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Theorem dvdsr 15527
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 15525 . . 3  |-  Rel  .||
3 brrelex12 4808 . . 3  |-  ( ( Rel  .||  /\  X  .||  Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
42, 3mpan 651 . 2  |-  ( X 
.||  Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
5 elex 2872 . . 3  |-  ( X  e.  B  ->  X  e.  _V )
6 id 19 . . . . 5  |-  ( ( z  .x.  X )  =  Y  ->  (
z  .x.  X )  =  Y )
7 ovex 5970 . . . . 5  |-  ( z 
.x.  X )  e. 
_V
86, 7syl6eqelr 2447 . . . 4  |-  ( ( z  .x.  X )  =  Y  ->  Y  e.  _V )
98rexlimivw 2739 . . 3  |-  ( E. z  e.  B  ( z  .x.  X )  =  Y  ->  Y  e.  _V )
105, 9anim12i 549 . 2  |-  ( ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y )  -> 
( X  e.  _V  /\  Y  e.  _V )
)
11 simpl 443 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
1211eleq1d 2424 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  B  <->  X  e.  B ) )
1311oveq2d 5961 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  .x.  x
)  =  ( z 
.x.  X ) )
14 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1513, 14eqeq12d 2372 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  .x.  x )  =  y  <-> 
( z  .x.  X
)  =  Y ) )
1615rexbidv 2640 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  ( z  .x.  X
)  =  Y ) )
1712, 16anbi12d 691 . . 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 15526 . . 3  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
2117, 20brabga 4361 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
224, 10, 21pm5.21nii 342 1  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   class class class wbr 4104   Rel wrel 4776   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .rcmulr 13306   ||rcdsr 15519
This theorem is referenced by:  dvdsr2  15528  dvdsrmul  15529  dvdsrcl  15530  dvdsrcl2  15531  dvdsrtr  15533  dvdsrmul1  15534  opprunit  15542  crngunit  15543  subrgdvds  15658  rhmdvdsr  23422
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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
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-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-dvdsr 15522
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