0.  } ) x  e.  E  <->  A. x  e.  B  ( x  =/=  .0.  ->  x  e.  E ) )
16 con34b 290 . . . . . . . . 9  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( -.  (
x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r `  R ) y )  =  .0.  ) )
17 impexp 444 . . . . . . . . . 10  |-  ( ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( -.  x  =  .0. 
->  ( -.  y  =  .0.  ->  -.  (
x ( .r `  R ) y )  =  .0.  ) ) )
18 ioran 488 . . . . . . . . . . 11  |-  ( -.  ( x  =  .0. 
\/  y  =  .0.  )  <->  ( -.  x  =  .0.  /\  -.  y  =  .0.  ) )
1918imbi1i 323 . . . . . . . . . 10  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
20 df-ne 2651 . . . . . . . . . . 11  |-  ( x  =/=  .0.  <->  -.  x  =  .0.  )
21 con34b 290 . . . . . . . . . . 11  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  )  <->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
2220, 21imbi12i 324 . . . . . . . . . 10  |-  ( ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) )  <-> 
( -.  x  =  .0.  ->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
) )
2317, 19, 223bitr4i 277 . . . . . . . . 9  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( x  =/=  .0.  ->  ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
2416, 23bitri 249 . . . . . . . 8  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( x  =/= 
.0.  ->  ( ( x ( .r `  R
) y )  =  .0.  ->  y  =  .0.  ) ) )
2524ralbii 2885 . . . . . . 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 2859 . . . . . . 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 249 . . . . . 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 2885 . . . . 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 277 . . . 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 250 . . 3  |-  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( B  \  {  .0.  } )  C_  E )
3130anbi2i 692 . 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 249 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  ( B  \  {  .0.  }
)  C_  E )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458    C_ wss 3461   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   0gc0g 14929  NzRingcnzr 18100  RLRegcrlreg 18122  Domncdomn 18123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-rlreg 18126  df-domn 18127
This theorem is referenced by:  domnrrg  18144  drngdomn  18147
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