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Theorem isdomn 16035
Description: Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
isdomn.b  |-  B  =  ( Base `  R
)
isdomn.t  |-  .x.  =  ( .r `  R )
isdomn.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
isdomn  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
Distinct variable groups:    x, B, y    x, R, y    x,  .0. , y
Allowed substitution hints:    .x. ( x, y)

Proof of Theorem isdomn
Dummy variables  b 
r  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . 4  |-  ( Base `  r )  e.  _V
21a1i 10 . . 3  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
3 fveq2 5525 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 isdomn.b . . . 4  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2333 . . 3  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fvex 5539 . . . . 5  |-  ( 0g
`  r )  e. 
_V
76a1i 10 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  e.  _V )
8 fveq2 5525 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
98adantr 451 . . . . 5  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
10 isdomn.z . . . . 5  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2333 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  .0.  )
12 simplr 731 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  b  =  B )
13 fveq2 5525 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
14 isdomn.t . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
1513, 14syl6eqr 2333 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1615oveqd 5875 . . . . . . . . 9  |-  ( r  =  R  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
1716adantr 451 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  ( x ( .r
`  r ) y )  =  ( x 
.x.  y ) )
18 id 19 . . . . . . . 8  |-  ( z  =  .0.  ->  z  =  .0.  )
1917, 18eqeqan12d 2298 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x ( .r
`  r ) y )  =  z  <->  ( x  .x.  y )  =  .0.  ) )
20 eqeq2 2292 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x  =  z  <->  x  =  .0.  ) )
21 eqeq2 2292 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
y  =  z  <->  y  =  .0.  ) )
2220, 21orbi12d 690 . . . . . . . 8  |-  ( z  =  .0.  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2322adantl 452 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2419, 23imbi12d 311 . . . . . 6  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( ( x ( .r `  r ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) ) )
2512, 24raleqbidv 2748 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  ( A. y  e.  b 
( ( x ( .r `  r ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) ) )
2612, 25raleqbidv 2748 . . . 4  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  ( A. x  e.  b  A. y  e.  b 
( ( x ( .r `  r ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) ) )
277, 11, 26sbcied2 3028 . . 3  |-  ( ( r  =  R  /\  b  =  B )  ->  ( [. ( 0g
`  r )  / 
z ]. A. x  e.  b  A. y  e.  b  ( ( x ( .r `  r
) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  ( x  =  .0. 
\/  y  =  .0.  ) ) ) )
282, 5, 27sbcied2 3028 . 2  |-  ( r  =  R  ->  ( [. ( Base `  r
)  /  b ]. [. ( 0g `  r
)  /  z ]. A. x  e.  b  A. y  e.  b 
( ( x ( .r `  r ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) ) )
29 df-domn 16025 . 2  |- Domn  =  {
r  e. NzRing  |  [. ( Base `  r )  / 
b ]. [. ( 0g
`  r )  / 
z ]. A. x  e.  b  A. y  e.  b  ( ( x ( .r `  r
) y )  =  z  ->  ( x  =  z  \/  y  =  z ) ) }
3028, 29elrab2 2925 1  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  B  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  NzRingcnzr 16009  Domncdomn 16021
This theorem is referenced by:  domnnzr  16036  domneq0  16038  isdomn2  16040  opprdomn  16042  abvn0b  16043  znfld  16514  ply1domn  19509  fta1b  19555  isdomn3  27523
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-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-iota 5219  df-fv 5263  df-ov 5861  df-domn 16025
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