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Theorem isdomn 17790
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 5881 . . . 4  |-  ( Base `  r )  e.  _V
21a1i 11 . . 3  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
3 fveq2 5871 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 isdomn.b . . . 4  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2526 . . 3  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fvex 5881 . . . . 5  |-  ( 0g
`  r )  e. 
_V
76a1i 11 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  e.  _V )
8 fveq2 5871 . . . . . 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 2526 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  .0.  )
12 simplr 754 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  b  =  B )
13 fveq2 5871 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
14 isdomn.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
1513, 14syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1615proplem3 14958 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  ( x ( .r
`  r ) y )  =  ( x 
.x.  y ) )
17 id 22 . . . . . . . 8  |-  ( z  =  .0.  ->  z  =  .0.  )
1816, 17eqeqan12d 2490 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x ( .r
`  r ) y )  =  z  <->  ( x  .x.  y )  =  .0.  ) )
19 eqeq2 2482 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x  =  z  <->  x  =  .0.  ) )
20 eqeq2 2482 . . . . . . . . 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 3077 . . . . 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 3077 . . . 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 3374 . . 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 3374 . 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 17779 . 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 3268 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 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   [.wsbc 3336   ` cfv 5593  (class class class)co 6294   Basecbs 14502   .rcmulr 14568   0gc0g 14707  NzRingcnzr 17752  Domncdomn 17775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6297  df-domn 17779
This theorem is referenced by:  domnnzr  17791  domneq0  17793  isdomn2  17795  opprdomn  17797  abvn0b  17798  znfld  18445  ply1domn  22369  fta1b  22415  isdomn3  31061
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