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Theorem dvdsr 15428
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
dvdsr.1  |-  B  =  ( Base `  R
)
dvdsr.2  |-  .||  =  (
||r `  R )
dvdsr.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
dvdsr  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
Distinct variable groups:    z, B    z, X    z, Y    z, R    z,  .x.
Allowed substitution hint:    .|| ( z)

Proof of Theorem dvdsr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . . 4  |-  .||  =  (
||r `  R )
21reldvdsr 15426 . . 3  |-  Rel  .||
3 brrelex12 4726 . . 3  |-  ( ( Rel  .||  /\  X  .||  Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
42, 3mpan 651 . 2  |-  ( X 
.||  Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
5 elex 2796 . . 3  |-  ( X  e.  B  ->  X  e.  _V )
6 id 19 . . . . 5  |-  ( ( z  .x.  X )  =  Y  ->  (
z  .x.  X )  =  Y )
7 ovex 5883 . . . . 5  |-  ( z 
.x.  X )  e. 
_V
86, 7syl6eqelr 2372 . . . 4  |-  ( ( z  .x.  X )  =  Y  ->  Y  e.  _V )
98rexlimivw 2663 . . 3  |-  ( E. z  e.  B  ( z  .x.  X )  =  Y  ->  Y  e.  _V )
105, 9anim12i 549 . 2  |-  ( ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y )  -> 
( X  e.  _V  /\  Y  e.  _V )
)
11 simpl 443 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
1211eleq1d 2349 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  B  <->  X  e.  B ) )
1311oveq2d 5874 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  .x.  x
)  =  ( z 
.x.  X ) )
14 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1513, 14eqeq12d 2297 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  .x.  x )  =  y  <-> 
( z  .x.  X
)  =  Y ) )
1615rexbidv 2564 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  ( z  .x.  X
)  =  Y ) )
1712, 16anbi12d 691 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
18 dvdsr.1 . . . 4  |-  B  =  ( Base `  R
)
19 dvdsr.3 . . . 4  |-  .x.  =  ( .r `  R )
2018, 1, 19dvdsrval 15427 . . 3  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
2117, 20brabga 4279 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
224, 10, 21pm5.21nii 342 1  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   ||rcdsr 15420
This theorem is referenced by:  dvdsr2  15429  dvdsrmul  15430  dvdsrcl  15431  dvdsrcl2  15432  dvdsrtr  15434  dvdsrmul1  15435  opprunit  15443  crngunit  15444  subrgdvds  15559
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-dvdsr 15423
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