Theorem 4sqleminfi 12591

Description: Lemma for 4sq 12604. A i^i ran F 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 ) }
4sqlemffi.f	|- F = ( v e. A |-> ( ( P - 1 ) - v ) )

Assertion

Ref	Expression
4sqleminfi	|- ( ph -> ( A i^i ran F ) e. Fin )

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

Allowed substitution hints:   A( u, m)   P( v)   F( v, u, m)   N( v)

Proof of Theorem 4sqleminfi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlemafi.n	. . 3 |- ( ph -> N e. NN )
2		4sqlemafi.p	. . 3 |- ( ph -> P e. NN )
3		4sqlemafi.a	. . 3 |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }
4	1, 2, 3	4sqlemafi 12589	. 2 |- ( ph -> A e. Fin )
5		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u = ( ( m ^ 2 ) mod P ) )
6		elfzelz 10117	. . . . . . . . . . . . . . . 16 |- ( m e. ( 0 ... N ) -> m e. ZZ )
7	6	ad2antlr 489	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> m e. ZZ )
8		zsqcl 10719	. . . . . . . . . . . . . . 15 |- ( m e. ZZ -> ( m ^ 2 ) e. ZZ )
9	7, 8	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( m ^ 2 ) e. ZZ )
10	2	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> P e. NN )
11	9, 10	zmodcld 10454	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
12	11	nn0zd 9463	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
13	5, 12	eqeltrd 2273	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
14	13	rexlimdva2 2617	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
15	14	abssdv 3258	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
16	3, 15	eqsstrid 3230	. . . . . . . 8 |- ( ph -> A C_ ZZ )
17	16	sselda 3184	. . . . . . 7 |- ( ( ph /\ x e. A ) -> x e. ZZ )
18	2	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. NN )
19	18	nnzd 9464	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. ZZ )
20		peano2zm 9381	. . . . . . . . 9 |- ( P e. ZZ -> ( P - 1 ) e. ZZ )
21	19, 20	syl 14	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( P - 1 ) e. ZZ )
22	16	sselda 3184	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> v e. ZZ )
23	22	adantlr 477	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> v e. ZZ )
24	21, 23	zsubcld 9470	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
25		zdceq 9418	. . . . . . 7 |- ( ( x e. ZZ /\ ( ( P - 1 ) - v ) e. ZZ ) -> DECID x = ( ( P - 1 ) - v ) )
26	17, 24, 25	syl2an2r 595	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> DECID x = ( ( P - 1 ) - v ) )
27	26	ralrimiva 2570	. . . . 5 |- ( ( ph /\ x e. A ) -> A. v e. A DECID x = ( ( P - 1 ) - v ) )
28		finexdc 6972	. . . . 5 |- ( ( A e. Fin /\ A. v e. A DECID x = ( ( P - 1 ) - v ) ) -> DECID E. v e. A x = ( ( P - 1 ) - v ) )
29	4, 27, 28	syl2an2r 595	. . . 4 |- ( ( ph /\ x e. A ) -> DECID E. v e. A x = ( ( P - 1 ) - v ) )
30		4sqlemffi.f	. . . . . . 7 |- F = ( v e. A |-> ( ( P - 1 ) - v ) )
31	30	elrnmpt 4916	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) ) )
32	31	elv 2767	. . . . 5 |- ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) )
33	32	dcbii 841	. . . 4 |- (DECID x e. ran F <-> DECID E. v e. A x = ( ( P - 1 ) - v ) )
34	29, 33	sylibr 134	. . 3 |- ( ( ph /\ x e. A ) -> DECID x e. ran F )
35	34	ralrimiva 2570	. 2 |- ( ph -> A. x e. A DECID x e. ran F )
36		infidc 7009	. 2 |- ( ( A e. Fin /\ A. x e. A DECID x e. ran F ) -> ( A i^i ran F ) e. Fin )
37	4, 35, 36	syl2anc 411	1 |- ( ph -> ( A i^i ran F ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   _Vcvv 2763   i^i cin 3156   |-> cmpt 4095   ran crn 4665  (class class class)co 5925   Fincfn 6808   0cc0 7896   1c1 7897   - cmin 8214   NNcn 9007   2c2 9058   ZZcz 9343   ...cfz 10100   mod cmo 10431   ^cexp 10647

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 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

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 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648

This theorem is referenced by:  4sqlem11  12595  4sqlem12  12596