Theorem 4sqlemafi 12934

Description: Lemma for 4sq 12949. A is finite. (Contributed by Jim Kingdon, 24-May-2025.)

Hypotheses

Ref	Expression
4sqlemafi.n	|- ( ph -> N e. NN )
4sqlemafi.p	|- ( ph -> P e. NN )
4sqlemafi.a	|- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }

Assertion

Ref	Expression
4sqlemafi	|- ( ph -> A e. Fin )

Distinct variable groups:   m, N, u   P, m, u   ph, m, u

Allowed substitution hints:   A( u, m)

Proof of Theorem 4sqlemafi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlemafi.a	. 2 |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }
2		0zd 9469	. . . 4 |- ( ph -> 0 e. ZZ )
3		4sqlemafi.p	. . . . 5 |- ( ph -> P e. NN )
4	3	nnzd 9579	. . . 4 |- ( ph -> P e. ZZ )
5		fzofig 10666	. . . 4 |- ( ( 0 e. ZZ /\ P e. ZZ ) -> ( 0..^ P ) e. Fin )
6	2, 4, 5	syl2anc 411	. . 3 |- ( ph -> ( 0..^ P ) e. Fin )
7		df-rex 2514	. . . . 5 |- ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) <-> E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) )
8	7	abbii 2345	. . . 4 |- { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } = { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) }
9		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> u = ( ( m ^ 2 ) mod P ) )
10		elfzelz 10233	. . . . . . . . . . 11 |- ( m e. ( 0 ... N ) -> m e. ZZ )
11	10	ad2antrl 490	. . . . . . . . . 10 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> m e. ZZ )
12		zsqcl 10844	. . . . . . . . . 10 |- ( m e. ZZ -> ( m ^ 2 ) e. ZZ )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> ( m ^ 2 ) e. ZZ )
14	3	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> P e. NN )
15		zmodfzo 10581	. . . . . . . . 9 |- ( ( ( m ^ 2 ) e. ZZ /\ P e. NN ) -> ( ( m ^ 2 ) mod P ) e. ( 0..^ P ) )
16	13, 14, 15	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> ( ( m ^ 2 ) mod P ) e. ( 0..^ P ) )
17	9, 16	eqeltrd 2306	. . . . . . 7 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> u e. ( 0..^ P ) )
18	17	ex 115	. . . . . 6 |- ( ph -> ( ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ( 0..^ P ) ) )
19	18	exlimdv 1865	. . . . 5 |- ( ph -> ( E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ( 0..^ P ) ) )
20	19	abssdv 3298	. . . 4 |- ( ph -> { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) } C_ ( 0..^ P ) )
21	8, 20	eqsstrid 3270	. . 3 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) )
22		0zd 9469	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> 0 e. ZZ )
23		4sqlemafi.n	. . . . . . . 8 |- ( ph -> N e. NN )
24	23	nnzd 9579	. . . . . . 7 |- ( ph -> N e. ZZ )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> N e. ZZ )
26		elfzoelz 10355	. . . . . . . 8 |- ( x e. ( 0..^ P ) -> x e. ZZ )
27	26	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> x e. ZZ )
28	10	adantl 277	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> m e. ZZ )
29	28, 12	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( m ^ 2 ) e. ZZ )
30	3	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> P e. NN )
31	29, 30	zmodcld 10579	. . . . . . . 8 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
32	31	nn0zd 9578	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
33		zdceq 9533	. . . . . . 7 |- ( ( x e. ZZ /\ ( ( m ^ 2 ) mod P ) e. ZZ ) -> DECID x = ( ( m ^ 2 ) mod P ) )
34	27, 32, 33	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> DECID x = ( ( m ^ 2 ) mod P ) )
35	22, 25, 34	exfzdc 10458	. . . . 5 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
36		vex 2802	. . . . . . 7 |- x e. _V
37		eqeq1 2236	. . . . . . . 8 |- ( u = x -> ( u = ( ( m ^ 2 ) mod P ) <-> x = ( ( m ^ 2 ) mod P ) ) )
38	37	rexbidv 2531	. . . . . . 7 |- ( u = x -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) <-> E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) ) )
39	36, 38	elab 2947	. . . . . 6 |- ( x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } <-> E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
40	39	dcbii 845	. . . . 5 |- (DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } <-> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
41	35, 40	sylibr 134	. . . 4 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } )
42	41	ralrimiva 2603	. . 3 |- ( ph -> A. x e. ( 0..^ P )DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } )
43		ssfidc 7110	. . 3 |- ( ( ( 0..^ P ) e. Fin /\ { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) /\ A. x e. ( 0..^ P )DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } ) -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } e. Fin )
44	6, 21, 42, 43	syl3anc 1271	. 2 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } e. Fin )
45	1, 44	eqeltrid 2316	1 |- ( ph -> A e. Fin )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 839   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   C_ wss 3197  (class class class)co 6007   Fincfn 6895   0cc0 8010   NNcn 9121   2c2 9172   ZZcz 9457   ...cfz 10216  ..^cfzo 10350   mod cmo 10556   ^cexp 10772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773

This theorem is referenced by:  4sqlemffi  12935  4sqleminfi  12936  4sqlem11  12940