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Theorem dvdsrcl 17493
Description: Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
dvdsr.1  |-  B  =  ( Base `  R
)
dvdsr.2  |-  .||  =  (
||r `  R )
Assertion
Ref Expression
dvdsrcl  |-  ( X 
.||  Y  ->  X  e.  B )

Proof of Theorem dvdsrcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvdsr.1 . . 3  |-  B  =  ( Base `  R
)
2 dvdsr.2 . . 3  |-  .||  =  (
||r `  R )
3 eqid 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
41, 2, 3dvdsr 17490 . 2  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. x  e.  B  ( x
( .r `  R
) X )  =  Y ) )
54simplbi 458 1  |-  ( X 
.||  Y  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   ` 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:  unitcl  17503
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