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Theorem dvdsrval 16749
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
dvdsr.1  |-  B  =  ( Base `  R
)
dvdsr.2  |-  .||  =  (
||r `  R )
dvdsr.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
dvdsrval  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
Distinct variable groups:    x, y,  .||    x, z, B, y    x, R, y, z    x,  .x. , y, z
Allowed substitution hint:    .|| ( z)

Proof of Theorem dvdsrval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . 3  |-  .||  =  (
||r `  R )
2 fveq2 5703 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvdsr.1 . . . . . . . . 9  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2493 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  B )
54eleq2d 2510 . . . . . . 7  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  <->  x  e.  B ) )
64rexeqdv 2936 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  ( Base `  r ) ( z ( .r `  r ) x )  =  y  <->  E. z  e.  B  ( z
( .r `  r
) x )  =  y ) )
75, 6anbi12d 710 . . . . . 6  |-  ( r  =  R  ->  (
( x  e.  (
Base `  r )  /\  E. z  e.  (
Base `  r )
( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z ( .r
`  r ) x )  =  y ) ) )
8 fveq2 5703 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvdsr.3 . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2493 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1110oveqd 6120 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( .r `  r ) x )  =  ( z  .x.  x ) )
1211eqeq1d 2451 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( .r
`  r ) x )  =  y  <->  ( z  .x.  x )  =  y ) )
1312rexbidv 2748 . . . . . . 7  |-  ( r  =  R  ->  ( E. z  e.  B  ( z ( .r
`  r ) x )  =  y  <->  E. z  e.  B  ( z  .x.  x )  =  y ) )
1413anbi2d 703 . . . . . 6  |-  ( r  =  R  ->  (
( x  e.  B  /\  E. z  e.  B  ( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z  .x.  x
)  =  y ) ) )
157, 14bitrd 253 . . . . 5  |-  ( r  =  R  ->  (
( x  e.  (
Base `  r )  /\  E. z  e.  (
Base `  r )
( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  B  /\  E. z  e.  B  ( z  .x.  x
)  =  y ) ) )
1615opabbidv 4367 . . . 4  |-  ( r  =  R  ->  { <. x ,  y >.  |  ( x  e.  ( Base `  r )  /\  E. z  e.  ( Base `  r ) ( z ( .r `  r
) x )  =  y ) }  =  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
17 df-dvdsr 16745 . . . 4  |-  ||r  =  (
r  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  r
)  /\  E. z  e.  ( Base `  r
) ( z ( .r `  r ) x )  =  y ) } )
18 fvex 5713 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2513 . . . . 5  |-  B  e. 
_V
20 eqcom 2445 . . . . . . . . 9  |-  ( ( z  .x.  x )  =  y  <->  y  =  ( z  .x.  x
) )
2120rexbii 2752 . . . . . . . 8  |-  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  y  =  ( z  .x.  x
) )
2221abbii 2561 . . . . . . 7  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  =  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }
2319abrexex 6563 . . . . . . 7  |-  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }  e.  _V
2422, 23eqeltri 2513 . . . . . 6  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V
2524a1i 11 . . . . 5  |-  ( x  e.  B  ->  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  e.  _V )
2619, 25opabex3 6568 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  e.  _V
2716, 17, 26fvmpt 5786 . . 3  |-  ( R  e.  _V  ->  ( ||r `  R )  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
281, 27syl5eq 2487 . 2  |-  ( R  e.  _V  ->  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
29 fvprc 5697 . . . 4  |-  ( -.  R  e.  _V  ->  (
||r `  R )  =  (/) )
301, 29syl5eq 2487 . . 3  |-  ( -.  R  e.  _V  ->  .||  =  (/) )
31 opabn0 4631 . . . . 5  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =/=  (/)  <->  E. x E. y ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) )
32 n0i 3654 . . . . . . . 8  |-  ( x  e.  B  ->  -.  B  =  (/) )
33 fvprc 5697 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
343, 33syl5eq 2487 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3532, 34nsyl2 127 . . . . . . 7  |-  ( x  e.  B  ->  R  e.  _V )
3635adantr 465 . . . . . 6  |-  ( ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3736exlimivv 1689 . . . . 5  |-  ( E. x E. y ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  ->  R  e.  _V )
3831, 37sylbi 195 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =/=  (/)  ->  R  e.  _V )
3938necon1bi 2666 . . 3  |-  ( -.  R  e.  _V  ->  {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =  (/) )
4030, 39eqtr4d 2478 . 2  |-  ( -.  R  e.  _V  ->  .||  =  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
4128, 40pm2.61i 164 1  |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2618   E.wrex 2728   _Vcvv 2984   (/)c0 3649   {copab 4361   ` cfv 5430  (class class class)co 6103   Basecbs 14186   .rcmulr 14251   ||rcdsr 16742
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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-dvdsr 16745
This theorem is referenced by:  dvdsr  16750  dvdsrpropd  16800  dvdsrzring  17913  dvdsrz  17914
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