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Theorem mzpcl34 30267
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 997 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  F  e.  P )
2 simp3 998 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  G  e.  P )
3 simp1 996 . . . 4  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  P  e.  (mzPolyCld `  V ) )
43elfvexd 5892 . . . . 5  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  V  e.  _V )
5 elmzpcl 30262 . . . . 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 16 . . . 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 210 . . 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 ) ) ) )
8 simprr 756 . . 3  |-  ( ( 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 ) ) )  ->  A. f  e.  P  A. g  e.  P  ( ( f  oF  +  g )  e.  P  /\  (
f  oF  x.  g )  e.  P
) )
97, 8syl 16 . 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 ) )
10 oveq1 6289 . . . . 5  |-  ( f  =  F  ->  (
f  oF  +  g )  =  ( F  oF  +  g ) )
1110eleq1d 2536 . . . 4  |-  ( f  =  F  ->  (
( f  oF  +  g )  e.  P  <->  ( F  oF  +  g )  e.  P ) )
12 oveq1 6289 . . . . 5  |-  ( f  =  F  ->  (
f  oF  x.  g )  =  ( F  oF  x.  g ) )
1312eleq1d 2536 . . . 4  |-  ( f  =  F  ->  (
( f  oF  x.  g )  e.  P  <->  ( F  oF  x.  g )  e.  P ) )
1411, 13anbi12d 710 . . 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
) ) )
15 oveq2 6290 . . . . 5  |-  ( g  =  G  ->  ( F  oF  +  g )  =  ( F  oF  +  G
) )
1615eleq1d 2536 . . . 4  |-  ( g  =  G  ->  (
( F  oF  +  g )  e.  P  <->  ( F  oF  +  G )  e.  P ) )
17 oveq2 6290 . . . . 5  |-  ( g  =  G  ->  ( F  oF  x.  g
)  =  ( F  oF  x.  G
) )
1817eleq1d 2536 . . . 4  |-  ( g  =  G  ->  (
( F  oF  x.  g )  e.  P  <->  ( F  oF  x.  G )  e.  P ) )
1916, 18anbi12d 710 . . 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 ) ) )
2014, 19rspc2va 3224 . 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 ) )
211, 2, 9, 20syl21anc 1227 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   {csn 4027    |-> cmpt 4505    X. cxp 4997   ` cfv 5586  (class class class)co 6282    oFcof 6520    ^m cmap 7417    + caddc 9491    x. cmul 9493   ZZcz 10860  mzPolyCldcmzpcl 30257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-mzpcl 30259
This theorem is referenced by:  mzpincl  30270  mzpadd  30274  mzpmul  30275
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