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Theorem rabdiophlem2 29145
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 2584 . . . . . 6  |-  F/_ a A
2 nfcsb1v 3309 . . . . . 6  |-  F/_ u [_ a  /  u ]_ A
3 csbeq1a 3302 . . . . . 6  |-  ( u  =  a  ->  A  =  [_ a  /  u ]_ A )
41, 2, 3cbvmpt 4387 . . . . 5  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
54fveq1i 5697 . . . 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 29059 . . . . . 6  |-  ( t  e.  ( ZZ  ^m  ( 1 ... M
) )  ->  (
t  |`  ( 1 ... N ) )  e.  ( ZZ  ^m  (
1 ... N ) ) )
87adantl 466 . . . . 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 29077 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A )
1110fmpt 5869 . . . . . . . 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 726 . . . . . 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 3309 . . . . . . . 8  |-  F/_ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A
1514nfel1 2594 . . . . . . 7  |-  F/ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ
16 csbeq1a 3302 . . . . . . . 8  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  A  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
1716eleq1d 2509 . . . . . . 7  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  ( A  e.  ZZ  <->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A  e.  ZZ ) )
1815, 17rspc 3072 . . . . . 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 3296 . . . . . 6  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  [_ a  /  u ]_ A  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
21 eqid 2443 . . . . . 6  |-  ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
2220, 21fvmptg 5777 . . . . 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 661 . . . 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 2488 . . 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 4383 . 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 6121 . . . 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 11506 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
296oveq2i 6107 . . . . 5  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3394 . . . 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 461 . . 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 29092 . . 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 1218 . 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 2517 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 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977   [_csb 3293    C_ wss 3333    e. cmpt 4355    |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   1c1 9288    + caddc 9290   NN0cn0 10584   ZZcz 10651   ...cfz 11442  mzPolycmzp 29063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-mzpcl 29064  df-mzp 29065
This theorem is referenced by:  elnn0rabdioph  29146  dvdsrabdioph  29153
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