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Theorem mzpcl34 29208
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 989 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  F  e.  P )
2 simp3 990 . 2  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  G  e.  P )
3 simp1 988 . . . 4  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  P  e.  (mzPolyCld `  V ) )
43elfvexd 5820 . . . . 5  |-  ( ( P  e.  (mzPolyCld `  V
)  /\  F  e.  P  /\  G  e.  P
)  ->  V  e.  _V )
5 elmzpcl 29203 . . . . 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 6200 . . . . 5  |-  ( f  =  F  ->  (
f  oF  +  g )  =  ( F  oF  +  g ) )
1110eleq1d 2520 . . . 4  |-  ( f  =  F  ->  (
( f  oF  +  g )  e.  P  <->  ( F  oF  +  g )  e.  P ) )
12 oveq1 6200 . . . . 5  |-  ( f  =  F  ->  (
f  oF  x.  g )  =  ( F  oF  x.  g ) )
1312eleq1d 2520 . . . 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 6201 . . . . 5  |-  ( g  =  G  ->  ( F  oF  +  g )  =  ( F  oF  +  G
) )
1615eleq1d 2520 . . . 4  |-  ( g  =  G  ->  (
( F  oF  +  g )  e.  P  <->  ( F  oF  +  G )  e.  P ) )
17 oveq2 6201 . . . . 5  |-  ( g  =  G  ->  ( F  oF  x.  g
)  =  ( F  oF  x.  G
) )
1817eleq1d 2520 . . . 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 3180 . 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 1218 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071    C_ wss 3429   {csn 3978    |-> cmpt 4451    X. cxp 4939   ` cfv 5519  (class class class)co 6193    oFcof 6421    ^m cmap 7317    + caddc 9389    x. cmul 9391   ZZcz 10750  mzPolyCldcmzpcl 29198
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-mzpcl 29200
This theorem is referenced by:  mzpincl  29211  mzpadd  29215  mzpmul  29216
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