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Theorem dvdsrval 17489
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 5848 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvdsr.1 . . . . . . . . 9  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2513 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  B )
54eleq2d 2524 . . . . . . 7  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  <->  x  e.  B ) )
64rexeqdv 3058 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  ( Base `  r ) ( z ( .r `  r ) x )  =  y  <->  E. z  e.  B  ( z
( .r `  r
) x )  =  y ) )
75, 6anbi12d 708 . . . . . 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 5848 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvdsr.3 . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2513 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1110oveqd 6287 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( .r `  r ) x )  =  ( z  .x.  x ) )
1211eqeq1d 2456 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( .r
`  r ) x )  =  y  <->  ( z  .x.  x )  =  y ) )
1312rexbidv 2965 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  B  ( z ( .r
`  r ) x )  =  y  <->  E. z  e.  B  ( z  .x.  x )  =  y ) )
1413anbi2d 701 . . . . . 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 253 . . . . 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 4502 . . . 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 17485 . . . 4  |-  ||r  =  (
r  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  r
)  /\  E. z  e.  ( Base `  r
) ( z ( .r `  r ) x )  =  y ) } )
18 fvex 5858 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2538 . . . . 5  |-  B  e. 
_V
20 eqcom 2463 . . . . . . . . 9  |-  ( ( z  .x.  x )  =  y  <->  y  =  ( z  .x.  x
) )
2120rexbii 2956 . . . . . . . 8  |-  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  y  =  ( z  .x.  x
) )
2221abbii 2588 . . . . . . 7  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  =  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }
2319abrexex 6747 . . . . . . 7  |-  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }  e.  _V
2422, 23eqeltri 2538 . . . . . 6  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V
2524a1i 11 . . . . 5  |-  ( x  e.  B  ->  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V )
2619, 25opabex3 6752 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  e.  _V
2716, 17, 26fvmpt 5931 . . 3  |-  ( R  e.  _V  ->  ( ||r `  R )  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
281, 27syl5eq 2507 . 2  |-  ( R  e.  _V  ->  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
29 fvprc 5842 . . . 4  |-  ( -.  R  e.  _V  ->  (
||r `  R )  =  (/) )
301, 29syl5eq 2507 . . 3  |-  ( -.  R  e.  _V  ->  .||  =  (/) )
31 opabn0 4767 . . . . 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 3788 . . . . . . . 8  |-  ( x  e.  B  ->  -.  B  =  (/) )
33 fvprc 5842 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
343, 33syl5eq 2507 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3532, 34nsyl2 127 . . . . . . 7  |-  ( x  e.  B  ->  R  e.  _V )
3635adantr 463 . . . . . 6  |-  ( ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3736exlimivv 1728 . . . . 5  |-  ( E. x E. y ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3831, 37sylbi 195 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =/=  (/)  ->  R  e.  _V )
3938necon1bi 2687 . . 3  |-  ( -.  R  e.  _V  ->  {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =  (/) )
4030, 39eqtr4d 2498 . 2  |-  ( -.  R  e.  _V  ->  .||  =  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
4128, 40pm2.61i 164 1  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439    =/= wne 2649   E.wrex 2805   _Vcvv 3106   (/)c0 3783   {copab 4496   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   ||rcdsr 17482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-dvdsr 17485
This theorem is referenced by:  dvdsr  17490  dvdsrpropd  17540  dvdsrzring  18696
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