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Theorem pf1rcl 17903
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 3745 . 2  |-  ( X  e.  Q  ->  -.  Q  =  (/) )
2 pf1rcl.q . . . 4  |-  Q  =  ran  (eval1 `  R )
3 eqid 2452 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
4 eqid 2452 . . . . . 6  |-  ( 1o eval  R )  =  ( 1o eval  R )
5 eqid 2452 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
63, 4, 5evl1fval 17882 . . . . 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 5169 . . . 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 5448 . . . 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 2485 . . 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 3674 . . . . 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 3750 . . . . . . 7  |-  ( -. 
ran  ( 1o eval  R
)  =  (/)  <->  E. x  x  e.  ran  ( 1o eval  R ) )
124, 5evlval 17729 . . . . . . . . . . 11  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  ( Base `  R
) )
1312rneqi 5169 . . . . . . . . . 10  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 ( Base `  R
) )
1413mpfrcl 17723 . . . . . . . . 9  |-  ( x  e.  ran  ( 1o eval  R )  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  ( Base `  R )  e.  (SubRing `  R ) ) )
1514simp2d 1001 . . . . . . . 8  |-  ( x  e.  ran  ( 1o eval  R )  ->  R  e.  CRing )
1615exlimiv 1689 . . . . . . 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 3772 . . . . 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 5291 . . . 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 2505 . 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 1370   E.wex 1587    e. wcel 1758   _Vcvv 3072    i^i cin 3430    C_ wss 3431   (/)c0 3740   {csn 3980    |-> cmpt 4453    X. cxp 4941   dom cdm 4943   ran crn 4944   "cima 4946    o. ccom 4947   ` cfv 5521  (class class class)co 6195   1oc1o 7018    ^m cmap 7319   Basecbs 14287   CRingccrg 16764  SubRingcsubrg 16979   evalSub ces 17705   eval cevl 17706  eval1ce1 17869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-evls 17707  df-evl 17708  df-evl1 17871
This theorem is referenced by:  pf1f  17904  pf1mpf  17906  pf1addcl  17907  pf1mulcl  17908  pf1ind  17909
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