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

Theorem mzpcompact2 26800
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 5750 . 2  |-  ( A  e.  (mzPoly `  B
)  ->  B  e.  _V )
2 fveq2 5720 . . . . 5  |-  ( d  =  B  ->  (mzPoly `  d )  =  (mzPoly `  B ) )
32eleq2d 2502 . . . 4  |-  ( d  =  B  ->  ( A  e.  (mzPoly `  d
)  <->  A  e.  (mzPoly `  B ) ) )
4 sseq2 3362 . . . . . 6  |-  ( d  =  B  ->  (
a  C_  d  <->  a  C_  B ) )
5 oveq2 6081 . . . . . . . 8  |-  ( d  =  B  ->  ( ZZ  ^m  d )  =  ( ZZ  ^m  B
) )
65mpteq1d 4282 . . . . . . 7  |-  ( d  =  B  ->  (
c  e.  ( ZZ 
^m  d )  |->  ( b `  ( c  |`  a ) ) )  =  ( c  e.  ( ZZ  ^m  B
)  |->  ( b `  ( c  |`  a
) ) ) )
76eqeq2d 2446 . . . . . 6  |-  ( d  =  B  ->  ( A  =  ( c  e.  ( ZZ  ^m  d
)  |->  ( b `  ( c  |`  a
) ) )  <->  A  =  ( c  e.  ( ZZ  ^m  B ) 
|->  ( b `  (
c  |`  a ) ) ) ) )
84, 7anbi12d 692 . . . . 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 2740 . . . 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 312 . . 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 2951 . . . 4  |-  d  e. 
_V
1211mzpcompact2lem 26799 . . 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 3003 . 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 34 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 359    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    C_ wss 3312    e. cmpt 4258    |` cres 4872   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   Fincfn 7101   ZZcz 10274  mzPolycmzp 26770
This theorem is referenced by:  eldioph2  26811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-mzpcl 26771  df-mzp 26772
  Copyright terms: Public domain W3C validator