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Theorem pf1rcl 19969
Description: Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypothesis
Ref Expression
pf1rcl.q  |-  Q  =  ran  (eval1 `  R )
Assertion
Ref Expression
pf1rcl  |-  ( X  e.  Q  ->  R  e.  CRing )

Proof of Theorem pf1rcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3633 . 2  |-  ( X  e.  Q  ->  -.  Q  =  (/) )
2 pf1rcl.q . . . 4  |-  Q  =  ran  (eval1 `  R )
3 eqid 2436 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
4 eqid 2436 . . . . . 6  |-  ( 1o eval  R )  =  ( 1o eval  R )
5 eqid 2436 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
63, 4, 5evl1fval 19947 . . . . 5  |-  (eval1 `  R
)  =  ( ( x  e.  ( (
Base `  R )  ^m  ( ( Base `  R
)  ^m  1o )
)  |->  ( x  o.  ( y  e.  (
Base `  R )  |->  ( 1o  X.  {
y } ) ) ) )  o.  ( 1o eval  R ) )
76rneqi 5096 . . . 4  |-  ran  (eval1 `  R )  =  ran  ( ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  R ) )
8 rnco2 5377 . . . 4  |-  ran  (
( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  o.  ( 1o eval  R ) )  =  ( ( x  e.  ( (
Base `  R )  ^m  ( ( Base `  R
)  ^m  1o )
)  |->  ( x  o.  ( y  e.  (
Base `  R )  |->  ( 1o  X.  {
y } ) ) ) ) " ran  ( 1o eval  R )
)
92, 7, 83eqtri 2460 . . 3  |-  Q  =  ( ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) ) " ran  ( 1o eval  R ) )
10 inss2 3562 . . . . 5  |-  ( dom  ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  i^i 
ran  ( 1o eval  R
) )  C_  ran  ( 1o eval  R )
11 neq0 3638 . . . . . . 7  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  <->  E. x  x  e.  ran  ( 1o eval  R ) )
124, 5evlval 19945 . . . . . . . . . . 11  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  ( Base `  R
) )
1312rneqi 5096 . . . . . . . . . 10  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 ( Base `  R
) )
1413mpfrcl 19939 . . . . . . . . 9  |-  ( x  e.  ran  ( 1o eval  R )  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  ( Base `  R )  e.  (SubRing `  R ) ) )
1514simp2d 970 . . . . . . . 8  |-  ( x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1615exlimiv 1644 . . . . . . 7  |-  ( E. x  x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1711, 16sylbi 188 . . . . . 6  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  ->  R  e.  CRing )
1817con1i 123 . . . . 5  |-  ( -.  R  e.  CRing  ->  ran  ( 1o eval  R )  =  (/) )
19 sseq0 3659 . . . . 5  |-  ( ( ( dom  ( x  e.  ( ( Base `  R )  ^m  (
( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  i^i 
ran  ( 1o eval  R
) )  C_  ran  ( 1o eval  R )  /\  ran  ( 1o eval  R
)  =  (/) )  -> 
( dom  ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  i^i 
ran  ( 1o eval  R
) )  =  (/) )
2010, 18, 19sylancr 645 . . . 4  |-  ( -.  R  e.  CRing  ->  ( dom  ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) )  i^i 
ran  ( 1o eval  R
) )  =  (/) )
21 imadisj 5223 . . . 4  |-  ( ( ( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) ) " ran  ( 1o eval  R ) )  =  (/)  <->  ( dom  ( x  e.  (
( Base `  R )  ^m  ( ( Base `  R
)  ^m  1o )
)  |->  ( x  o.  ( y  e.  (
Base `  R )  |->  ( 1o  X.  {
y } ) ) ) )  i^i  ran  ( 1o eval  R )
)  =  (/) )
2220, 21sylibr 204 . . 3  |-  ( -.  R  e.  CRing  ->  (
( x  e.  ( ( Base `  R
)  ^m  ( ( Base `  R )  ^m  1o ) )  |->  ( x  o.  ( y  e.  ( Base `  R
)  |->  ( 1o  X.  { y } ) ) ) ) " ran  ( 1o eval  R ) )  =  (/) )
239, 22syl5eq 2480 . 2  |-  ( -.  R  e.  CRing  ->  Q  =  (/) )
241, 23nsyl2 121 1  |-  ( X  e.  Q  ->  R  e.  CRing )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814    e. cmpt 4266    X. cxp 4876   dom cdm 4878   ran crn 4879   "cima 4881    o. ccom 4882   ` cfv 5454  (class class class)co 6081   1oc1o 6717    ^m cmap 7018   Basecbs 13469   CRingccrg 15661  SubRingcsubrg 15864   evalSub ces 16409   eval cevl 16410  eval1ce1 16573
This theorem is referenced by:  pf1f  19970  pf1mpf  19972  pf1addcl  19973  pf1mulcl  19974  pf1ind  19975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-evls 16420  df-evl 16421  df-evl1 16580
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