Theorem 4sqlemffi 13032

Description: Lemma for 4sq 13046. 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 5372	. . 3 |- Fun F
3		funrel 5350	. . 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 9645	. . . . . . . . 9 |- ( ph -> P e. ZZ )
7		peano2zm 9561	. . . . . . . . 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 10305	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 0 ... N ) -> m e. ZZ )
13	12	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. ( 0 ... N ) ) -> m e. ZZ )
14		zsqcl 10918	. . . . . . . . . . . . . . . 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 10653	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
18	17	nn0zd 9644	. . . . . . . . . . . . 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 2308	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
21	20	rexlimdva2 2654	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
22	21	abssdv 3302	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
23	10, 22	eqsstrid 3274	. . . . . . . 8 |- ( ph -> A C_ ZZ )
24	23	sselda 3228	. . . . . . 7 |- ( ( ph /\ v e. A ) -> v e. ZZ )
25	9, 24	zsubcld 9651	. . . . . 6 |- ( ( ph /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
26	25	ralrimiva 2606	. . . . 5 |- ( ph -> A. v e. A ( ( P - 1 ) - v ) e. ZZ )
27	8	zcnd 9647	. . . . . . . . 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 9647	. . . . . . . 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 533	. . . . . . . . . . 11 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> x e. A )
34	32, 33	sseldd 3229	. . . . . . . . . 10 |- ( ( ph /\ ( v e. A /\ x e. A ) ) -> x e. ZZ )
35	34	zcnd 9647	. . . . . . . . 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 8573	. . . . . . 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 2615	. . . . 5 |- ( ph -> A. v e. A A. x e. A ( ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) -> v = x ) )
41		oveq2 6036	. . . . . 6 |- ( v = x -> ( ( P - 1 ) - v ) = ( ( P - 1 ) - x ) )
42	1, 41	f1mpt 5922	. . . . 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 5338	. . . 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 5470	. . . 4 |- ( ph -> dom F = A )
48		4sqlemafi.n	. . . . 5 |- ( ph -> N e. NN )
49	48, 5, 10	4sqlemafi 13031	. . . 4 |- ( ph -> A e. Fin )
50	47, 49	eqeltrd 2308	. . 3 |- ( ph -> dom F e. Fin )
51		fundmfibi 7180	. . . 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 7184	. 2 |- ( ( Rel F /\ Fun `' F /\ F e. Fin ) -> ran F e. Fin )
55	4, 46, 53, 54	mp3an2i 1379	1 |- ( ph -> ran F e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   C_ wss 3201   |-> cmpt 4155   `'ccnv 4730   dom cdm 4731   ran crn 4732   Rel wrel 4736   Fun wfun 5327   -->wf 5329   -1-1->wf1 5330  (class class class)co 6028   Fincfn 6952   CCcc 8073   0cc0 8075   1c1 8076   - cmin 8392   NNcn 9185   2c2 9236   ZZcz 9523   ...cfz 10288   mod cmo 10630   ^cexp 10846

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847

This theorem is referenced by:  4sqlem11  13037