Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabdiophlem2 Structured version   Unicode version

Theorem rabdiophlem2 26862
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 2572 . . . . . 6  |-  F/_ a A
2 nfcsb1v 3283 . . . . . 6  |-  F/_ u [_ a  /  u ]_ A
3 csbeq1a 3259 . . . . . 6  |-  ( u  =  a  ->  A  =  [_ a  /  u ]_ A )
41, 2, 3cbvmpt 4299 . . . . 5  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
54fveq1i 5729 . . . 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 26774 . . . . . 6  |-  ( t  e.  ( ZZ  ^m  ( 1 ... M
) )  ->  (
t  |`  ( 1 ... N ) )  e.  ( ZZ  ^m  (
1 ... N ) ) )
87adantl 453 . . . . 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 26793 . . . . . . . 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 2436 . . . . . . . . 9  |-  ( u  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  A )  =  ( u  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  A )
1110fmpt 5890 . . . . . . . 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 204 . . . . . . 7  |-  ( ( u  e.  ( ZZ 
^m  ( 1 ... N ) )  |->  A )  e.  (mzPoly `  ( 1 ... N
) )  ->  A. u  e.  ( ZZ  ^m  (
1 ... N ) ) A  e.  ZZ )
1312ad2antlr 708 . . . . . 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 3283 . . . . . . . 8  |-  F/_ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A
1514nfel1 2582 . . . . . . 7  |-  F/ u [_ ( t  |`  (
1 ... N ) )  /  u ]_ A  e.  ZZ
16 csbeq1a 3259 . . . . . . . 8  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  A  =  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
1716eleq1d 2502 . . . . . . 7  |-  ( u  =  ( t  |`  ( 1 ... N
) )  ->  ( A  e.  ZZ  <->  [_ ( t  |`  ( 1 ... N
) )  /  u ]_ A  e.  ZZ ) )
1815, 17rspc 3046 . . . . . 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 58 . . . . 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 3254 . . . . . 6  |-  ( a  =  ( t  |`  ( 1 ... N
) )  ->  [_ a  /  u ]_ A  = 
[_ ( t  |`  ( 1 ... N
) )  /  u ]_ A )
21 eqid 2436 . . . . . 6  |-  ( a  e.  ( ZZ  ^m  ( 1 ... N
) )  |->  [_ a  /  u ]_ A )  =  ( a  e.  ( ZZ  ^m  (
1 ... N ) ) 
|->  [_ a  /  u ]_ A )
2220, 21fvmptg 5804 . . . . 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 643 . . . 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 2481 . . 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 4295 . 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 6106 . . . 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 11095 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
296oveq2i 6092 . . . . 5  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3381 . . . 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 448 . . 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 26807 . . 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 1184 . 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 2510 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 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   [_csb 3251    C_ wss 3320    e. cmpt 4266    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   1c1 8991    + caddc 8993   NN0cn0 10221   ZZcz 10282   ...cfz 11043  mzPolycmzp 26779
This theorem is referenced by:  elnn0rabdioph  26863  dvdsrabdioph  26870
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-mzpcl 26780  df-mzp 26781
  Copyright terms: Public domain W3C validator