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Theorem rrgval 17699
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 5857 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 rrgval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2519 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5857 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
6 rrgval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
75, 6syl6eqr 2519 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
87oveqd 6292 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
9 fveq2 5857 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
10 rrgval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2519 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
128, 11eqeq12d 2482 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( .r
`  r ) y )  =  ( 0g
`  r )  <->  ( x  .x.  y )  =  .0.  ) )
1311eqeq2d 2474 . . . . . . 7  |-  ( r  =  R  ->  (
y  =  ( 0g
`  r )  <->  y  =  .0.  ) )
1412, 13imbi12d 320 . . . . . 6  |-  ( r  =  R  ->  (
( ( x ( .r `  r ) y )  =  ( 0g `  r )  ->  y  =  ( 0g `  r ) )  <->  ( ( x 
.x.  y )  =  .0.  ->  y  =  .0.  ) ) )
154, 14raleqbidv 3065 . . . . 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 3101 . . . 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 17695 . . . 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 5867 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2544 . . . . 5  |-  B  e. 
_V
2019rabex 4591 . . . 4  |-  { x  e.  B  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) }  e.  _V
2116, 17, 20fvmpt 5941 . . 3  |-  ( R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
22 fvprc 5851 . . . 4  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  (/) )
23 fvprc 5851 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
243, 23syl5eq 2513 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
25 rabeq 3100 . . . . . 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 3799 . . . . 5  |-  { x  e.  (/)  |  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  y  =  .0.  ) }  =  (/)
2826, 27syl6eq 2517 . . . 4  |-  ( -.  R  e.  _V  ->  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }  =  (/) )
2922, 28eqtr4d 2504 . . 3  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
3021, 29pm2.61i 164 . 2  |-  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
311, 30eqtri 2489 1  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106   (/)c0 3778   ` cfv 5579  (class class class)co 6275   Basecbs 14479   .rcmulr 14545   0gc0g 14684  RLRegcrlreg 17691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-rlreg 17695
This theorem is referenced by:  isrrg  17700  rrgeq0  17702  rrgss  17705
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