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Theorem eldioph 37321
Description: Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
eldioph |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )
Distinct variable groups:   t, N, u   t, K, u   t, P, u
Proof of Theorem eldioph
Dummy variables k p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1061 . 2 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> N e. NN0 )
2 simp2 1062 . . 3 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> K e. ( ZZ>= ` N ) )
3 simp3 1063 . . . 4 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> P e. (mzPoly ` ( 1 ... K ) ) )
4 eqidd 2623 . . . 4 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
5 fveq1 6190 . . . . . . . . . 10 |- ( p = P -> ( p ` u ) = ( P ` u ) )
65eqeq1d 2624 . . . . . . . . 9 |- ( p = P -> ( ( p ` u ) = 0 <-> ( P ` u ) = 0 ) )
76anbi2d 740 . . . . . . . 8 |- ( p = P -> ( ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
87rexbidv 3052 . . . . . . 7 |- ( p = P -> ( E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
98abbidv 2741 . . . . . 6 |- ( p = P -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
109eqeq2d 2632 . . . . 5 |- ( p = P -> ( { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } <-> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } ) )
1110rspcev 3309 . . . 4 |- ( ( P e. (mzPoly ` ( 1 ... K ) ) /\ { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } ) -> E. p e. (mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
123, 4, 11syl2anc 693 . . 3 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> E. p e. (mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
13 oveq2 6658 . . . . . 6 |- ( k = K -> ( 1 ... k ) = ( 1 ... K ) )
1413fveq2d 6195 . . . . 5 |- ( k = K -> (mzPoly ` ( 1 ... k ) ) = (mzPoly ` ( 1 ... K ) ) )
1513oveq2d 6666 . . . . . . . 8 |- ( k = K -> ( NN0 ^m ( 1 ... k ) ) = ( NN0 ^m ( 1 ... K ) ) )
1615rexeqdv 3145 . . . . . . 7 |- ( k = K -> ( E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) ) )
1716abbidv 2741 . . . . . 6 |- ( k = K -> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
1817eqeq2d 2632 . . . . 5 |- ( k = K -> ( { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } <-> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
1914, 18rexeqbidv 3153 . . . 4 |- ( k = K -> ( E. p e. (mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } <-> E. p e. (mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
2019rspcev 3309 . . 3 |- ( ( K e. ( ZZ>= ` N ) /\ E. p e. (mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) -> E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
212, 12, 20syl2anc 693 . 2 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
22 eldiophb 37320 . 2 |- ( { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) <-> ( N e. NN0 /\ E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
231, 21, 22sylanbrc 698 1 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. (mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  mzPolycmzp 37285  Diophcdioph 37318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-dioph 37319
This theorem is referenced by:  eldioph2  37325  eq0rabdioph  37340
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