Theorem diophrex 37339

Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Assertion

Ref	Expression
diophrex	|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) -> { t | E. u e. S t = ( u |` ( 1 ... N ) ) } e. (Dioph ` N ) )

Distinct variable groups:   t, N, u   t, S, u

Allowed substitution hints:   M( u, t)

Proof of Theorem diophrex

Dummy variables a b c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . 5 |- ( a = t -> ( a = ( b |` ( 1 ... N ) ) <-> t = ( b |` ( 1 ... N ) ) ) )
2	1	rexbidv 3052	. . . 4 |- ( a = t -> ( E. b e. S a = ( b |` ( 1 ... N ) ) <-> E. b e. S t = ( b |` ( 1 ... N ) ) ) )
3		reseq1 5390	. . . . . 6 |- ( b = u -> ( b |` ( 1 ... N ) ) = ( u |` ( 1 ... N ) ) )
4	3	eqeq2d 2632	. . . . 5 |- ( b = u -> ( t = ( b |` ( 1 ... N ) ) <-> t = ( u |` ( 1 ... N ) ) ) )
5	4	cbvrexv 3172	. . . 4 |- ( E. b e. S t = ( b |` ( 1 ... N ) ) <-> E. u e. S t = ( u |` ( 1 ... N ) ) )
6	2, 5	syl6bb 276	. . 3 |- ( a = t -> ( E. b e. S a = ( b |` ( 1 ... N ) ) <-> E. u e. S t = ( u |` ( 1 ... N ) ) ) )
7	6	cbvabv 2747	. 2 |- { a | E. b e. S a = ( b |` ( 1 ... N ) ) } = { t | E. u e. S t = ( u |` ( 1 ... N ) ) }
8		rexeq 3139	. . . . . 6 |- ( S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } -> ( E. b e. S a = ( b |` ( 1 ... N ) ) <-> E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) ) )
9	8	abbidv 2741	. . . . 5 |- ( S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } -> { a | E. b e. S a = ( b |` ( 1 ... N ) ) } = { a | E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) } )
10	9	adantl 482	. . . 4 |- ( ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) /\ S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } ) -> { a | E. b e. S a = ( b |` ( 1 ... N ) ) } = { a | E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) } )
11		eqeq1 2626	. . . . . . . . . . 11 |- ( d = b -> ( d = ( e |` ( 1 ... M ) ) <-> b = ( e |` ( 1 ... M ) ) ) )
12	11	anbi1d 741	. . . . . . . . . 10 |- ( d = b -> ( ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) <-> ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) ) )
13	12	rexbidv 3052	. . . . . . . . 9 |- ( d = b -> ( E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) <-> E. e e. ( NN0 ^m NN ) ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) ) )
14	13	rexab 3369	. . . . . . . 8 |- ( E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) <-> E. b ( E. e e. ( NN0 ^m NN ) ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) )
15		r19.41v 3089	. . . . . . . . . 10 |- ( E. e e. ( NN0 ^m NN ) ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> ( E. e e. ( NN0 ^m NN ) ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) )
16	15	exbii 1774	. . . . . . . . 9 |- ( E. b E. e e. ( NN0 ^m NN ) ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. b ( E. e e. ( NN0 ^m NN ) ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) )
17		rexcom4 3225	. . . . . . . . . 10 |- ( E. e e. ( NN0 ^m NN ) E. b ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. b E. e e. ( NN0 ^m NN ) ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) )
18		anass 681	. . . . . . . . . . . . . 14 |- ( ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> ( b = ( e |` ( 1 ... M ) ) /\ ( ( c ` e ) = 0 /\ a = ( b |` ( 1 ... N ) ) ) ) )
19	18	exbii 1774	. . . . . . . . . . . . 13 |- ( E. b ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. b ( b = ( e |` ( 1 ... M ) ) /\ ( ( c ` e ) = 0 /\ a = ( b |` ( 1 ... N ) ) ) ) )
20		vex 3203	. . . . . . . . . . . . . . 15 |- e e. _V
21	20	resex 5443	. . . . . . . . . . . . . 14 |- ( e |` ( 1 ... M ) ) e. _V
22		reseq1 5390	. . . . . . . . . . . . . . . 16 |- ( b = ( e |` ( 1 ... M ) ) -> ( b |` ( 1 ... N ) ) = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) )
23	22	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( b = ( e |` ( 1 ... M ) ) -> ( a = ( b |` ( 1 ... N ) ) <-> a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) )
24	23	anbi2d 740	. . . . . . . . . . . . . 14 |- ( b = ( e |` ( 1 ... M ) ) -> ( ( ( c ` e ) = 0 /\ a = ( b |` ( 1 ... N ) ) ) <-> ( ( c ` e ) = 0 /\ a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) ) )
25	21, 24	ceqsexv 3242	. . . . . . . . . . . . 13 |- ( E. b ( b = ( e |` ( 1 ... M ) ) /\ ( ( c ` e ) = 0 /\ a = ( b |` ( 1 ... N ) ) ) ) <-> ( ( c ` e ) = 0 /\ a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) )
26	19, 25	bitri 264	. . . . . . . . . . . 12 |- ( E. b ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> ( ( c ` e ) = 0 /\ a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) )
