Theorem 4sqlemafi 12575

Description: Lemma for 4sq 12590. 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 9341	. . . 4 |- ( ph -> 0 e. ZZ )
3		4sqlemafi.p	. . . . 5 |- ( ph -> P e. NN )
4	3	nnzd 9450	. . . 4 |- ( ph -> P e. ZZ )
5		fzofig 10527	. . . 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 2481	. . . . 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 2312	. . . 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 10103	. . . . . . . . . . 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 10705	. . . . . . . . . 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 10442	. . . . . . . . 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 2273	. . . . . . 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 1833	. . . . 5 |- ( ph -> ( E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ( 0..^ P ) ) )
20	19	abssdv 3258	. . . 4 |- ( ph -> { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) } C_ ( 0..^ P ) )
21	8, 20	eqsstrid 3230	. . 3 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) )
22		0zd 9341	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> 0 e. ZZ )
23		4sqlemafi.n	. . . . . . . 8 |- ( ph -> N e. NN )
24	23	nnzd 9450	. . . . . . 7 |- ( ph -> N e. ZZ )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> N e. ZZ )
26		elfzoelz 10225	. . . . . . . 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 10440	. . . . . . . 8 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
32	31	nn0zd 9449	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
33		zdceq 9404	. . . . . . 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 10319	. . . . 5 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
36		vex 2766	. . . . . . 7 |- x e. _V
37		eqeq1 2203	. . . . . . . 8 |- ( u = x -> ( u = ( ( m ^ 2 ) mod P ) <-> x = ( ( m ^ 2 ) mod P ) ) )
38	37	rexbidv 2498	. . . . . . 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 2908	. . . . . 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 841	. . . . 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 2570	. . 3 |- ( ph -> A. x e. ( 0..^ P )DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } )
43		ssfidc 7000	. . 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 1249	. 2 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } e. Fin )
45	1, 44	eqeltrid 2283	1 |- ( ph -> A e. Fin )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   C_ wss 3157  (class class class)co 5923   Fincfn 6801   0cc0 7882   NNcn 8993   2c2 9044   ZZcz 9329   ...cfz 10086  ..^cfzo 10220   mod cmo 10417   ^cexp 10633

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7973  ax-resscn 7974  ax-1cn 7975  ax-1re 7976  ax-icn 7977  ax-addcl 7978  ax-addrcl 7979  ax-mulcl 7980  ax-mulrcl 7981  ax-addcom 7982  ax-mulcom 7983  ax-addass 7984  ax-mulass 7985  ax-distr 7986  ax-i2m1 7987  ax-0lt1 7988  ax-1rid 7989  ax-0id 7990  ax-rnegex 7991  ax-precex 7992  ax-cnre 7993  ax-pre-ltirr 7994  ax-pre-ltwlin 7995  ax-pre-lttrn 7996  ax-pre-apti 7997  ax-pre-ltadd 7998  ax-pre-mulgt0 7999  ax-pre-mulext 8000  ax-arch 8001

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6200  df-2nd 6201  df-recs 6365  df-frec 6451  df-1o 6476  df-er 6594  df-en 6802  df-fin 6804  df-pnf 8066  df-mnf 8067  df-xr 8068  df-ltxr 8069  df-le 8070  df-sub 8202  df-neg 8203  df-reap 8605  df-ap 8612  df-div 8703  df-inn 8994  df-2 9052  df-n0 9253  df-z 9330  df-uz 9605  df-q 9697  df-rp 9732  df-fz 10087  df-fzo 10221  df-fl 10363  df-mod 10418  df-seqfrec 10543  df-exp 10634

This theorem is referenced by:  4sqlemffi  12576  4sqleminfi  12577  4sqlem11  12581