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Theorem pf1rcl 19536
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 3536 . 2  |-  ( X  e.  Q  ->  -.  Q  =  (/) )
2 pf1rcl.q . . . 4  |-  Q  =  ran  (eval1 `  R )
3 eqid 2358 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
4 eqid 2358 . . . . . 6  |-  ( 1o eval  R )  =  ( 1o eval  R )
5 eqid 2358 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
63, 4, 5evl1fval 19514 . . . . 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 4987 . . . 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 5262 . . . 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 2382 . . 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 3466 . . . . 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 3541 . . . . . . 7  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  <->  E. x  x  e.  ran  ( 1o eval  R ) )
124, 5evlval 19512 . . . . . . . . . . 11  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  ( Base `  R
) )
1312rneqi 4987 . . . . . . . . . 10  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 ( Base `  R
) )
1413mpfrcl 19506 . . . . . . . . 9  |-  ( x  e.  ran  ( 1o eval  R )  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  ( Base `  R )  e.  (SubRing `  R ) ) )
1514simp2d 968 . . . . . . . 8  |-  ( x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1615exlimiv 1634 . . . . . . 7  |-  ( E. x  x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1711, 16sylbi 187 . . . . . 6  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  ->  R  e.  CRing )
1817con1i 121 . . . . 5  |-  ( -.  R  e.  CRing  ->  ran  ( 1o eval  R )  =  (/) )
19 sseq0 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 )  /\  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 644 . . . 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 5114 . . . 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 203 . . 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 2402 . 2  |-  ( -.  R  e.  CRing  ->  Q  =  (/) )
241, 23nsyl2 119 1  |-  ( X  e.  Q  ->  R  e.  CRing )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864    i^i cin 3227    C_ wss 3228   (/)c0 3531   {csn 3716    e. cmpt 4158    X. cxp 4769   dom cdm 4771   ran crn 4772   "cima 4774    o. ccom 4775   ` cfv 5337  (class class class)co 5945   1oc1o 6559    ^m cmap 6860   Basecbs 13245   CRingccrg 15437  SubRingcsubrg 15640   evalSub ces 16189   eval cevl 16190  eval1ce1 16353
This theorem is referenced by:  pf1f  19537  pf1mpf  19539  pf1addcl  19540  pf1mulcl  19541  pf1ind  19542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-evls 16200  df-evl 16201  df-evl1 16360
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