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Theorem isdomn 16134
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 5622 . . . 4  |-  ( Base `  r )  e.  _V
21a1i 10 . . 3  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
3 fveq2 5608 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 isdomn.b . . . 4  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2408 . . 3  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fvex 5622 . . . . 5  |-  ( 0g
`  r )  e. 
_V
76a1i 10 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  e.  _V )
8 fveq2 5608 . . . . . 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 2408 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  .0.  )
12 simplr 731 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  b  =  B )
13 fveq2 5608 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
14 isdomn.t . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
1513, 14syl6eqr 2408 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1615oveqd 5962 . . . . . . . . 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 2373 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x ( .r
`  r ) y )  =  z  <->  ( x  .x.  y )  =  .0.  ) )
20 eqeq2 2367 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x  =  z  <->  x  =  .0.  ) )
21 eqeq2 2367 . . . . . . . . 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 2824 . . . . 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 2824 . . . 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 3104 . . 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 3104 . 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 16124 . 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 3001 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   [.wsbc 3067   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .rcmulr 13306   0gc0g 13499  NzRingcnzr 16108  Domncdomn 16120
This theorem is referenced by:  domnnzr  16135  domneq0  16137  isdomn2  16139  opprdomn  16141  abvn0b  16142  znfld  16620  ply1domn  19613  fta1b  19659  isdomn3  26846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-domn 16124
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