Theorem 4sqlemafi 12833

Description: Lemma for 4sq 12848. A 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 ) }

Assertion

Ref	Expression
4sqlemafi	|- ( ph -> A e. Fin )

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

Allowed substitution hints:   A( u, m)

Proof of Theorem 4sqlemafi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlemafi.a	. 2 |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }
2		0zd 9419	. . . 4 |- ( ph -> 0 e. ZZ )
3		4sqlemafi.p	. . . . 5 |- ( ph -> P e. NN )
4	3	nnzd 9529	. . . 4 |- ( ph -> P e. ZZ )
5		fzofig 10614	. . . 4 |- ( ( 0 e. ZZ /\ P e. ZZ ) -> ( 0..^ P ) e. Fin )
6	2, 4, 5	syl2anc 411	. . 3 |- ( ph -> ( 0..^ P ) e. Fin )
7		df-rex 2492	. . . . 5 |- ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) <-> E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) )
8	7	abbii 2323	. . . 4 |- { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } = { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) }
9		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> u = ( ( m ^ 2 ) mod P ) )
10		elfzelz 10182	. . . . . . . . . . 11 |- ( m e. ( 0 ... N ) -> m e. ZZ )
11	10	ad2antrl 490	. . . . . . . . . 10 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> m e. ZZ )
12		zsqcl 10792	. . . . . . . . . 10 |- ( m e. ZZ -> ( m ^ 2 ) e. ZZ )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> ( m ^ 2 ) e. ZZ )
14	3	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> P e. NN )
15		zmodfzo 10529	. . . . . . . . 9 |- ( ( ( m ^ 2 ) e. ZZ /\ P e. NN ) -> ( ( m ^ 2 ) mod P ) e. ( 0..^ P ) )
16	13, 14, 15	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> ( ( m ^ 2 ) mod P ) e. ( 0..^ P ) )
17	9, 16	eqeltrd 2284	. . . . . . 7 |- ( ( ph /\ ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) ) -> u e. ( 0..^ P ) )
18	17	ex 115	. . . . . 6 |- ( ph -> ( ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ( 0..^ P ) ) )
19	18	exlimdv 1843	. . . . 5 |- ( ph -> ( E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ( 0..^ P ) ) )
20	19	abssdv 3275	. . . 4 |- ( ph -> { u | E. m ( m e. ( 0 ... N ) /\ u = ( ( m ^ 2 ) mod P ) ) } C_ ( 0..^ P ) )
21	8, 20	eqsstrid 3247	. . 3 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) )
22		0zd 9419	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> 0 e. ZZ )
23		4sqlemafi.n	. . . . . . . 8 |- ( ph -> N e. NN )
24	23	nnzd 9529	. . . . . . 7 |- ( ph -> N e. ZZ )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ P ) ) -> N e. ZZ )
26		elfzoelz 10304	. . . . . . . 8 |- ( x e. ( 0..^ P ) -> x e. ZZ )
27	26	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> x e. ZZ )
28	10	adantl 277	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> m e. ZZ )
29	28, 12	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( m ^ 2 ) e. ZZ )
30	3	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> P e. NN )
31	29, 30	zmodcld 10527	. . . . . . . 8 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
32	31	nn0zd 9528	. . . . . . 7 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
33		zdceq 9483	. . . . . . 7 |- ( ( x e. ZZ /\ ( ( m ^ 2 ) mod P ) e. ZZ ) -> DECID x = ( ( m ^ 2 ) mod P ) )
34	27, 32, 33	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ x e. ( 0..^ P ) ) /\ m e. ( 0 ... N ) ) -> DECID x = ( ( m ^ 2 ) mod P ) )
35	22, 25, 34	exfzdc 10406	. . . . 5 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
36		vex 2779	. . . . . . 7 |- x e. _V
37		eqeq1 2214	. . . . . . . 8 |- ( u = x -> ( u = ( ( m ^ 2 ) mod P ) <-> x = ( ( m ^ 2 ) mod P ) ) )
38	37	rexbidv 2509	. . . . . . 7 |- ( u = x -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) <-> E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) ) )
39	36, 38	elab 2924	. . . . . 6 |- ( x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } <-> E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
40	39	dcbii 842	. . . . 5 |- (DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } <-> DECID E. m e. ( 0 ... N ) x = ( ( m ^ 2 ) mod P ) )
41	35, 40	sylibr 134	. . . 4 |- ( ( ph /\ x e. ( 0..^ P ) ) -> DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } )
42	41	ralrimiva 2581	. . 3 |- ( ph -> A. x e. ( 0..^ P )DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } )
43		ssfidc 7060	. . 3 |- ( ( ( 0..^ P ) e. Fin /\ { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ( 0..^ P ) /\ A. x e. ( 0..^ P )DECID x e. { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } ) -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } e. Fin )
44	6, 21, 42, 43	syl3anc 1250	. 2 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } e. Fin )
45	1, 44	eqeltrid 2294	1 |- ( ph -> A e. Fin )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 836   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   C_ wss 3174  (class class class)co 5967   Fincfn 6850   0cc0 7960   NNcn 9071   2c2 9122   ZZcz 9407   ...cfz 10165  ..^cfzo 10299   mod cmo 10504   ^cexp 10720

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  4sqlemffi  12834  4sqleminfi  12835  4sqlem11  12839