Theorem 4sqleminfi 13120

Description: Lemma for 4sq 13133. 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 13118	. 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 10378	. . . . . . . . . . . . . . . 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 10996	. . . . . . . . . . . . . . 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 10731	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
12	11	nn0zd 9716	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
13	5, 12	eqeltrd 2311	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
14	13	rexlimdva2 2665	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
15	14	abssdv 3316	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
16	3, 15	eqsstrid 3288	. . . . . . . 8 |- ( ph -> A C_ ZZ )
17	16	sselda 3242	. . . . . . 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 9717	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. ZZ )
20		peano2zm 9632	. . . . . . . . 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 3242	. . . . . . . . 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 9723	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
25		zdceq 9670	. . . . . . 7 |- ( ( x e. ZZ /\ ( ( P - 1 ) - v ) e. ZZ ) -> DECID x = ( ( P - 1 ) - v ) )
26	17, 24, 25	syl2an2r 599	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> DECID x = ( ( P - 1 ) - v ) )
27	26	ralrimiva 2617	. . . . 5 |- ( ( ph /\ x e. A ) -> A. v e. A DECID x = ( ( P - 1 ) - v ) )
28		finexdc 7173	. . . . 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 599	. . . 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 5011	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) ) )
32	31	elv 2819	. . . . 5 |- ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) )
33	32	dcbii 848	. . . 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 2617	. 2 |- ( ph -> A. x e. A DECID x e. ran F )
36		infidc 7214	. 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 842   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   i^i cin 3213   |-> cmpt 4176   ran crn 4755  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   - cmin 8460   NNcn 9254   2c2 9305   ZZcz 9594   ...cfz 10361   mod cmo 10708   ^cexp 10924

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  4sqlem11  13124  4sqlem12  13125