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Theorem dvdszrcl 12633
Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
dvdszrcl  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )

Proof of Theorem dvdszrcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvds 12629 . . 3  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x
)  =  y ) }
2 opabssxp 4844 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x )  =  y ) }  C_  ( ZZ  X.  ZZ )
31, 2eqsstri 3284 . 2  |-  ||  C_  ( ZZ  X.  ZZ )
43brel 4819 1  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   class class class wbr 4104   {copab 4157    X. cxp 4769  (class class class)co 5945    x. cmul 8832   ZZcz 10116    || cdivides 12628
This theorem is referenced by:  dvdsmulgcd  12830  oddvdsi  14962  odmulg  14968  gexdvdsi  14993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-dvds 12629
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