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Theorem pf1rcl 17752
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 3639 . 2  |-  ( X  e.  Q  ->  -.  Q  =  (/) )
2 pf1rcl.q . . . 4  |-  Q  =  ran  (eval1 `  R )
3 eqid 2441 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
4 eqid 2441 . . . . . 6  |-  ( 1o eval  R )  =  ( 1o eval  R )
5 eqid 2441 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
63, 4, 5evl1fval 17732 . . . . 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 5062 . . . 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 5342 . . . 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 2465 . . 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 3568 . . . . 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 3644 . . . . . . 7  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  <->  E. x  x  e.  ran  ( 1o eval  R ) )
124, 5evlval 17586 . . . . . . . . . . 11  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  ( Base `  R
) )
1312rneqi 5062 . . . . . . . . . 10  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 ( Base `  R
) )
1413mpfrcl 17580 . . . . . . . . 9  |-  ( x  e.  ran  ( 1o eval  R )  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  ( Base `  R )  e.  (SubRing `  R ) ) )
1514simp2d 996 . . . . . . . 8  |-  ( x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1615exlimiv 1693 . . . . . . 7  |-  ( E. x  x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1711, 16sylbi 195 . . . . . 6  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  ->  R  e.  CRing )
1817con1i 129 . . . . 5  |-  ( -.  R  e.  CRing  ->  ran  ( 1o eval  R )  =  (/) )
19 sseq0 3666 . . . . 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 658 . . . 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 5185 . . . 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 212 . . 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 2485 . 2  |-  ( -.  R  e.  CRing  ->  Q  =  (/) )
241, 23nsyl2 127 1  |-  ( X  e.  Q  ->  R  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364   E.wex 1591    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ran crn 4837   "cima 4839    o. ccom 4840   ` cfv 5415  (class class class)co 6090   1oc1o 6909    ^m cmap 7210   Basecbs 14170   CRingccrg 16636  SubRingcsubrg 16841   evalSub ces 17562   eval cevl 17563  eval1ce1 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-evls 17564  df-evl 17565  df-evl1 17724
This theorem is referenced by:  pf1f  17753  pf1mpf  17755  pf1addcl  17756  pf1mulcl  17757  pf1ind  17758
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