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Theorem rabdiophlem2 26986
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 2432 . . . . . 6  |-  F/_ a A
2 nfcsb1v 3126 . . . . . 6  |-  F/_ u [_ a  /  u ]_ A
3 csbeq1a 3102 . . . . . 6  |-  ( u  =  a  ->  A  =  [_ a  /  u ]_ A )
41, 2, 3cbvmpt 4126 . . . . 5  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
54fveq1i 5542 . . . 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 26898 . . . . . 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 26917 . . . . . . . 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 2296 . . . . . . . . 9  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A )
1110fmpt 5697 . . . . . . . 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 3126 . . . . . . . 8  |-  F/_ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A
1514nfel1 2442 . . . . . . 7  |-  F/ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ
16 csbeq1a 3102 . . . . . . . 8  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  A  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
1716eleq1d 2362 . . . . . . 7  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  ( A  e.  ZZ  <->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A  e.  ZZ ) )
1815, 17rspc 2891 . . . . . 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 3097 . . . . . 6  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  [_ a  /  u ]_ A  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
21 eqid 2296 . . . . . 6  |-  ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
2220, 21fvmptg 5616 . . . . 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 2341 . . 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 4122 . 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 5899 . . . 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 10850 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
296oveq2i 5885 . . . . 5  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3224 . . . 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 26931 . . 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 2370 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   [_csb 3094    C_ wss 3165    e. cmpt 4093    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   1c1 8754    + caddc 8756   NN0cn0 9981   ZZcz 10040   ...cfz 10798  mzPolycmzp 26903
This theorem is referenced by:  elnn0rabdioph  26987  dvdsrabdioph  26994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-mzpcl 26904  df-mzp 26905
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