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Theorem dvdsrvald 13267
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
dvdsrvald.1  |-  ( ph  ->  B  =  ( Base `  R ) )
dvdsrvald.2  |-  ( ph  -> 
.||  =  ( ||r `  R
) )
dvdsrvald.r  |-  ( ph  ->  R  e. SRing )
dvdsrvald.3  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
Assertion
Ref Expression
dvdsrvald  |-  ( ph  -> 
.||  =  { <. 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    ph, x, y, z
Allowed substitution hint:    .|| ( z)

Proof of Theorem dvdsrvald
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 df-dvdsr 13263 . . 3  |-  ||r  =  (
r  e.  _V  |->  {
<. x ,  y >.  |  ( x  e.  ( Base `  r
)  /\  E. z  e.  ( Base `  r
) ( z ( .r `  r ) x )  =  y ) } )
2 fveq2 5517 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
32eleq2d 2247 . . . . 5  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  <->  x  e.  ( Base `  R )
) )
4 fveq2 5517 . . . . . . . 8  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
54oveqd 5894 . . . . . . 7  |-  ( r  =  R  ->  (
z ( .r `  r ) x )  =  ( z ( .r `  R ) x ) )
65eqeq1d 2186 . . . . . 6  |-  ( r  =  R  ->  (
( z ( .r
`  r ) x )  =  y  <->  ( z
( .r `  R
) x )  =  y ) )
72, 6rexeqbidv 2686 . . . . 5  |-  ( r  =  R  ->  ( E. z  e.  ( Base `  r ) ( z ( .r `  r ) x )  =  y  <->  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) )
83, 7anbi12d 473 . . . 4  |-  ( r  =  R  ->  (
( x  e.  (
Base `  r )  /\  E. z  e.  (
Base `  r )
( z ( .r
`  r ) x )  =  y )  <-> 
( x  e.  (
Base `  R )  /\  E. z  e.  (
Base `  R )
( z ( .r
`  R ) x )  =  y ) ) )
98opabbidv 4071 . . 3  |-  ( r  =  R  ->  { <. x ,  y >.  |  ( x  e.  ( Base `  r )  /\  E. z  e.  ( Base `  r ) ( z ( .r `  r
) x )  =  y ) }  =  { <. x ,  y
>.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) } )
10 dvdsrvald.r . . . 4  |-  ( ph  ->  R  e. SRing )
1110elexd 2752 . . 3  |-  ( ph  ->  R  e.  _V )
12 basfn 12522 . . . . . 6  |-  Base  Fn  _V
13 funfvex 5534 . . . . . . 7  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
1413funfni 5318 . . . . . 6  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
1512, 11, 14sylancr 414 . . . . 5  |-  ( ph  ->  ( Base `  R
)  e.  _V )
16 xpexg 4742 . . . . 5  |-  ( ( ( Base `  R
)  e.  _V  /\  ( Base `  R )  e.  _V )  ->  (
( Base `  R )  X.  ( Base `  R
) )  e.  _V )
1715, 15, 16syl2anc 411 . . . 4  |-  ( ph  ->  ( ( Base `  R
)  X.  ( Base `  R ) )  e. 
_V )
18 simprr 531 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  ( z ( .r `  R ) x )  =  y )
1910ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  R  e. SRing )
20 simprl 529 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  z  e.  (
Base `  R )
)
21 simplr 528 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  x  e.  (
Base `  R )
)
22 eqid 2177 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
23 eqid 2177 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
2422, 23srgcl 13158 . . . . . . . . . 10  |-  ( ( R  e. SRing  /\  z  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( z
( .r `  R
) x )  e.  ( Base `  R
) )
2519, 20, 21, 24syl3anc 1238 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  ( z ( .r `  R ) x )  e.  (
Base `  R )
)
2618, 25eqeltrrd 2255 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( Base `  R
) )  /\  (
z  e.  ( Base `  R )  /\  (
z ( .r `  R ) x )  =  y ) )  ->  y  e.  (
Base `  R )
)
2726rexlimdvaa 2595 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  R )
)  ->  ( E. z  e.  ( Base `  R ) ( z ( .r `  R
) x )  =  y  ->  y  e.  ( Base `  R )
) )
2827imdistanda 448 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y )  ->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) ) )
2928ssopab2dv 4280 . . . . 5  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) }  C_  { <. x ,  y >.  |  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) } )
30 df-xp 4634 . . . . 5  |-  ( (
Base `  R )  X.  ( Base `  R
) )  =  { <. x ,  y >.  |  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) }
3129, 30sseqtrrdi 3206 . . . 4  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) }  C_  (
( Base `  R )  X.  ( Base `  R
) ) )
3217, 31ssexd 4145 . . 3  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) }  e.  _V )
331, 9, 11, 32fvmptd3 5611 . 2  |-  ( ph  ->  ( ||r `
 R )  =  { <. x ,  y
>.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) } )
34 dvdsrvald.2 . 2  |-  ( ph  -> 
.||  =  ( ||r `  R
) )
35 dvdsrvald.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  R ) )
3635eleq2d 2247 . . . 4  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
37 dvdsrvald.3 . . . . . . 7  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
3837oveqd 5894 . . . . . 6  |-  ( ph  ->  ( z  .x.  x
)  =  ( z ( .r `  R
) x ) )
3938eqeq1d 2186 . . . . 5  |-  ( ph  ->  ( ( z  .x.  x )  =  y  <-> 
( z ( .r
`  R ) x )  =  y ) )
4035, 39rexeqbidv 2686 . . . 4  |-  ( ph  ->  ( E. z  e.  B  ( z  .x.  x )  =  y  <->  E. z  e.  ( Base `  R ) ( z ( .r `  R ) x )  =  y ) )
4136, 40anbi12d 473 . . 3  |-  ( ph  ->  ( ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y )  <->  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) ) )
4241opabbidv 4071 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) }  =  { <. x ,  y >.  |  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) } )
4333, 34, 423eqtr4d 2220 1  |-  ( ph  -> 
.||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  ( z  .x.  x )  =  y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   _Vcvv 2739   {copab 4065    X. cxp 4626    Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   .rcmulr 12539  SRingcsrg 13151   ||rcdsr 13260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mgp 13136  df-srg 13152  df-dvdsr 13263
This theorem is referenced by:  dvdsrd  13268  dvdsrex  13272  dvdsrpropdg  13321  dvdsrzring  13532
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