Theorem 4sqleminfi 12753

Description: Lemma for 4sq 12766. 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 12751	. 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 10149	. . . . . . . . . . . . . . . 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 10757	. . . . . . . . . . . . . . 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 10492	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
12	11	nn0zd 9495	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
13	5, 12	eqeltrd 2282	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
14	13	rexlimdva2 2626	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
15	14	abssdv 3267	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
16	3, 15	eqsstrid 3239	. . . . . . . 8 |- ( ph -> A C_ ZZ )
17	16	sselda 3193	. . . . . . 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 9496	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. ZZ )
20		peano2zm 9412	. . . . . . . . 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 3193	. . . . . . . . 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 9502	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
25		zdceq 9450	. . . . . . 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 2579	. . . . 5 |- ( ( ph /\ x e. A ) -> A. v e. A DECID x = ( ( P - 1 ) - v ) )
28		finexdc 7001	. . . . 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 4928	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) ) )
32	31	elv 2776	. . . . 5 |- ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) )
33	32	dcbii 842	. . . 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 2579	. 2 |- ( ph -> A. x e. A DECID x e. ran F )
36		infidc 7038	. 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 836   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   _Vcvv 2772   i^i cin 3165   |-> cmpt 4106   ran crn 4677  (class class class)co 5946   Fincfn 6829   0cc0 7927   1c1 7928   - cmin 8245   NNcn 9038   2c2 9089   ZZcz 9374   ...cfz 10132   mod cmo 10469   ^cexp 10685

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-er 6622  df-en 6830  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686

This theorem is referenced by:  4sqlem11  12757  4sqlem12  12758