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Theorem mzpcompact2 30847
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 5899 . 2  |-  ( A  e.  (mzPoly `  B
)  ->  B  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( d  =  B  ->  (mzPoly `  d )  =  (mzPoly `  B ) )
32eleq2d 2527 . . . 4  |-  ( d  =  B  ->  ( A  e.  (mzPoly `  d
)  <->  A  e.  (mzPoly `  B ) ) )
4 sseq2 3521 . . . . . 6  |-  ( d  =  B  ->  (
a  C_  d  <->  a  C_  B ) )
5 oveq2 6304 . . . . . . . 8  |-  ( d  =  B  ->  ( ZZ  ^m  d )  =  ( ZZ  ^m  B
) )
65mpteq1d 4538 . . . . . . 7  |-  ( d  =  B  ->  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) )  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) )
76eqeq2d 2471 . . . . . 6  |-  ( d  =  B  ->  ( A  =  ( c  e.  ( ZZ  ^m  d
)  |->  ( b `  ( c  |`  a
) ) )  <->  A  =  ( c  e.  ( ZZ  ^m  B ) 
|->  ( b `  (
c  |`  a ) ) ) ) )
84, 7anbi12d 710 . . . . 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 ) ) ) ) ) )
982rexbidv 2975 . . . 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
) ) ) ) ) )
103, 9imbi12d 320 . . 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
) ) ) ) ) ) )
11 vex 3112 . . . 4  |-  d  e. 
_V
1211mzpcompact2lem 30846 . . 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 ) ) ) ) )
1310, 12vtoclg 3167 . 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 ) ) ) ) ) )
141, 13mpcom 36 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C_ wss 3471    |-> cmpt 4515    |` cres 5010   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Fincfn 7535   ZZcz 10885  mzPolycmzp 30816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-mzpcl 30817  df-mzp 30818
This theorem is referenced by:  eldioph2  30857
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