Theorem eldiophb 37320

Description: Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)

Assertion

Ref	Expression
eldiophb	|- ( D e. (Dioph ` N ) <-> ( N e. NN0 /\ E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) D = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )

Distinct variable groups:   D, k, p   k, N, p, t, u

Allowed substitution hints:   D( u, t)

Proof of Theorem eldiophb

Dummy variables n d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dioph 37319	. . . 4 |- Dioph = ( n e. NN0 |-> ran ( k e. ( ZZ>= ` n ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) } ) )
2	1	dmmptss 5631	. . 3 |- dom Dioph C_ NN0
3		elfvdm 6220	. . 3 |- ( D e. (Dioph ` N ) -> N e. dom Dioph )
4	2, 3	sseldi 3601	. 2 |- ( D e. (Dioph ` N ) -> N e. NN0 )
5		fveq2 6191	. . . . . . 7 |- ( n = N -> ( ZZ>= ` n ) = ( ZZ>= ` N ) )
6		eqidd 2623	. . . . . . 7 |- ( n = N -> (mzPoly ` ( 1 ... k ) ) = (mzPoly ` ( 1 ... k ) ) )
7		oveq2 6658	. . . . . . . . . . . 12 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
8	7	reseq2d 5396	. . . . . . . . . . 11 |- ( n = N -> ( u |` ( 1 ... n ) ) = ( u |` ( 1 ... N ) ) )
9	8	eqeq2d 2632	. . . . . . . . . 10 |- ( n = N -> ( t = ( u |` ( 1 ... n ) ) <-> t = ( u |` ( 1 ... N ) ) ) )
10	9	anbi1d 741	. . . . . . . . 9 |- ( n = N -> ( ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) <-> ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) ) )
11	10	rexbidv 3052	. . . . . . . 8 |- ( n = N -> ( 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 ) ) )
12	11	abbidv 2741	. . . . . . 7 |- ( n = N -> { 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	5, 6, 12	mpt2eq123dv 6717	. . . . . 6 |- ( n = N -> ( k e. ( ZZ>= ` n ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) } ) = ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
14	13	rneqd 5353	. . . . 5 |- ( n = N -> ran ( k e. ( ZZ>= ` n ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... n ) ) /\ ( p ` u ) = 0 ) } ) = ran ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
15		ovex 6678	. . . . . . 7 |- ( NN0 ^m ( 1 ... N ) ) e. _V
16	15	pwex 4848	. . . . . 6 |- ~P ( NN0 ^m ( 1 ... N ) ) e. _V
17		eqid 2622	. . . . . . . 8 |- ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) = ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
18	17	rnmpt2 6770	. . . . . . 7 |- ran ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) = { d | E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } }
19		elmapi 7879	. . . . . . . . . . . . . . . . 17 |- ( u e. ( NN0 ^m ( 1 ... k ) ) -> u : ( 1 ... k ) --> NN0 )
20		fzss2 12381	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ZZ>= ` N ) -> ( 1 ... N ) C_ ( 1 ... k ) )
21		fssres 6070	. . . . . . . . . . . . . . . . 17 |- ( ( u : ( 1 ... k ) --> NN0 /\ ( 1 ... N ) C_ ( 1 ... k ) ) -> ( u |` ( 1 ... N ) ) : ( 1 ... N ) --> NN0 )
22	19, 20, 21	syl2anr 495	. . . . . . . . . . . . . . . 16 |- ( ( k e. ( ZZ>= ` N ) /\ u e. ( NN0 ^m ( 1 ... k ) ) ) -> ( u |` ( 1 ... N ) ) : ( 1 ... N ) --> NN0 )
23		nn0ex 11298	. . . . . . . . . . . . . . . . 17 |- NN0 e. _V
24		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( 1 ... N ) e. _V
25	23, 24	elmap 7886	. . . . . . . . . . . . . . . 16 |- ( ( u |` ( 1 ... N ) ) e. ( NN0 ^m ( 1 ... N ) ) <-> ( u |` ( 1 ... N ) ) : ( 1 ... N ) --> NN0 )
26	22, 25	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( k e. ( ZZ>= ` N ) /\ u e. ( NN0 ^m ( 1 ... k ) ) ) -> ( u |` ( 1 ... N ) ) e. ( NN0 ^m ( 1 ... N ) ) )
27		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( t = ( u |` ( 1 ... N ) ) -> ( t e. ( NN0 ^m ( 1 ... N ) ) <-> ( u |` ( 1 ... N ) ) e. ( NN0 ^m ( 1 ... N ) ) ) )
28	27	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) -> ( t e. ( NN0 ^m ( 1 ... N ) ) <-> ( u |` ( 1 ... N ) ) e. ( NN0 ^m ( 1 ... N ) ) ) )
