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Theorem rabdiophlem2 26883
Description: Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Hypothesis
Ref Expression
rabdiophlem2.1  |-  M  =  ( N  +  1 )
Assertion
Ref Expression
rabdiophlem2  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( t  e.  ( ZZ  ^m  ( 1 ... M ) ) 
|->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )  e.  (mzPoly `  ( 1 ... M
) ) )
Distinct variable groups:    u, N, t    u, M, t    t, A
Allowed substitution hint:    A( u)

Proof of Theorem rabdiophlem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 nfcv 2419 . . . . . 6  |-  F/_ a A
2 nfcsb1v 3113 . . . . . 6  |-  F/_ u [_ a  /  u ]_ A
3 csbeq1a 3089 . . . . . 6  |-  ( u  =  a  ->  A  =  [_ a  /  u ]_ A )
41, 2, 3cbvmpt 4110 . . . . 5  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
54fveq1i 5526 . . . 4  |-  ( ( u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A ) `  ( t  |`  ( 1 ... N
) ) )  =  ( ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A ) `  (
t  |`  ( 1 ... N ) ) )
6 rabdiophlem2.1 . . . . . . 7  |-  M  =  ( N  +  1 )
76mapfzcons1cl 26795 . . . . . 6  |-  ( t  e.  ( ZZ  ^m  ( 1 ... M
) )  ->  (
t  |`  ( 1 ... N ) )  e.  ( ZZ  ^m  (
1 ... N ) ) )
87adantl 452 . . . . 5  |-  ( ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  /\  t  e.  ( ZZ  ^m  ( 1 ... M
) ) )  -> 
( t  |`  (
1 ... N ) )  e.  ( ZZ  ^m  ( 1 ... N
) ) )
9 mzpf 26814 . . . . . . . 8  |-  ( ( u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) )  ->  (
u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A ) : ( ZZ 
^m  ( 1 ... N ) ) --> ZZ )
10 eqid 2283 . . . . . . . . 9  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A )
1110fmpt 5681 . . . . . . . 8  |-  ( A. u  e.  ( ZZ  ^m  ( 1 ... N
) ) A  e.  ZZ  <->  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A ) : ( ZZ  ^m  ( 1 ... N ) ) --> ZZ )
129, 11sylibr 203 . . . . . . 7  |-  ( ( u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) )  ->  A. u  e.  ( ZZ  ^m  (
1 ... N ) ) A  e.  ZZ )
1312ad2antlr 707 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  /\  t  e.  ( ZZ  ^m  ( 1 ... M
) ) )  ->  A. u  e.  ( ZZ  ^m  ( 1 ... N ) ) A  e.  ZZ )
14 nfcsb1v 3113 . . . . . . . 8  |-  F/_ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A
1514nfel1 2429 . . . . . . 7  |-  F/ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ
16 csbeq1a 3089 . . . . . . . 8  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  A  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
1716eleq1d 2349 . . . . . . 7  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  ( A  e.  ZZ  <->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A  e.  ZZ ) )
1815, 17rspc 2878 . . . . . 6  |-  ( ( t  |`  ( 1 ... N ) )  e.  ( ZZ  ^m  ( 1 ... N
) )  ->  ( A. u  e.  ( ZZ  ^m  ( 1 ... N ) ) A  e.  ZZ  ->  [_ (
t  |`  ( 1 ... N ) )  /  u ]_ A  e.  ZZ ) )
198, 13, 18sylc 56 . . . . 5  |-  ( ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  /\  t  e.  ( ZZ  ^m  ( 1 ... M
) ) )  ->  [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ )
20 csbeq1 3084 . . . . . 6  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  [_ a  /  u ]_ A  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
21 eqid 2283 . . . . . 6  |-  ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
2220, 21fvmptg 5600 . . . . 5  |-  ( ( ( t  |`  (
1 ... N ) )  e.  ( ZZ  ^m  ( 1 ... N
) )  /\  [_ (
t  |`  ( 1 ... N ) )  /  u ]_ A  e.  ZZ )  ->  ( ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A ) `
 ( t  |`  ( 1 ... N
) ) )  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
238, 19, 22syl2anc 642 . . . 4  |-  ( ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  /\  t  e.  ( ZZ  ^m  ( 1 ... M
) ) )  -> 
( ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A ) `  (
t  |`  ( 1 ... N ) ) )  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
245, 23syl5req 2328 . . 3  |-  ( ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  /\  t  e.  ( ZZ  ^m  ( 1 ... M
) ) )  ->  [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  =  ( ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A ) `
 ( t  |`  ( 1 ... N
) ) ) )
2524mpteq2dva 4106 . 2  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( t  e.  ( ZZ  ^m  ( 1 ... M ) ) 
|->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )  =  ( t  e.  ( ZZ 
^m  ( 1 ... M ) )  |->  ( ( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A ) `  (
t  |`  ( 1 ... N ) ) ) ) )
26 ovex 5883 . . . 4  |-  ( 1 ... M )  e. 
_V
2726a1i 10 . . 3  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( 1 ... M
)  e.  _V )
28 fzssp1 10834 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
296oveq2i 5869 . . . . 5  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3211 . . . 4  |-  ( 1 ... N )  C_  ( 1 ... M
)
3130a1i 10 . . 3  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( 1 ... N
)  C_  ( 1 ... M ) )
32 simpr 447 . . 3  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( u  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )
33 mzpresrename 26828 . . 3  |-  ( ( ( 1 ... M
)  e.  _V  /\  ( 1 ... N
)  C_  ( 1 ... M )  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( t  e.  ( ZZ  ^m  ( 1 ... M ) ) 
|->  ( ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A ) `  (
t  |`  ( 1 ... N ) ) ) )  e.  (mzPoly `  ( 1 ... M
) ) )
3427, 31, 32, 33syl3anc 1182 . 2  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( t  e.  ( ZZ  ^m  ( 1 ... M ) ) 
|->  ( ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A ) `  (
t  |`  ( 1 ... N ) ) ) )  e.  (mzPoly `  ( 1 ... M
) ) )
3525, 34eqeltrd 2357 1  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( t  e.  ( ZZ  ^m  ( 1 ... M ) ) 
|->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )  e.  (mzPoly `  ( 1 ... M
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [_csb 3081    C_ wss 3152    e. cmpt 4077    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   1c1 8738    + caddc 8740   NN0cn0 9965   ZZcz 10024   ...cfz 10782  mzPolycmzp 26800
This theorem is referenced by:  elnn0rabdioph  26884  dvdsrabdioph  26891
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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  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-uz 10231  df-fz 10783  df-mzpcl 26801  df-mzp 26802
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