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Theorem dvdsr 17167
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 17165 . . 3  |-  Rel  .||
3 brrelex12 5043 . . 3  |-  ( ( Rel  .||  /\  X  .||  Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
42, 3mpan 670 . 2  |-  ( X 
.||  Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
5 elex 3127 . . 3  |-  ( X  e.  B  ->  X  e.  _V )
6 id 22 . . . . 5  |-  ( ( z  .x.  X )  =  Y  ->  (
z  .x.  X )  =  Y )
7 ovex 6320 . . . . 5  |-  ( z 
.x.  X )  e. 
_V
86, 7syl6eqelr 2564 . . . 4  |-  ( ( z  .x.  X )  =  Y  ->  Y  e.  _V )
98rexlimivw 2956 . . 3  |-  ( E. z  e.  B  ( z  .x.  X )  =  Y  ->  Y  e.  _V )
105, 9anim12i 566 . 2  |-  ( ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y )  -> 
( X  e.  _V  /\  Y  e.  _V )
)
11 simpl 457 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
1211eleq1d 2536 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  B  <->  X  e.  B ) )
1311oveq2d 6311 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  .x.  x
)  =  ( z 
.x.  X ) )
14 simpr 461 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1513, 14eqeq12d 2489 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  .x.  x )  =  y  <-> 
( z  .x.  X
)  =  Y ) )
1615rexbidv 2978 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  ( z  .x.  X
)  =  Y ) )
1712, 16anbi12d 710 . . 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 17166 . . 3  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
2117, 20brabga 4767 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) ) )
224, 10, 21pm5.21nii 353 1  |-  ( X 
.||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   class class class wbr 4453   Rel wrel 5010   ` cfv 5594  (class class class)co 6295   Basecbs 14507   .rcmulr 14573   ||rcdsr 17159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-dvdsr 17162
This theorem is referenced by:  dvdsr2  17168  dvdsrmul  17169  dvdsrcl  17170  dvdsrcl2  17171  dvdsrtr  17173  dvdsrmul1  17174  opprunit  17182  crngunit  17183  subrgdvds  17314  rhmdvdsr  27633
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