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Theorem dvdsrcl 16865
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 16862 . 2  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. x  e.  B  ( x
( .r `  R
) X )  =  Y ) )
54simplbi 460 1  |-  ( X 
.||  Y  ->  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   .rcmulr 14359   ||rcdsr 16854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-dvdsr 16857
This theorem is referenced by:  unitcl  16875
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