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Theorem mzpcl34 35038
Description: Defining properties 3 and 4 of a polynomially closed function set  P: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
mzpcl34  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  ( ( F  oF  +  G
)  e.  P  /\  ( F  oF  x.  G )  e.  P
) )

Proof of Theorem mzpcl34
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1000 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  F  e.  P )
2 simp3 1001 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  G  e.  P )
3 simp1 999 . . . 4  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  P  e.  (mzPolyCld `  V ) )
43elfvexd 5879 . . . . 5  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  V  e.  _V )
5 elmzpcl 35033 . . . . 5  |-  ( V  e.  _V  ->  ( P  e.  (mzPolyCld `  V
)  <->  ( P  C_  ( ZZ  ^m  ( ZZ  ^m  V ) )  /\  ( ( A. f  e.  ZZ  (
( ZZ  ^m  V
)  X.  { f } )  e.  P  /\  A. f  e.  V  ( g  e.  ( ZZ  ^m  V ) 
|->  ( g `  f
) )  e.  P
)  /\  A. f  e.  P  A. g  e.  P  ( (
f  oF  +  g )  e.  P  /\  ( f  oF  x.  g )  e.  P ) ) ) ) )
64, 5syl 17 . . . 4  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  ( P  e.  (mzPolyCld `  V )  <->  ( P  C_  ( ZZ  ^m  ( ZZ  ^m  V
) )  /\  (
( A. f  e.  ZZ  ( ( ZZ 
^m  V )  X. 
{ f } )  e.  P  /\  A. f  e.  V  (
g  e.  ( ZZ 
^m  V )  |->  ( g `  f ) )  e.  P )  /\  A. f  e.  P  A. g  e.  P  ( ( f  oF  +  g )  e.  P  /\  ( f  oF  x.  g )  e.  P ) ) ) ) )
73, 6mpbid 212 . . 3  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  ( P  C_  ( ZZ  ^m  ( ZZ  ^m  V ) )  /\  ( ( A. f  e.  ZZ  (
( ZZ  ^m  V
)  X.  { f } )  e.  P  /\  A. f  e.  V  ( g  e.  ( ZZ  ^m  V ) 
|->  ( g `  f
) )  e.  P
)  /\  A. f  e.  P  A. g  e.  P  ( (
f  oF  +  g )  e.  P  /\  ( f  oF  x.  g )  e.  P ) ) ) )
87simprrd 761 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  A. f  e.  P  A. g  e.  P  ( (
f  oF  +  g )  e.  P  /\  ( f  oF  x.  g )  e.  P ) )
9 oveq1 6287 . . . . 5  |-  ( f  =  F  ->  (
f  oF  +  g )  =  ( F  oF  +  g ) )
109eleq1d 2473 . . . 4  |-  ( f  =  F  ->  (
( f  oF  +  g )  e.  P  <->  ( F  oF  +  g )  e.  P ) )
11 oveq1 6287 . . . . 5  |-  ( f  =  F  ->  (
f  oF  x.  g )  =  ( F  oF  x.  g ) )
1211eleq1d 2473 . . . 4  |-  ( f  =  F  ->  (
( f  oF  x.  g )  e.  P  <->  ( F  oF  x.  g )  e.  P ) )
1310, 12anbi12d 711 . . 3  |-  ( f  =  F  ->  (
( ( f  oF  +  g )  e.  P  /\  (
f  oF  x.  g )  e.  P
)  <->  ( ( F  oF  +  g )  e.  P  /\  ( F  oF  x.  g )  e.  P
) ) )
14 oveq2 6288 . . . . 5  |-  ( g  =  G  ->  ( F  oF  +  g )  =  ( F  oF  +  G
) )
1514eleq1d 2473 . . . 4  |-  ( g  =  G  ->  (
( F  oF  +  g )  e.  P  <->  ( F  oF  +  G )  e.  P ) )
16 oveq2 6288 . . . . 5  |-  ( g  =  G  ->  ( F  oF  x.  g
)  =  ( F  oF  x.  G
) )
1716eleq1d 2473 . . . 4  |-  ( g  =  G  ->  (
( F  oF  x.  g )  e.  P  <->  ( F  oF  x.  G )  e.  P ) )
1815, 17anbi12d 711 . . 3  |-  ( g  =  G  ->  (
( ( F  oF  +  g )  e.  P  /\  ( F  oF  x.  g
)  e.  P )  <-> 
( ( F  oF  +  G )  e.  P  /\  ( F  oF  x.  G
)  e.  P ) ) )
1913, 18rspc2va 3172 . 2  |-  ( ( ( F  e.  P  /\  G  e.  P
)  /\  A. f  e.  P  A. g  e.  P  ( (
f  oF  +  g )  e.  P  /\  ( f  oF  x.  g )  e.  P ) )  -> 
( ( F  oF  +  G )  e.  P  /\  ( F  oF  x.  G
)  e.  P ) )
201, 2, 8, 19syl21anc 1231 1  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  ( ( F  oF  +  G
)  e.  P  /\  ( F  oF  x.  G )  e.  P
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061    C_ wss 3416   {csn 3974    |-> cmpt 4455    X. cxp 4823   ` cfv 5571  (class class class)co 6280    oFcof 6521    ^m cmap 7459    + caddc 9527    x. cmul 9529   ZZcz 10907  mzPolyCldcmzpcl 35028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283  df-mzpcl 35030
This theorem is referenced by:  mzpincl  35041  mzpadd  35045  mzpmul  35046
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