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Theorem rrgval 17356
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 5689 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 rrgval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2491 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
5 fveq2 5689 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
6 rrgval.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
75, 6syl6eqr 2491 . . . . . . . . 9  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
87oveqd 6106 . . . . . . . 8  |-  ( r  =  R  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
9 fveq2 5689 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
10 rrgval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
119, 10syl6eqr 2491 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
128, 11eqeq12d 2455 . . . . . . 7  |-  ( r  =  R  ->  (
( x ( .r
`  r ) y )  =  ( 0g
`  r )  <->  ( x  .x.  y )  =  .0.  ) )
1311eqeq2d 2452 . . . . . . 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 2929 . . . . 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 2965 . . . 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 17352 . . . 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 5699 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2511 . . . . 5  |-  B  e. 
_V
2019rabex 4441 . . . 4  |-  { x  e.  B  |  A. y  e.  B  (
( x  .x.  y
)  =  .0.  ->  y  =  .0.  ) }  e.  _V
2116, 17, 20fvmpt 5772 . . 3  |-  ( R  e.  _V  ->  (RLReg `  R )  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) } )
22 fvprc 5683 . . . 4  |-  ( -.  R  e.  _V  ->  (RLReg `  R )  =  (/) )
23 fvprc 5683 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
243, 23syl5eq 2485 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
25 rabeq 2964 . . . . . 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 3656 . . . . 5  |-  { x  e.  (/)  |  A. y  e.  B  ( (
x  .x.  y )  =  .0.  ->  y  =  .0.  ) }  =  (/)
2826, 27syl6eq 2489 . . . 4  |-  ( -.  R  e.  _V  ->  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }  =  (/) )
2922, 28eqtr4d 2476 . . 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 2461 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 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970   (/)c0 3635   ` cfv 5416  (class class class)co 6089   Basecbs 14172   .rcmulr 14237   0gc0g 14376  RLRegcrlreg 17348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-rlreg 17352
This theorem is referenced by:  isrrg  17357  rrgeq0  17359  rrgss  17362
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