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Theorem dvdszrcl 12862
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 12858 . . 3  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x
)  =  y ) }
2 opabssxp 4953 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x )  =  y ) }  C_  ( ZZ  X.  ZZ )
31, 2eqsstri 3380 . 2  |-  ||  C_  ( ZZ  X.  ZZ )
43brel 4929 1  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4215   {copab 4268    X. cxp 4879  (class class class)co 6084    x. cmul 9000   ZZcz 10287    || cdivides 12857
This theorem is referenced by:  dvdsmulgcd  13059  oddvdsi  15191  odmulg  15197  gexdvdsi  15222
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 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-dvds 12858
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