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Theorem dvdsrval 15443
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
dvdsr.1  |-  B  =  ( Base `  R
)
dvdsr.2  |-  .||  =  (
||r `  R )
dvdsr.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
dvdsrval  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
Distinct variable groups:    x, y,  .||    x, z, B, y    x, R, y, z    x,  .x. , y, z
Allowed substitution hint:    .|| ( z)

Proof of Theorem dvdsrval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . 3  |-  .||  =  (
||r `  R )
2 fveq2 5541 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvdsr.1 . . . . . . . . 9  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2346 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  B )
54eleq2d 2363 . . . . . . 7  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  <->  x  e.  B ) )
64rexeqdv 2756 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  ( Base `  r ) ( z ( .r `  r ) x )  =  y  <->  E. z  e.  B  ( z
( .r `  r
) x )  =  y ) )
75, 6anbi12d 691 . . . . . 6  |-  ( r  =  R  ->  (
( x  e.  (
Base `  r )  /\  E. z  e.  (
Base `  r )
( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z ( .r
`  r ) x )  =  y ) ) )
8 fveq2 5541 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvdsr.3 . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1110oveqd 5891 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( .r `  r ) x )  =  ( z  .x.  x ) )
1211eqeq1d 2304 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( .r
`  r ) x )  =  y  <->  ( z  .x.  x )  =  y ) )
1312rexbidv 2577 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  B  ( z ( .r
`  r ) x )  =  y  <->  E. z  e.  B  ( z  .x.  x )  =  y ) )
1413anbi2d 684 . . . . . 6  |-  ( r  =  R  ->  (
( x  e.  B  /\  E. z  e.  B  ( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z  .x.  x
)  =  y ) ) )
157, 14bitrd 244 . . . . 5  |-  ( r  =  R  ->  (
( x  e.  (
Base `  r )  /\  E. z  e.  (
Base `  r )
( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z  .x.  x
)  =  y ) ) )
1615opabbidv 4098 . . . 4  |-  ( r  =  R  ->  { <. x ,  y >.  |  ( x  e.  ( Base `  r )  /\  E. z  e.  ( Base `  r ) ( z ( .r `  r
) x )  =  y ) }  =  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
17 df-dvdsr 15439 . . . 4  |-  ||r  =  (
r  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  r
)  /\  E. z  e.  ( Base `  r
) ( z ( .r `  r ) x )  =  y ) } )
18 fvex 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2366 . . . . 5  |-  B  e. 
_V
20 eqcom 2298 . . . . . . . . 9  |-  ( ( z  .x.  x )  =  y  <->  y  =  ( z  .x.  x
) )
2120rexbii 2581 . . . . . . . 8  |-  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  y  =  ( z  .x.  x
) )
2221abbii 2408 . . . . . . 7  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  =  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }
2319abrexex 5779 . . . . . . 7  |-  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }  e.  _V
2422, 23eqeltri 2366 . . . . . 6  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V
2524a1i 10 . . . . 5  |-  ( x  e.  B  ->  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V )
2619, 25opabex3 5785 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  e.  _V
2716, 17, 26fvmpt 5618 . . 3  |-  ( R  e.  _V  ->  ( ||r `  R )  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
281, 27syl5eq 2340 . 2  |-  ( R  e.  _V  ->  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
29 fvprc 5535 . . . 4  |-  ( -.  R  e.  _V  ->  (
||r `  R )  =  (/) )
301, 29syl5eq 2340 . . 3  |-  ( -.  R  e.  _V  ->  .||  =  (/) )
31 opabn0 4311 . . . . 5  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =/=  (/)  <->  E. x E. y ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) )
32 n0i 3473 . . . . . . . 8  |-  ( x  e.  B  ->  -.  B  =  (/) )
33 fvprc 5535 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
343, 33syl5eq 2340 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3532, 34nsyl2 119 . . . . . . 7  |-  ( x  e.  B  ->  R  e.  _V )
3635adantr 451 . . . . . 6  |-  ( ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3736exlimivv 1625 . . . . 5  |-  ( E. x E. y ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3831, 37sylbi 187 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =/=  (/)  ->  R  e.  _V )
3938necon1bi 2502 . . 3  |-  ( -.  R  e.  _V  ->  {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =  (/) )
4030, 39eqtr4d 2331 . 2  |-  ( -.  R  e.  _V  ->  .||  =  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
4128, 40pm2.61i 156 1  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   _Vcvv 2801   (/)c0 3468   {copab 4092   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   ||rcdsr 15436
This theorem is referenced by:  dvdsr  15444  dvdsrpropd  15494  dvdsrz  16456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-dvdsr 15439
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