Theorem 4sqleminfi 13095

Description: Lemma for 4sq 13108. 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 13093	. 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 10359	. . . . . . . . . . . . . . . 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 10972	. . . . . . . . . . . . . . 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 10707	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
12	11	nn0zd 9698	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
13	5, 12	eqeltrd 2309	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
14	13	rexlimdva2 2663	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
15	14	abssdv 3312	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
16	3, 15	eqsstrid 3284	. . . . . . . 8 |- ( ph -> A C_ ZZ )
17	16	sselda 3238	. . . . . . 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 9699	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. ZZ )
20		peano2zm 9615	. . . . . . . . 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 3238	. . . . . . . . 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 9705	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
25		zdceq 9653	. . . . . . 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 2615	. . . . 5 |- ( ( ph /\ x e. A ) -> A. v e. A DECID x = ( ( P - 1 ) - v ) )
28		finexdc 7160	. . . . 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 5006	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) ) )
32	31	elv 2817	. . . . 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 2615	. 2 |- ( ph -> A. x e. A DECID x e. ran F )
36		infidc 7201	. 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 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2813   i^i cin 3210   |-> cmpt 4171   ran crn 4750  (class class class)co 6050   Fincfn 6975   0cc0 8127   1c1 8128   - cmin 8444   NNcn 9237   2c2 9288   ZZcz 9577   ...cfz 10342   mod cmo 10684   ^cexp 10900

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901

This theorem is referenced by:  4sqlem11  13099  4sqlem12  13100