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Theorem dvdsrval 17107
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 5866 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
3 dvdsr.1 . . . . . . . . 9  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2526 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  B )
54eleq2d 2537 . . . . . . 7  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  <->  x  e.  B ) )
64rexeqdv 3065 . . . . . . 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 5866 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
9 dvdsr.3 . . . . . . . . . . 11  |-  .x.  =  ( .r `  R )
108, 9syl6eqr 2526 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
1110oveqd 6302 . . . . . . . . 9  |-  ( r  =  R  ->  (
z ( .r `  r ) x )  =  ( z  .x.  x ) )
1211eqeq1d 2469 . . . . . . . 8  |-  ( r  =  R  ->  (
( z ( .r
`  r ) x )  =  y  <->  ( z  .x.  x )  =  y ) )
1312rexbidv 2973 . . . . . . 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 4510 . . . 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 17103 . . . 4  |-  ||r  =  (
r  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  r
)  /\  E. z  e.  ( Base `  r
) ( z ( .r `  r ) x )  =  y ) } )
18 fvex 5876 . . . . . 6  |-  ( Base `  R )  e.  _V
193, 18eqeltri 2551 . . . . 5  |-  B  e. 
_V
20 eqcom 2476 . . . . . . . . 9  |-  ( ( z  .x.  x )  =  y  <->  y  =  ( z  .x.  x
) )
2120rexbii 2965 . . . . . . . 8  |-  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  B  y  =  ( z  .x.  x
) )
2221abbii 2601 . . . . . . 7  |-  { y  |  E. z  e.  B  ( z  .x.  x )  =  y }  =  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }
2319abrexex 6759 . . . . . . 7  |-  { y  |  E. z  e.  B  y  =  ( z  .x.  x ) }  e.  _V
2422, 23eqeltri 2551 . . . . . 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 6764 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  e.  _V
2716, 17, 26fvmpt 5951 . . 3  |-  ( R  e.  _V  ->  ( ||r `  R )  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
281, 27syl5eq 2520 . 2  |-  ( R  e.  _V  ->  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
29 fvprc 5860 . . . 4  |-  ( -.  R  e.  _V  ->  (
||r `  R )  =  (/) )
301, 29syl5eq 2520 . . 3  |-  ( -.  R  e.  _V  ->  .||  =  (/) )
31 opabn0 4778 . . . . 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 3790 . . . . . . . 8  |-  ( x  e.  B  ->  -.  B  =  (/) )
33 fvprc 5860 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
343, 33syl5eq 2520 . . . . . . . 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 1699 . . . . 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 2700 . . 3  |-  ( -.  R  e.  _V  ->  {
<. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =  (/) )
4030, 39eqtr4d 2511 . 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 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2815   _Vcvv 3113   (/)c0 3785   {copab 4504   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   ||rcdsr 17100
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-dvdsr 17103
This theorem is referenced by:  dvdsr  17108  dvdsrpropd  17158  dvdsrzring  18314  dvdsrz  18315
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