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Theorem pf1rcl 18256
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 3773 . 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 18235 . . . . 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 5216 . . . 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 5501 . . . 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 2474 . . 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 3702 . . . . 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 3778 . . . . . . 7  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  <->  E. x  x  e.  ran  ( 1o eval  R ) )
124, 5evlval 18064 . . . . . . . . . . 11  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  ( Base `  R
) )
1312rneqi 5216 . . . . . . . . . 10  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 ( Base `  R
) )
1413mpfrcl 18058 . . . . . . . . 9  |-  ( x  e.  ran  ( 1o eval  R )  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  ( Base `  R )  e.  (SubRing `  R ) ) )
1514simp2d 1008 . . . . . . . 8  |-  ( x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1615exlimiv 1707 . . . . . . 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 3800 . . . . 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 663 . . . 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 5343 . . . 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 2494 . 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 1381   E.wex 1597    e. wcel 1802   _Vcvv 3093    i^i cin 3458    C_ wss 3459   (/)c0 3768   {csn 4011    |-> cmpt 4492    X. cxp 4984   dom cdm 4986   ran crn 4987   "cima 4989    o. ccom 4990   ` cfv 5575  (class class class)co 6278   1oc1o 7122    ^m cmap 7419   Basecbs 14506   CRingccrg 17070  SubRingcsubrg 17296   evalSub ces 18040   eval cevl 18041  eval1ce1 18222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-evls 18042  df-evl 18043  df-evl1 18224
This theorem is referenced by:  pf1f  18257  pf1mpf  18259  pf1addcl  18260  pf1mulcl  18261  pf1ind  18262
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