MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdomn Unicode version

Theorem isdomn 16309
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 5701 . . . 4  |-  ( Base `  r )  e.  _V
21a1i 11 . . 3  |-  ( r  =  R  ->  ( Base `  r )  e. 
_V )
3 fveq2 5687 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 isdomn.b . . . 4  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2454 . . 3  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fvex 5701 . . . . 5  |-  ( 0g
`  r )  e. 
_V
76a1i 11 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  e.  _V )
8 fveq2 5687 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
98adantr 452 . . . . 5  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  ( 0g
`  R ) )
10 isdomn.z . . . . 5  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2454 . . . 4  |-  ( ( r  =  R  /\  b  =  B )  ->  ( 0g `  r
)  =  .0.  )
12 simplr 732 . . . . 5  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  b  =  B )
13 fveq2 5687 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
14 isdomn.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
1513, 14syl6eqr 2454 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1615proplem3 13871 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  ( x ( .r
`  r ) y )  =  ( x 
.x.  y ) )
17 id 20 . . . . . . . 8  |-  ( z  =  .0.  ->  z  =  .0.  )
1816, 17eqeqan12d 2419 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x ( .r
`  r ) y )  =  z  <->  ( x  .x.  y )  =  .0.  ) )
19 eqeq2 2413 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x  =  z  <->  x  =  .0.  ) )
20 eqeq2 2413 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
y  =  z  <->  y  =  .0.  ) )
2119, 20orbi12d 691 . . . . . . . 8  |-  ( z  =  .0.  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2221adantl 453 . . . . . . 7  |-  ( ( ( r  =  R  /\  b  =  B )  /\  z  =  .0.  )  ->  (
( x  =  z  \/  y  =  z )  <->  ( x  =  .0.  \/  y  =  .0.  ) ) )
2318, 22imbi12d 312 . . . . . 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 2876 . . . . 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 2876 . . . 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 3158 . . 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 3158 . 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 16299 . 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 3054 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [.wsbc 3121   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485   0gc0g 13678  NzRingcnzr 16283  Domncdomn 16295
This theorem is referenced by:  domnnzr  16310  domneq0  16312  isdomn2  16314  opprdomn  16316  abvn0b  16317  znfld  16796  ply1domn  19999  fta1b  20045  isdomn3  27391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-domn 16299
  Copyright terms: Public domain W3C validator