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Theorem mzpcompact2 29086
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 5715 . 2  |-  ( A  e.  (mzPoly `  B
)  ->  B  e.  _V )
2 fveq2 5689 . . . . 5  |-  ( d  =  B  ->  (mzPoly `  d )  =  (mzPoly `  B ) )
32eleq2d 2508 . . . 4  |-  ( d  =  B  ->  ( A  e.  (mzPoly `  d
)  <->  A  e.  (mzPoly `  B ) ) )
4 sseq2 3376 . . . . . 6  |-  ( d  =  B  ->  (
a  C_  d  <->  a  C_  B ) )
5 oveq2 6097 . . . . . . . 8  |-  ( d  =  B  ->  ( ZZ  ^m  d )  =  ( ZZ  ^m  B
) )
65mpteq1d 4371 . . . . . . 7  |-  ( d  =  B  ->  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) )  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) )
76eqeq2d 2452 . . . . . 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 2756 . . . 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 2973 . . . 4  |-  d  e. 
_V
1211mzpcompact2lem 29085 . . 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 3028 . 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 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970    C_ wss 3326    e. cmpt 4348    |` cres 4840   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   Fincfn 7308   ZZcz 10644  mzPolycmzp 29055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-mzpcl 29056  df-mzp 29057
This theorem is referenced by:  eldioph2  29097
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