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Theorem isdomn3 27523
Description: Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
Hypotheses
Ref Expression
isdomn3.b  |-  B  =  ( Base `  R
)
isdomn3.z  |-  .0.  =  ( 0g `  R )
isdomn3.u  |-  U  =  (mulGrp `  R )
Assertion
Ref Expression
isdomn3  |-  ( R  e. Domn 
<->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U ) ) )

Proof of Theorem isdomn3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdomn3.b . . 3  |-  B  =  ( Base `  R
)
2 eqid 2283 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 isdomn3.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 eqid 2283 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
65, 3isnzr 16011 . . . . 5  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  .0.  )
)
76anbi1i 676 . . . 4  |-  ( ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) )  <->  ( ( R  e.  Ring  /\  ( 1r `  R )  =/= 
.0.  )  /\  A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
8 anass 630 . . . 4  |-  ( ( ( R  e.  Ring  /\  ( 1r `  R
)  =/=  .0.  )  /\  A. x  e.  B  A. y  e.  B  ( ( x ( .r `  R ) y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) ) )  <->  ( R  e.  Ring  /\  ( ( 1r `  R )  =/= 
.0.  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) ) ) )
97, 8bitri 240 . . 3  |-  ( ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) )  <->  ( R  e.  Ring  /\  ( ( 1r `  R )  =/= 
.0.  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) ) ) )
101, 5rngidcl 15361 . . . . . . 7  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
11 eldifsn 3749 . . . . . . . 8  |-  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  <->  ( ( 1r `  R )  e.  B  /\  ( 1r
`  R )  =/= 
.0.  ) )
1211baibr 872 . . . . . . 7  |-  ( ( 1r `  R )  e.  B  ->  (
( 1r `  R
)  =/=  .0.  <->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) ) )
1310, 12syl 15 . . . . . 6  |-  ( R  e.  Ring  ->  ( ( 1r `  R )  =/=  .0.  <->  ( 1r `  R )  e.  ( B  \  {  .0.  } ) ) )
141, 2rngcl 15354 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( .r `  R ) y )  e.  B )
15143expb 1152 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  R
) y )  e.  B )
1615biantrurd 494 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
x ( .r `  R ) y )  =/=  .0.  <->  ( (
x ( .r `  R ) y )  e.  B  /\  (
x ( .r `  R ) y )  =/=  .0.  ) ) )
17 eldifsn 3749 . . . . . . . . . 10  |-  ( ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } )  <->  ( (
x ( .r `  R ) y )  e.  B  /\  (
x ( .r `  R ) y )  =/=  .0.  ) )
1816, 17syl6bbr 254 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
x ( .r `  R ) y )  =/=  .0.  <->  ( x
( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) )
1918imbi2d 307 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  =/=  .0.  )  <->  ( ( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) ) )
20192ralbidva 2583 . . . . . . 7  |-  ( R  e.  Ring  ->  ( A. x  e.  B  A. y  e.  B  (
( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  =/=  .0.  )  <->  A. x  e.  B  A. y  e.  B  (
( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) ) )
21 con34b 283 . . . . . . . . 9  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( -.  (
x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r `  R ) y )  =  .0.  ) )
22 neanior 2531 . . . . . . . . . 10  |-  ( ( x  =/=  .0.  /\  y  =/=  .0.  )  <->  -.  (
x  =  .0.  \/  y  =  .0.  )
)
23 df-ne 2448 . . . . . . . . . 10  |-  ( ( x ( .r `  R ) y )  =/=  .0.  <->  -.  (
x ( .r `  R ) y )  =  .0.  )
2422, 23imbi12i 316 . . . . . . . . 9  |-  ( ( ( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  =/=  .0.  )  <->  ( -.  ( x  =  .0.  \/  y  =  .0.  )  ->  -.  ( x ( .r
`  R ) y )  =  .0.  )
)
2521, 24bitr4i 243 . . . . . . . 8  |-  ( ( ( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  ( ( x  =/=  .0.  /\  y  =/=  .0.  )  ->  (
x ( .r `  R ) y )  =/=  .0.  ) )
26252ralbii 2569 . . . . . . 7  |-  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  A. x  e.  B  A. y  e.  B  ( ( x  =/= 
.0.  /\  y  =/=  .0.  )  ->  ( x ( .r `  R
) y )  =/= 
.0.  ) )
27 impexp 433 . . . . . . . . . 10  |-  ( ( ( ( x  e.  B  /\  y  e.  B )  /\  (
x  =/=  .0.  /\  y  =/=  .0.  ) )  ->  ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) )  <->  ( (
x  e.  B  /\  y  e.  B )  ->  ( ( x  =/= 
.0.  /\  y  =/=  .0.  )  ->  ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) ) )
28 an4 797 . . . . . . . . . . . 12  |-  ( ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  =/=  .0.  /\  y  =/= 
.0.  ) )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  ) ) )
29 eldifsn 3749 . . . . . . . . . . . . 13  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
30 eldifsn 3749 . . . . . . . . . . . . 13  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
3129, 30anbi12i 678 . . . . . . . . . . . 12  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  <->  ( (
x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  ) ) )
3228, 31bitr4i 243 . . . . . . . . . . 11  |-  ( ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  =/=  .0.  /\  y  =/= 
.0.  ) )  <->  ( x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) ) )
