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Theorem mzpcompact2 26933
Description: Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
mzpcompact2  |-  ( A  e.  (mzPoly `  B
)  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
( a  C_  B  /\  A  =  (
c  e.  ( ZZ 
^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) )
Distinct variable groups:    A, a,
b    B, a, b, c
Allowed substitution hint:    A( c)

Proof of Theorem mzpcompact2
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 elfvex 5571 . 2  |-  ( A  e.  (mzPoly `  B
)  ->  B  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( d  =  B  ->  (mzPoly `  d )  =  (mzPoly `  B ) )
32eleq2d 2363 . . . 4  |-  ( d  =  B  ->  ( A  e.  (mzPoly `  d
)  <->  A  e.  (mzPoly `  B ) ) )
4 sseq2 3213 . . . . . 6  |-  ( d  =  B  ->  (
a  C_  d  <->  a  C_  B ) )
5 oveq2 5882 . . . . . . . 8  |-  ( d  =  B  ->  ( ZZ  ^m  d )  =  ( ZZ  ^m  B
) )
6 mpteq1 4116 . . . . . . . 8  |-  ( ( ZZ  ^m  d )  =  ( ZZ  ^m  B )  ->  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) )  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) )
75, 6syl 15 . . . . . . 7  |-  ( d  =  B  ->  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) )  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) )
87eqeq2d 2307 . . . . . 6  |-  ( d  =  B  ->  ( A  =  ( c  e.  ( ZZ  ^m  d
)  |->  ( b `  ( c  |`  a
) ) )  <->  A  =  ( c  e.  ( ZZ  ^m  B ) 
|->  ( b `  (
c  |`  a ) ) ) ) )
94, 8anbi12d 691 . . . . 5  |-  ( d  =  B  ->  (
( a  C_  d  /\  A  =  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) ) )  <->  ( a  C_  B  /\  A  =  ( c  e.  ( ZZ 
^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) ) )
1092rexbidv 2599 . . . 4  |-  ( d  =  B  ->  ( E. a  e.  Fin  E. b  e.  (mzPoly `  a ) ( a 
C_  d  /\  A  =  ( c  e.  ( ZZ  ^m  d
)  |->  ( b `  ( c  |`  a
) ) ) )  <->  E. a  e.  Fin  E. b  e.  (mzPoly `  a ) ( a 
C_  B  /\  A  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) ) ) )
113, 10imbi12d 311 . . 3  |-  ( d  =  B  ->  (
( A  e.  (mzPoly `  d )  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
( a  C_  d  /\  A  =  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) ) ) )  <->  ( A  e.  (mzPoly `  B )  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a ) ( a 
C_  B  /\  A  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) ) ) ) )
12 vex 2804 . . . 4  |-  d  e. 
_V
1312mzpcompact2lem 26932 . . 3  |-  ( A  e.  (mzPoly `  d
)  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
( a  C_  d  /\  A  =  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) ) ) )
1411, 13vtoclg 2856 . 2  |-  ( B  e.  _V  ->  ( A  e.  (mzPoly `  B
)  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
( a  C_  B  /\  A  =  (
c  e.  ( ZZ 
^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) ) )
151, 14mpcom 32 1  |-  ( A  e.  (mzPoly `  B
)  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
( a  C_  B  /\  A  =  (
c  e.  ( ZZ 
^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165    e. cmpt 4093    |` cres 4707   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   ZZcz 10040  mzPolycmzp 26903
This theorem is referenced by:  eldioph2  26944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-mzpcl 26904  df-mzp 26905
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