Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mzpcompact2 Unicode version

Theorem mzpcompact2 26830
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 5555 . 2  |-  ( A  e.  (mzPoly `  B
)  ->  B  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( d  =  B  ->  (mzPoly `  d )  =  (mzPoly `  B ) )
32eleq2d 2350 . . . 4  |-  ( d  =  B  ->  ( A  e.  (mzPoly `  d
)  <->  A  e.  (mzPoly `  B ) ) )
4 sseq2 3200 . . . . . 6  |-  ( d  =  B  ->  (
a  C_  d  <->  a  C_  B ) )
5 oveq2 5866 . . . . . . . 8  |-  ( d  =  B  ->  ( ZZ  ^m  d )  =  ( ZZ  ^m  B
) )
6 mpteq1 4100 . . . . . . . 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 2294 . . . . . 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 2586 . . . 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 2791 . . . 4  |-  d  e. 
_V
1312mzpcompact2lem 26829 . . 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 2843 . 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 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152    e. cmpt 4077    |` cres 4691   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   ZZcz 10024  mzPolycmzp 26800
This theorem is referenced by:  eldioph2  26841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-mzpcl 26801  df-mzp 26802
  Copyright terms: Public domain W3C validator