3332imbi1i 315 . . . . . . . . . 10  |-  ( ( ( ( x  e.  B  /\  y  e.  B )  /\  (
x  =/=  .0.  /\  y  =/=  .0.  ) )  ->  ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) )  <->  ( (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  ->  (
x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) ) )
3427, 33bitr3i 242 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  B
)  ->  ( (
x  =/=  .0.  /\  y  =/=  .0.  )  -> 
( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) )  <->  ( (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  ->  (
x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) ) )
35342albii 1554 . . . . . . . 8  |-  ( A. x A. y ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  =/= 
.0.  /\  y  =/=  .0.  )  ->  ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) )  <->  A. x A. y ( ( x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) )
36 r2al 2580 . . . . . . . 8  |-  ( A. x  e.  B  A. y  e.  B  (
( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) )  <->  A. x A. y ( ( x  e.  B  /\  y  e.  B )  ->  (
( x  =/=  .0.  /\  y  =/=  .0.  )  ->  ( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) ) )
37 r2al 2580 . . . . . . . 8  |-  ( A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } )  <->  A. x A. y
( ( x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x ( .r
`  R ) y )  e.  ( B 
\  {  .0.  }
) ) )
3835, 36, 373bitr4ri 269 . . . . . . 7  |-  ( A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } )  <->  A. x  e.  B  A. y  e.  B  ( ( x  =/= 
.0.  /\  y  =/=  .0.  )  ->  ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) )
3920, 26, 383bitr4g 279 . . . . . 6  |-  ( R  e.  Ring  ->  ( A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
)  <->  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) )
4013, 39anbi12d 691 . . . . 5  |-  ( R  e.  Ring  ->  ( ( ( 1r `  R
)  =/=  .0.  /\  A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) )  <->  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) ) ) )
41 isdomn3.u . . . . . . 7  |-  U  =  (mulGrp `  R )
4241rngmgp 15347 . . . . . 6  |-  ( R  e.  Ring  ->  U  e. 
Mnd )
4341, 1mgpbas 15331 . . . . . . . . 9  |-  B  =  ( Base `  U
)
4441, 5rngidval 15343 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 0g `  U
)
4541, 2mgpplusg 15329 . . . . . . . . 9  |-  ( .r
`  R )  =  ( +g  `  U
)
4643, 44, 45issubm 14425 . . . . . . . 8  |-  ( U  e.  Mnd  ->  (
( B  \  {  .0.  } )  e.  (SubMnd `  U )  <->  ( ( B  \  {  .0.  }
)  C_  B  /\  ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B 
\  {  .0.  }
) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) ) )
47 3anass 938 . . . . . . . 8  |-  ( ( ( B  \  {  .0.  } )  C_  B  /\  ( 1r `  R
)  e.  ( B 
\  {  .0.  }
)  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) )  <->  ( ( B  \  {  .0.  }
)  C_  B  /\  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) ) )
4846, 47syl6bb 252 . . . . . . 7  |-  ( U  e.  Mnd  ->  (
( B  \  {  .0.  } )  e.  (SubMnd `  U )  <->  ( ( B  \  {  .0.  }
)  C_  B  /\  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) ) ) )
49 difss 3303 . . . . . . . 8  |-  ( B 
\  {  .0.  }
)  C_  B
5049biantrur 492 . . . . . . 7  |-  ( ( ( 1r `  R
)  e.  ( B 
\  {  .0.  }
)  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) )  <->  ( ( B  \  {  .0.  }
)  C_  B  /\  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R
) y )  e.  ( B  \  {  .0.  } ) ) ) )
5148, 50syl6bbr 254 . . . . . 6  |-  ( U  e.  Mnd  ->  (
( B  \  {  .0.  } )  e.  (SubMnd `  U )  <->  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) ) ) )
5242, 51syl 15 . . . . 5  |-  ( R  e.  Ring  ->  ( ( B  \  {  .0.  } )  e.  (SubMnd `  U )  <->  ( ( 1r `  R )  e.  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( x ( .r `  R ) y )  e.  ( B  \  {  .0.  } ) ) ) )
5340, 52bitr4d 247 . . . 4  |-  ( R  e.  Ring  ->  ( ( ( 1r `  R
)  =/=  .0.  /\  A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) )  <->  ( B  \  {  .0.  } )  e.  (SubMnd `  U
) ) )
5453pm5.32i 618 . . 3  |-  ( ( R  e.  Ring  /\  (
( 1r `  R
)  =/=  .0.  /\  A. x  e.  B  A. y  e.  B  (
( x ( .r
`  R ) y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )  <->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U
) ) )
559, 54bitri 240 . 2  |-  ( ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  ( (
x ( .r `  R ) y )  =  .0.  ->  (
x  =  .0.  \/  y  =  .0.  )
) )  <->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U
) ) )
564, 55bitri 240 1  |-  ( R  e. Domn 
<->  ( R  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  U ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934   A.wal 1527    = 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   Mndcmnd 14361  SubMndcsubmnd 14414  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339  NzRingcnzr 16009  Domncdomn 16021
This theorem is referenced by:  deg1mhm  27526
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  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-mgp 15326  df-rng 15340  df-ur 15342  df-nzr 16010  df-domn 16025
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