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Theorem isdomn 18070
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 5882 . . . 4  |-  ( Base `  r )  e.  _V
21a1i 11 . . 3  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
3 fveq2 5872 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 isdomn.b . . . 4  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2516 . . 3  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fvex 5882 . . . . 5  |-  ( 0g
`  r )  e. 
_V
76a1i 11 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  e.  _V )
8 fveq2 5872 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
98adantr 465 . . . . 5  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
10 isdomn.z . . . . 5  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2516 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  .0.  )
12 simplr 755 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  b  =  B )
13 fveq2 5872 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
14 isdomn.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
1513, 14syl6eqr 2516 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1615oveqdr 6320 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  ( x ( .r
`  r ) y )  =  ( x 
.x.  y ) )
17 id 22 . . . . . . . 8  |-  ( z  =  .0.  ->  z  =  .0.  )
1816, 17eqeqan12d 2480 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x ( .r
`  r ) y )  =  z  <->  ( x  .x.  y )  =  .0.  ) )
19 eqeq2 2472 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x  =  z  <->  x  =  .0.  ) )
20 eqeq2 2472 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
y  =  z  <->  y  =  .0.  ) )
2119, 20orbi12d 709 . . . . . . . 8  |-  ( z  =  .0.  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2221adantl 466 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2318, 22imbi12d 320 . . . . . 6  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( ( x ( .r `  r ) y )  =  z  ->  ( x  =  z  \/  y  =  z ) )  <->  ( (
x  .x.  y )  =  .0.  ->  ( x  =  .0.  \/  y  =  .0.  ) ) ) )
2412, 23raleqbidv 3068 . . . . 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.  ) ) ) )
2512, 24raleqbidv 3068 . . . 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.  ) ) ) )
267, 11, 25sbcied2 3365 . . 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.  ) ) ) )
272, 5, 26sbcied2 3365 . 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.  ) ) ) )
28 df-domn 18059 . 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 ) ) }
2927, 28elrab2 3259 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327   ` cfv 5594  (class class class)co 6296   Basecbs 14644   .rcmulr 14713   0gc0g 14857  NzRingcnzr 18032  Domncdomn 18055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-domn 18059
This theorem is referenced by:  domnnzr  18071  domneq0  18073  isdomn2  18075  opprdomn  18077  abvn0b  18078  znfld  18726  ply1domn  22650  fta1b  22696  isdomn3  31368
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