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Theorem rabdiophlem2 29049
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 2577 . . . . . 6  |-  F/_ a A
2 nfcsb1v 3301 . . . . . 6  |-  F/_ u [_ a  /  u ]_ A
3 csbeq1a 3294 . . . . . 6  |-  ( u  =  a  ->  A  =  [_ a  /  u ]_ A )
41, 2, 3cbvmpt 4379 . . . . 5  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
54fveq1i 5689 . . . 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 28963 . . . . . 6  |-  ( t  e.  ( ZZ  ^m  ( 1 ... M
) )  ->  (
t  |`  ( 1 ... N ) )  e.  ( ZZ  ^m  (
1 ... N ) ) )
87adantl 463 . . . . 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 28981 . . . . . . . 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 2441 . . . . . . . . 9  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A )
1110fmpt 5861 . . . . . . . 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 212 . . . . . . 7  |-  ( ( u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) )  ->  A. u  e.  ( ZZ  ^m  (
1 ... N ) ) A  e.  ZZ )
1312ad2antlr 721 . . . . . 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 3301 . . . . . . . 8  |-  F/_ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A
1514nfel1 2587 . . . . . . 7  |-  F/ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ
16 csbeq1a 3294 . . . . . . . 8  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  A  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
1716eleq1d 2507 . . . . . . 7  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  ( A  e.  ZZ  <->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A  e.  ZZ ) )
1815, 17rspc 3064 . . . . . 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 60 . . . . 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 3288 . . . . . 6  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  [_ a  /  u ]_ A  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
21 eqid 2441 . . . . . 6  |-  ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
2220, 21fvmptg 5769 . . . . 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 656 . . . 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 2486 . . 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 4375 . 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 6115 . . . 4  |-  ( 1 ... M )  e. 
_V
2726a1i 11 . . 3  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( 1 ... M
)  e.  _V )
28 fzssp1 11497 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
296oveq2i 6101 . . . . 5  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3386 . . . 4  |-  ( 1 ... N )  C_  ( 1 ... M
)
3130a1i 11 . . 3  |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( 1 ... N
)  C_  ( 1 ... M ) )
32 simpr 458 . . 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 28996 . . 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 1213 . 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 2515 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   [_csb 3285    C_ wss 3325    e. cmpt 4347    |` cres 4838   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   1c1 9279    + caddc 9281   NN0cn0 10575   ZZcz 10642   ...cfz 11433  mzPolycmzp 28967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  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 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-mzpcl 28968  df-mzp 28969
This theorem is referenced by:  elnn0rabdioph  29050  dvdsrabdioph  29057
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