27		ancom 466	. . . . . . . . . . . . 13 |- ( ( ( c ` e ) = 0 /\ a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) <-> ( a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) )
28		simpl2 1065	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> M e. ( ZZ>= ` N ) )
29		fzss2 12381	. . . . . . . . . . . . . . . 16 |- ( M e. ( ZZ>= ` N ) -> ( 1 ... N ) C_ ( 1 ... M ) )
30		resabs1 5427	. . . . . . . . . . . . . . . 16 |- ( ( 1 ... N ) C_ ( 1 ... M ) -> ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) = ( e |` ( 1 ... N ) ) )
31	28, 29, 30	3syl 18	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) = ( e |` ( 1 ... N ) ) )
32	31	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) <-> a = ( e |` ( 1 ... N ) ) ) )
33	32	anbi1d 741	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( ( a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) <-> ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
34	27, 33	syl5bb 272	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( ( ( c ` e ) = 0 /\ a = ( ( e |` ( 1 ... M ) ) |` ( 1 ... N ) ) ) <-> ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
35	26, 34	syl5bb 272	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( E. b ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
36	35	rexbidv 3052	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( E. e e. ( NN0 ^m NN ) E. b ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
37	17, 36	syl5bbr 274	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( E. b E. e e. ( NN0 ^m NN ) ( ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
38	16, 37	syl5bbr 274	. . . . . . . 8 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( E. b ( E. e e. ( NN0 ^m NN ) ( b = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) /\ a = ( b |` ( 1 ... N ) ) ) <-> E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
39	14, 38	syl5bb 272	. . . . . . 7 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> ( E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) <-> E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) ) )
40	39	abbidv 2741	. . . . . 6 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> { a | E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) } = { a | E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) } )
41		eldioph3 37329	. . . . . . 7 |- ( ( N e. NN0 /\ c e. (mzPoly ` NN ) ) -> { a | E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) } e. (Dioph ` N ) )
42	41	3ad2antl1 1223	. . . . . 6 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> { a | E. e e. ( NN0 ^m NN ) ( a = ( e |` ( 1 ... N ) ) /\ ( c ` e ) = 0 ) } e. (Dioph ` N ) )
43	40, 42	eqeltrd 2701	. . . . 5 |- ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) -> { a | E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) } e. (Dioph ` N ) )
44	43	adantr 481	. . . 4 |- ( ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) /\ S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } ) -> { a | E. b e. { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } a = ( b |` ( 1 ... N ) ) } e. (Dioph ` N ) )
45	10, 44	eqeltrd 2701	. . 3 |- ( ( ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) /\ c e. (mzPoly ` NN ) ) /\ S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } ) -> { a | E. b e. S a = ( b |` ( 1 ... N ) ) } e. (Dioph ` N ) )
46		eldioph3b 37328	. . . . 5 |- ( S e. (Dioph ` M ) <-> ( M e. NN0 /\ E. c e. (mzPoly ` NN ) S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } ) )
47	46	simprbi 480	. . . 4 |- ( S e. (Dioph ` M ) -> E. c e. (mzPoly ` NN ) S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } )
48	47	3ad2ant3 1084	. . 3 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) -> E. c e. (mzPoly ` NN ) S = { d | E. e e. ( NN0 ^m NN ) ( d = ( e |` ( 1 ... M ) ) /\ ( c ` e ) = 0 ) } )
49	45, 48	r19.29a 3078	. 2 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) -> { a | E. b e. S a = ( b |` ( 1 ... N ) ) } e. (Dioph ` N ) )
50	7, 49	syl5eqelr 2706	1 |- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) /\ S e. (Dioph ` M ) ) -> { t | E. u e. S t = ( u |` ( 1 ... N ) ) } e. (Dioph ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   NNcn 11020   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-inf2 8538  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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-rmo 2920  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-int 4476  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  rexrabdioph  37358