Theorem 4sqlemffi 12934

Description: Lemma for 4sq 12948. 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
4sqlemffi	|- ( ph -> ran F e. Fin )

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

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

Proof of Theorem 4sqlemffi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlemffi.f	. . . 4 |- F = ( v e. A |-> ( ( P - 1 ) - v ) )
2	1	funmpt2 5357	. . 3 |- Fun F
3		funrel 5335	. . 3 |- ( Fun F -> Rel F )
4	2, 3	ax-mp 5	. 2 |- Rel F
5		4sqlemafi.p	. . . . . . . . . 10 |- ( ph -> P e. NN )
6	5	nnzd 9579	. . . . . . . . 9 |- ( ph -> P e. ZZ )
7		peano2zm 9495	. . . . . . . . 9 |- ( P e. ZZ -> ( P - 1 ) e. ZZ )
8	6, 7	syl 14	. . . . . . . 8 |- ( ph -> ( P - 1 ) e. ZZ )
9	8	adantr 276	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( P - 1 ) e. ZZ )
10		4sqlemafi.a	. . . . . . . . 9 |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }
11		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u = ( ( m ^ 2 ) mod P ) )
12		elfzelz 10233	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 0 ... N ) -> m e. ZZ )
13	12	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. ( 0 ... N ) ) -> m e. ZZ )
14		zsqcl 10844	. . . . . . . . . . . . . . . 16 |- ( m e. ZZ -> ( m ^ 2 ) e. ZZ )
15	13, 14	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. ( 0 ... N ) ) -> ( m ^ 2 ) e. ZZ )
16	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. ( 0 ... N ) ) -> P e. NN )
17	15, 16	zmodcld 10579	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
18	17	nn0zd 9578	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
19	18	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
20	11, 19	eqeltrd 2306	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
21	20	rexlimdva2 2651	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
22	21	abssdv 3298	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
23	10, 22	eqsstrid 3270	. . . . . . . 8 |- ( ph -> A C_ ZZ )
24	23	sselda 3224	. . . . . . 7 |- ( ( ph /\ v e. A ) -> v e. ZZ )
25	9, 24	zsubcld 9585	. . . . . 6 |- ( ( ph /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
26	25	ralrimiva 2603	. . . . 5 |- ( ph -> A. v e. A ( ( P - 1 ) - v ) e. ZZ )
27	8	zcnd 9581	. . . . . . . . 9 |- ( ph -> ( P - 1 ) e. CC )
28	27	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> ( P - 1 ) e. CC )
29	24	adantrr 479	. . . . . . . . . 10 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> v e. ZZ )
30	29	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> v e. ZZ )
31	30	zcnd 9581	. . . . . . . 8 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> v e. CC )
32	23	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> A C_ ZZ )
33		simprr 531	. . . . . . . . . . 11 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> x e. A )
34	32, 33	sseldd 3225	. . . . . . . . . 10 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> x e. ZZ )
35	34	zcnd 9581	. . . . . . . . 9 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> x e. CC )
36	35	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> x e. CC )
37		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) )
38	28, 31, 36, 37	subcand 8509	. . . . . . 7 |- ( ( ( ph /\ ( v e. A /\ x e. A ) ) /\ ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) ) -> v = x )
39	38	ex 115	. . . . . 6 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> ( ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) -> v = x ) )
40	39	ralrimivva 2612	. . . . 5 |- ( ph -> A. v e. A A. x e. A ( ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) -> v = x ) )
41		oveq2 6015	. . . . . 6 |- ( v = x -> ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) )
42	1, 41	f1mpt 5901	. . . . 5 |- ( F : A -1-1-> ZZ <-> ( A. v e. A ( ( P - 1 ) - v ) e. ZZ /\ A. v e. A A. x e. A ( ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) -> v = x ) ) )
43	26, 40, 42	sylanbrc 417	. . . 4 |- ( ph -> F : A -1-1-> ZZ )
44		df-f1 5323	. . . 4 |- ( F : A -1-1-> ZZ <-> ( F : A --> ZZ /\ Fun `' F ) )
45	43, 44	sylib 122	. . 3 |- ( ph -> ( F : A --> ZZ /\ Fun `' F ) )
46	45	simprd 114	. 2 |- ( ph -> Fun `' F )
47	1, 25	dmmptd 5454	. . . 4 |- ( ph -> dom F = A )
48		4sqlemafi.n	. . . . 5 |- ( ph -> N e. NN )
49	48, 5, 10	4sqlemafi 12933	. . . 4 |- ( ph -> A e. Fin )
50	47, 49	eqeltrd 2306	. . 3 |- ( ph -> dom F e. Fin )
51		fundmfibi 7116	. . . 4 |- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) )
52	2, 51	ax-mp 5	. . 3 |- ( F e. Fin <-> dom F e. Fin )
53	50, 52	sylibr 134	. 2 |- ( ph -> F e. Fin )
54		funrnfi 7120	. 2 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
55	4, 46, 53, 54	mp3an2i 1376	1 |- ( ph -> ran F e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   C_ wss 3197   |-> cmpt 4145   `'ccnv 4718   dom cdm 4719   ran crn 4720   Rel wrel 4724   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315  (class class class)co 6007   Fincfn 6895   CCcc 8008   0cc0 8010   1c1 8011   - cmin 8328   NNcn 9121   2c2 9172   ZZcz 9457   ...cfz 10216   mod cmo 10556   ^cexp 10772

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773

This theorem is referenced by:  4sqlem11  12939