29	26, 28	syl5ibrcom 237	. . . . . . . . . . . . . 14 |- ( ( k e. ( ZZ>= ` N ) /\ u e. ( NN0 ^m ( 1 ... k ) ) ) -> ( ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) -> t e. ( NN0 ^m ( 1 ... N ) ) ) )
30	29	rexlimdva 3031	. . . . . . . . . . . . 13 |- ( k e. ( ZZ>= ` N ) -> ( E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) -> t e. ( NN0 ^m ( 1 ... N ) ) ) )
31	30	abssdv 3676	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` N ) -> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } C_ ( NN0 ^m ( 1 ... N ) ) )
32	15	elpw2 4828	. . . . . . . . . . . 12 |- ( { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } e. ~P ( NN0 ^m ( 1 ... N ) ) <-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } C_ ( NN0 ^m ( 1 ... N ) ) )
33	31, 32	sylibr 224	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` N ) -> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } e. ~P ( NN0 ^m ( 1 ... N ) ) )
34		eleq1 2689	. . . . . . . . . . 11 |- ( d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } -> ( d e. ~P ( NN0 ^m ( 1 ... N ) ) <-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } e. ~P ( NN0 ^m ( 1 ... N ) ) ) )
35	33, 34	syl5ibrcom 237	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` N ) -> ( d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } -> d e. ~P ( NN0 ^m ( 1 ... N ) ) ) )
36	35	rexlimdvw 3034	. . . . . . . . 9 |- ( k e. ( ZZ>= ` N ) -> ( E. p e. (mzPoly ` ( 1 ... k ) ) d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } -> d e. ~P ( NN0 ^m ( 1 ... N ) ) ) )
37	36	rexlimiv 3027	. . . . . . . 8 |- ( E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } -> d e. ~P ( NN0 ^m ( 1 ... N ) ) )
38	37	abssi 3677	. . . . . . 7 |- { d | E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) d = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } } C_ ~P ( NN0 ^m ( 1 ... N ) )
39	18, 38	eqsstri 3635	. . . . . 6 |- ran ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) C_ ~P ( NN0 ^m ( 1 ... N ) )
40	16, 39	ssexi 4803	. . . . 5 |- ran ( 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. _V
41	14, 1, 40	fvmpt 6282	. . . 4 |- ( N e. NN0 -> (Dioph ` N ) = ran ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
42	41	eleq2d 2687	. . 3 |- ( N e. NN0 -> ( D e. (Dioph ` N ) <-> D e. ran ( k e. ( ZZ>= ` N ) , p e. (mzPoly ` ( 1 ... k ) ) |-> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) ) )
43		ovex 6678	. . . . . 6 |- ( NN0 ^m ( 1 ... k ) ) e. _V
44	43	abrexex 7141	. . . . 5 |- { t | E. u e. ( NN0 ^m ( 1 ... k ) ) t = ( u |` ( 1 ... N ) ) } e. _V
45		simpl 473	. . . . . . 7 |- ( ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) -> t = ( u |` ( 1 ... N ) ) )
46	45	reximi 3011	. . . . . 6 |- ( 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 ) ) )
47	46	ss2abi 3674	. . . . 5 |- { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } C_ { t | E. u e. ( NN0 ^m ( 1 ... k ) ) t = ( u |` ( 1 ... N ) ) }
48	44, 47	ssexi 4803	. . . 4 |- { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } e. _V
49	17, 48	elrnmpt2 6773	. . 3 |- ( D e. ran ( 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. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) D = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
50	42, 49	syl6bb 276	. 2 |- ( N e. NN0 -> ( D e. (Dioph ` N ) <-> E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) D = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
51	4, 50	biadan2 674	1 |- ( D e. (Dioph ` N ) <-> ( N e. NN0 /\ E. k e. ( ZZ>= ` N ) E. p e. (mzPoly ` ( 1 ... k ) ) D = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^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:  eldioph  37321  eldioph2b  37326  eldiophelnn0  37327