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Theorem dvdsr 16726
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 16724 . . 3  |-  Rel  .||
3 brrelex12 4871 . . 3  |-  ( ( Rel  .||  /\  X  .||  Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
42, 3mpan 670 . 2  |-  ( X 
.||  Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
5 elex 2976 . . 3  |-  ( X  e.  B  ->  X  e.  _V )
6 id 22 . . . . 5  |-  ( ( z  .x.  X )  =  Y  ->  (
z  .x.  X )  =  Y )
7 ovex 6111 . . . . 5  |-  ( z 
.x.  X )  e. 
_V
86, 7syl6eqelr 2527 . . . 4  |-  ( ( z  .x.  X )  =  Y  ->  Y  e.  _V )
98rexlimivw 2832 . . 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 2504 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  B  <->  X  e.  B ) )
1311oveq2d 6102 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  .x.  x
)  =  ( z 
.x.  X ) )
14 simpr 461 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1513, 14eqeq12d 2452 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  .x.  x )  =  y  <-> 
( z  .x.  X
)  =  Y ) )
1615rexbidv 2731 . . . 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 16725 . . 3  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
2117, 20brabga 4598 . 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 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967   class class class wbr 4287   Rel wrel 4840   ` cfv 5413  (class class class)co 6086   Basecbs 14166   .rcmulr 14231   ||rcdsr 16718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-dvdsr 16721
This theorem is referenced by:  dvdsr2  16727  dvdsrmul  16728  dvdsrcl  16729  dvdsrcl2  16730  dvdsrtr  16732  dvdsrmul1  16733  opprunit  16741  crngunit  16742  subrgdvds  16857  rhmdvdsr  26237
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