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Theorem rrgval 16274
Description: Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e  |-  E  =  (RLReg `  R )
rrgval.b  |-  B  =  ( Base `  R
)
rrgval.t  |-  .x.  =  ( .r `  R )
rrgval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
rrgval  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
Distinct variable groups:    x, B, y    x, R, y
Allowed substitution hints:    .x. ( x, y)    E( x, y)    .0. ( x, y)

Proof of Theorem rrgval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . 2  |-  E  =  (RLReg `  R )
2 fveq2 5668 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 rrgval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2437 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5668 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
6 rrgval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
75, 6syl6eqr 2437 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
87oveqd 6037 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
9 fveq2 5668 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
10 rrgval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2437 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
128, 11eqeq12d 2401 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( .r
`  r ) y )  =  ( 0g
`  r )  <->  ( x  .x.  y )  =  .0.  ) )
1311eqeq2d 2398 . . . . . . 7  |-  ( r  =  R  ->  (
y  =  ( 0g
`  r )  <->  y  =  .0.  ) )
1412, 13imbi12d 312 . . . . . 6  |-  ( r  =  R  ->  (
( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) )  <->  ( ( x 
.x.  y )  =  .0.  ->  y  =  .0.  ) ) )
154, 14raleqbidv 2859 . . . . 5  |-  ( r  =  R  ->  ( A. y  e.  ( Base `  r ) ( ( x ( .r
`  r ) y )  =  ( 0g
`  r )  -> 
y  =  ( 0g
`  r ) )  <->  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) ) )
164, 15rabeqbidv 2894 . . . 4  |-  ( r  =  R  ->  { x  e.  ( Base `  r
)  |  A. y  e.  ( Base `  r
) ( ( x ( .r `  r
) y )  =  ( 0g `  r
)  ->  y  =  ( 0g `  r ) ) }  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
17 df-rlreg 16270 . . . 4  |- RLReg  =  ( r  e.  _V  |->  { x  e.  ( Base `  r )  |  A. y  e.  ( Base `  r ) ( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) ) } )
18 fvex 5682 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2457 . . . . 5  |-  B  e. 
_V
2019rabex 4295 . . . 4  |-  { x  e.  B  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) }  e.  _V
2116, 17, 20fvmpt 5745 . . 3  |-  ( R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
22 fvprc 5662 . . . 4  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  (/) )
23 fvprc 5662 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
243, 23syl5eq 2431 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
25 rabeq 2893 . . . . . 6  |-  ( B  =  (/)  ->  { x  e.  B  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) }  =  { x  e.  (/)  |  A. y  e.  B  ( ( x 
.x.  y )  =  .0.  ->  y  =  .0.  ) } )
2624, 25syl 16 . . . . 5  |-  ( -.  R  e.  _V  ->  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }  =  {
x  e.  (/)  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) } )
27 rab0 3591 . . . . 5  |-  { x  e.  (/)  |  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  y  =  .0.  ) }  =  (/)
2826, 27syl6eq 2435 . . . 4  |-  ( -.  R  e.  _V  ->  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }  =  (/) )
2922, 28eqtr4d 2422 . . 3  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
3021, 29pm2.61i 158 . 2  |-  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
311, 30eqtri 2407 1  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899   (/)c0 3571   ` cfv 5394  (class class class)co 6020   Basecbs 13396   .rcmulr 13457   0gc0g 13650  RLRegcrlreg 16266
This theorem is referenced by:  isrrg  16275  rrgeq0  16277  rrgss  16279
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-rlreg 16270
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