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Theorem rrgval 16028
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 5525 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 rrgval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5525 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
6 rrgval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
75, 6syl6eqr 2333 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
87oveqd 5875 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
9 fveq2 5525 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
10 rrgval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2333 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
128, 11eqeq12d 2297 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( .r
`  r ) y )  =  ( 0g
`  r )  <->  ( x  .x.  y )  =  .0.  ) )
1311eqeq2d 2294 . . . . . . 7  |-  ( r  =  R  ->  (
y  =  ( 0g
`  r )  <->  y  =  .0.  ) )
1412, 13imbi12d 311 . . . . . 6  |-  ( r  =  R  ->  (
( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) )  <->  ( ( x 
.x.  y )  =  .0.  ->  y  =  .0.  ) ) )
154, 14raleqbidv 2748 . . . . 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 2783 . . . 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 16024 . . . 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 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2353 . . . . 5  |-  B  e. 
_V
2019rabex 4165 . . . 4  |-  { x  e.  B  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) }  e.  _V
2116, 17, 20fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
22 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  (/) )
23 fvprc 5519 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
243, 23syl5eq 2327 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
25 rabeq 2782 . . . . . 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 15 . . . . 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 3475 . . . . 5  |-  { x  e.  (/)  |  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  y  =  .0.  ) }  =  (/)
2826, 27syl6eq 2331 . . . 4  |-  ( -.  R  e.  _V  ->  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }  =  (/) )
2922, 28eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
3021, 29pm2.61i 156 . 2  |-  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
311, 30eqtri 2303 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  RLRegcrlreg 16020
This theorem is referenced by:  isrrg  16029  rrgeq0  16031  rrgss  16033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-rlreg 16024
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