Theorem 4sqlemafi 12958

Description: Lemma for 4sq 12973. 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 9481	. . . 4 |- ( ph -> 0 e. ZZ )
3		4sqlemafi.p	. . . . 5 |- ( ph -> P e. NN )
4	3	nnzd 9591	. . . 4 |- ( ph -> P e. ZZ )
5		fzofig 10684	. . . 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 10250	. . . . . . . . . . 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 10862	. . . . . . . . . 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 10599	. . . . . . . . 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 3299	. . . 4 |- ( ph -> { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) } C_ ( 0..^ P ) )
21	8, 20	eqsstrid 3271	. . 3 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) )
22		0zd 9481	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> 0 e. ZZ )
23		4sqlemafi.n	. . . . . . . 8 |- ( ph -> N e. NN )
24	23	nnzd 9591	. . . . . . 7 |- ( ph -> N e. ZZ )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> N e. ZZ )
26		elfzoelz 10372	. . . . . . . 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 10597	. . . . . . . 8 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
32	31	nn0zd 9590	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
33		zdceq 9545	. . . . . . 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 10476	. . . . 5 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
36		vex 2803	. . . . . . 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 2948	. . . . . 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 7122	. . 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 3198  (class class class)co 6013   Fincfn 6904   0cc0 8022   NNcn 9133   2c2 9184   ZZcz 9469   ...cfz 10233  ..^cfzo 10367   mod cmo 10574   ^cexp 10790

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791

This theorem is referenced by:  4sqlemffi  12959  4sqleminfi  12960  4sqlem11  12964