Theorem eldioph3 37329

Description: Inference version of eldioph3b 37328 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Assertion

Ref	Expression
eldioph3	|- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )

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

Proof of Theorem eldioph3

Dummy variables a b p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> N e. NN0 )
2		simpr 477	. . 3 |- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> P e. (mzPoly ` NN ) )
3		eqidd 2623	. . 3 |- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
4		fveq1 6190	. . . . . . . . . 10 |- ( p = P -> ( p ` b ) = ( P ` b ) )
5	4	eqeq1d 2624	. . . . . . . . 9 |- ( p = P -> ( ( p ` b ) = 0 <-> ( P ` b ) = 0 ) )
6	5	anbi2d 740	. . . . . . . 8 |- ( p = P -> ( ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) <-> ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) ) )
7	6	rexbidv 3052	. . . . . . 7 |- ( p = P -> ( E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) <-> E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) ) )
8	7	abbidv 2741	. . . . . 6 |- ( p = P -> { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } = { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) } )
9		eqeq1 2626	. . . . . . . . . 10 |- ( a = t -> ( a = ( b |` ( 1 ... N ) ) <-> t = ( b |` ( 1 ... N ) ) ) )
10	9	anbi1d 741	. . . . . . . . 9 |- ( a = t -> ( ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) <-> ( t = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) ) )
11	10	rexbidv 3052	. . . . . . . 8 |- ( a = t -> ( E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) <-> E. b e. ( NN0 ^m NN ) ( t = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) ) )
12		reseq1 5390	. . . . . . . . . . 11 |- ( b = u -> ( b |` ( 1 ... N ) ) = ( u |` ( 1 ... N ) ) )
13	12	eqeq2d 2632	. . . . . . . . . 10 |- ( b = u -> ( t = ( b |` ( 1 ... N ) ) <-> t = ( u |` ( 1 ... N ) ) ) )
14		fveq2 6191	. . . . . . . . . . 11 |- ( b = u -> ( P ` b ) = ( P ` u ) )
15	14	eqeq1d 2624	. . . . . . . . . 10 |- ( b = u -> ( ( P ` b ) = 0 <-> ( P ` u ) = 0 ) )
16	13, 15	anbi12d 747	. . . . . . . . 9 |- ( b = u -> ( ( t = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) <-> ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
17	16	cbvrexv 3172	. . . . . . . 8 |- ( E. b e. ( NN0 ^m NN ) ( t = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) <-> E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) )
18	11, 17	syl6bb 276	. . . . . . 7 |- ( a = t -> ( E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) <-> E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
19	18	cbvabv 2747	. . . . . 6 |- { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( P ` b ) = 0 ) } = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) }
20	8, 19	syl6eq 2672	. . . . 5 |- ( p = P -> { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
21	20	eqeq2d 2632	. . . 4 |- ( p = P -> ( { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } <-> { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } ) )
22	21	rspcev 3309	. . 3 |- ( ( P e. (mzPoly ` NN ) /\ { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } ) -> E. p e. (mzPoly ` NN ) { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } )
23	2, 3, 22	syl2anc 693	. 2 |- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> E. p e. (mzPoly ` NN ) { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } )
24		eldioph3b 37328	. 2 |- ( { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) <-> ( N e. NN0 /\ E. p e. (mzPoly ` NN ) { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { a | E. b e. ( NN0 ^m NN ) ( a = ( b |` ( 1 ... N ) ) /\ ( p ` b ) = 0 ) } ) )
25	1, 23, 24	sylanbrc 698	1 |- ( ( N e. NN0 /\ P e. (mzPoly ` NN ) ) -> { t | E. u e. ( NN0 ^m NN ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. (Dioph ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   NNcn 11020   NN0cn0 11292   ...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:  diophrex  37339