Theorem 4sqleminfi 13154

Description: Lemma for 4sq 13167. 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 13152	. 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 10407	. . . . . . . . . . . . . . . 16 |- ( m e. ( 0 ... N ) -> m e. ZZ )
7	6	ad2antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> m e. ZZ )
8		zsqcl 11025	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> P e. NN )
11	9, 10	zmodcld 10760	. . . . . . . . . . . . 13 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. NN0 )
12	11	nn0zd 9745	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> ( ( m ^ 2 ) mod P ) e. ZZ )
13	5, 12	eqeltrd 2315	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... N ) ) /\ u = ( ( m ^ 2 ) mod P ) ) -> u e. ZZ )
14	13	rexlimdva2 2671	. . . . . . . . . 10 |- ( ph -> ( E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) -> u e. ZZ ) )
15	14	abssdv 3322	. . . . . . . . 9 |- ( ph -> { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) } C_ ZZ )
16	3, 15	eqsstrid 3294	. . . . . . . 8 |- ( ph -> A C_ ZZ )
17	16	sselda 3248	. . . . . . 7 |- ( ( ph /\ x e. A ) -> x e. ZZ )
18	2	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. NN )
19	18	nnzd 9746	. . . . . . . . 9 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> P e. ZZ )
20		peano2zm 9661	. . . . . . . . 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 3248	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> v e. ZZ )
23	22	adantlr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> v e. ZZ )
24	21, 23	zsubcld 9752	. . . . . . 7 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> ( ( P - 1 ) - v ) e. ZZ )
25		zdceq 9699	. . . . . . 7 |- ( ( x e. ZZ /\ ( ( P - 1 ) - v ) e. ZZ ) -> DECID x = ( ( P - 1 ) - v ) )
26	17, 24, 25	syl2an2r 603	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ v e. A ) -> DECID x = ( ( P - 1 ) - v ) )
27	26	ralrimiva 2623	. . . . 5 |- ( ( ph /\ x e. A ) -> A. v e. A DECID x = ( ( P - 1 ) - v ) )
28		finexdc 7197	. . . . 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 603	. . . 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 5026	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) ) )
32	31	elv 2825	. . . . 5 |- ( x e. ran F <-> E. v e. A x = ( ( P - 1 ) - v ) )
33	32	dcbii 852	. . . 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 2623	. 2 |- ( ph -> A. x e. A DECID x e. ran F )
36		infidc 7238	. 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 415	1 |- ( ph -> ( A i^i ran F ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   i^i cin 3219   |-> cmpt 4187   ran crn 4770  (class class class)co 6075   Fincfn 7012   0cc0 8169   1c1 8170   - cmin 8487   NNcn 9283   2c2 9334   ZZcz 9623   ...cfz 10390   mod cmo 10737   ^cexp 10953

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954

This theorem is referenced by:  4sqlem11  13158  4sqlem12  13159