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Theorem dvdsr 15782
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 15780 . . 3  |-  Rel  .||
3 brrelex12 4944 . . 3  |-  ( ( Rel  .||  /\  X  .||  Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
42, 3mpan 653 . 2  |-  ( X 
.||  Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
5 elex 2970 . . 3  |-  ( X  e.  B  ->  X  e.  _V )
6 id 21 . . . . 5  |-  ( ( z  .x.  X )  =  Y  ->  (
z  .x.  X )  =  Y )
7 ovex 6135 . . . . 5  |-  ( z 
.x.  X )  e. 
_V
86, 7syl6eqelr 2531 . . . 4  |-  ( ( z  .x.  X )  =  Y  ->  Y  e.  _V )
98rexlimivw 2832 . . 3  |-  ( E. z  e.  B  ( z  .x.  X )  =  Y  ->  Y  e.  _V )
105, 9anim12i 551 . 2  |-  ( ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y )  -> 
( X  e.  _V  /\  Y  e.  _V )
)
11 simpl 445 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
1211eleq1d 2508 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  B  <->  X  e.  B ) )
1311oveq2d 6126 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  .x.  x
)  =  ( z 
.x.  X ) )
14 simpr 449 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1513, 14eqeq12d 2456 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  .x.  x )  =  y  <-> 
( z  .x.  X
)  =  Y ) )
1615rexbidv 2732 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  ( z  .x.  X
)  =  Y ) )
1712, 16anbi12d 693 . . 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 15781 . . 3  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
2117, 20brabga 4498 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
224, 10, 21pm5.21nii 344 1  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   E.wrex 2712   _Vcvv 2962   class class class wbr 4237   Rel wrel 4912   ` cfv 5483  (class class class)co 6110   Basecbs 13500   .rcmulr 13561   ||rcdsr 15774
This theorem is referenced by:  dvdsr2  15783  dvdsrmul  15784  dvdsrcl  15785  dvdsrcl2  15786  dvdsrtr  15788  dvdsrmul1  15789  opprunit  15797  crngunit  15798  subrgdvds  15913  rhmdvdsr  24287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-dvdsr 15777
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