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Theorem isdomn2 17495
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 17490 . 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 3455 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  E  <->  A. x  e.  ( B  \  {  .0.  } ) x  e.  E )
6 isdomn2.t . . . . . . . . 9  |-  E  =  (RLReg `  R )
76, 1, 2, 3isrrg 17483 . . . . . . . 8  |-  ( x  e.  E  <->  ( x  e.  B  /\  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
87baib 896 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  E  <->  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  y  =  .0.  ) ) )
98imbi2d 316 . . . . . 6  |-  ( x  e.  B  ->  (
( x  =/=  .0.  ->  x  e.  E )  <-> 
( x  =/=  .0.  ->  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  y  =  .0.  ) ) ) )
109ralbiia 2838 . . . . 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 4109 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
1211imbi1i 325 . . . . . . 7  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  ->  x  e.  E )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  ->  x  e.  E )
)
13 impexp 446 . . . . . . 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 2837 . . . . 5  |-  ( A. x  e.  ( B  \  {  .0.  } ) x  e.  E  <->  A. x  e.  B  ( x  =/=  .0.  ->  x  e.  E ) )
16 con34b 292 . . . . . . . . 9  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( -.  (
x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r `  R ) y )  =  .0.  ) )
17 impexp 446 . . . . . . . . . 10  |-  ( ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( -.  x  =  .0. 
->  ( -.  y  =  .0.  ->  -.  (
x ( .r `  R ) y )  =  .0.  ) ) )
18 ioran 490 . . . . . . . . . . 11  |-  ( -.  ( x  =  .0. 
\/  y  =  .0.  )  <->  ( -.  x  =  .0.  /\  -.  y  =  .0.  ) )
1918imbi1i 325 . . . . . . . . . 10  |-  ( ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )  <->  ( ( -.  x  =  .0.  /\  -.  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
20 df-ne 2650 . . . . . . . . . . 11  |-  ( x  =/=  .0.  <->  -.  x  =  .0.  )
21 con34b 292 . . . . . . . . . . 11  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  y  =  .0.  )  <->  ( -.  y  =  .0.  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
2220, 21imbi12i 326 . . . . . . . . . 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 2839 . . . . . . 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 2909 . . . . . . 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 2839 . . . . 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 694 . 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 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    \ cdif 3434    C_ wss 3437   {csn 3986   ` cfv 5527  (class class class)co 6201   Basecbs 14293   .rcmulr 14359   0gc0g 14498  NzRingcnzr 17463  RLRegcrlreg 17474  Domncdomn 17475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-rlreg 17478  df-domn 17479
This theorem is referenced by:  domnrrg  17496  drngdomn  17499
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