Theorem lgsfcl2 16005

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 15998). (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
lgsval.1	|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
lgsfcl2.z	|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }

Assertion

Ref	Expression
lgsfcl2	|- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Distinct variable groups:   A, n, x   x, F   n, N, x   n, Z

Allowed substitution hints:   F( n)   Z( x)

Proof of Theorem lgsfcl2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9605	. . . . . . . . 9 |- 0 e. ZZ
2		0le1 8772	. . . . . . . . 9 |- 0 <_ 1
3		fveq2 5675	. . . . . . . . . . . 12 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 11768	. . . . . . . . . . . 12 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2283	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4124	. . . . . . . . . 10 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . . 10 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 2979	. . . . . . . . 9 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 951	. . . . . . . 8 |- 0 e. Z
10	9	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 0 e. Z )
11		1z 9620	. . . . . . . . . 10 |- 1 e. ZZ
12		1le1 8863	. . . . . . . . . 10 |- 1 <_ 1
13		fveq2 5675	. . . . . . . . . . . . 13 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
14		abs1 11782	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
15	13, 14	eqtrdi 2283	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = 1 )
16	15	breq1d 4124	. . . . . . . . . . 11 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
17	16, 7	elrab2 2979	. . . . . . . . . 10 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
18	11, 12, 17	mpbir2an 951	. . . . . . . . 9 |- 1 e. Z
19	18	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. Z )
20		neg1z 9626	. . . . . . . . . 10 |- -u 1 e. ZZ
21		fveq2 5675	. . . . . . . . . . . . 13 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
22		ax-1cn 8236	. . . . . . . . . . . . . . 15 |- 1 e. CC
23	22	absnegi 11857	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
24	23, 14	eqtri 2255	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
25	21, 24	eqtrdi 2283	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = 1 )
26	25	breq1d 4124	. . . . . . . . . . 11 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
27	26, 7	elrab2 2979	. . . . . . . . . 10 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
28	20, 12, 27	mpbir2an 951	. . . . . . . . 9 |- -u 1 e. Z
29	28	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. Z )
30		simp1 1024	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
31		8nn 9422	. . . . . . . . . . . . . 14 |- 8 e. NN
32	31	a1i 9	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 8 e. NN )
33	30, 32	zmodcld 10731	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. NN0 )
34	33	nn0zd 9716	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. ZZ )
35		zdceq 9670	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
36	34, 11, 35	sylancl 413	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 1 )
37		7nn 9421	. . . . . . . . . . . 12 |- 7 e. NN
38	37	nnzi 9615	. . . . . . . . . . 11 |- 7 e. ZZ
39		zdceq 9670	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
40	34, 38, 39	sylancl 413	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 7 )
41		dcor 944	. . . . . . . . . 10 |- (DECID ( A mod 8 ) = 1 -> (DECID ( A mod 8 ) = 7 -> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
42	36, 40, 41	sylc 62	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
43		elprg 3714	. . . . . . . . . . 11 |- ( ( A mod 8 ) e. NN0 -> ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
44	33, 43	syl 14	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
45	44	dcbid 846	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> (DECID ( A mod 8 ) e. { 1 , 7 } <-> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
46	42, 45	mpbird 167	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
47	19, 29, 46	ifcldcd 3664	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z )
48		2nn 9416	. . . . . . . . 9 |- 2 e. NN
49	48	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 2 e. NN )
50		dvdsdc 12509	. . . . . . . 8 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
51	49, 30, 50	syl2anc 411	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID 2 || A )
52	10, 47, 51	ifcldcd 3664	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) e. Z )
53	52	ad3antrrr 492	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ n = 2 ) -> if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) e. Z )
54		simpl1 1027	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> A e. ZZ )
55	54	ad2antrr 488	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> A e. ZZ )
56		simplr 529	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. Prime )
57		simpr 110	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> -. n = 2 )
58	57	neqned 2421	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
59		eldifsn 3825	. . . . . . 7 |- ( n e. ( Prime \ { 2 } ) <-> ( n e. Prime /\ n =/= 2 ) )
60	56, 58, 59	sylanbrc 417	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. ( Prime \ { 2 } ) )
61	7	lgslem4 16002	. . . . . 6 |- ( ( A e. ZZ /\ n e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
62	55, 60, 61	syl2anc 411	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
63		simplr 529	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. NN )
64	63	nnzd 9717	. . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. ZZ )
65		2z 9622	. . . . . 6 |- 2 e. ZZ
66		zdceq 9670	. . . . . 6 |- ( ( n e. ZZ /\ 2 e. ZZ ) -> DECID n = 2 )
67	64, 65, 66	sylancl 413	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> DECID n = 2 )
68	53, 62, 67	ifcldadc 3656	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) e. Z )
69		simpr 110	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. Prime )
70		simpll2 1064	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
71		simpll3 1065	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
72		pczcl 13021	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
73	69, 70, 71, 72	syl12anc 1272	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
74	7	ssrab3 3328	. . . . . 6 |- Z C_ ZZ
75		zsscn 9602	. . . . . 6 |- ZZ C_ CC
76	74, 75	sstri 3251	. . . . 5 |- Z C_ CC
77	7	lgslem3 16001	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
78	76, 77, 18	expcllem 10936	. . . 4 |- ( ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) e. Z /\ ( n pCnt N ) e. NN0 ) -> ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) e. Z )
79	68, 73, 78	syl2anc 411	. . 3 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) e. Z )
80	18	a1i 9	. . 3 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ -. n e. Prime ) -> 1 e. Z )
81		simpr 110	. . . 4 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> n e. NN )
82		prmdc 12852	. . . 4 |- ( n e. NN -> DECID n e. Prime )
83	81, 82	syl 14	. . 3 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> DECID n e. Prime )
84	79, 80, 83	ifcldadc 3656	. 2 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) e. Z )
85		lgsval.1	. 2 |- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
86	84, 85	fmptd 5836	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   \ cdif 3211   ifcif 3624   {csn 3694   {cpr 3695   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   <_ cle 8325   - cmin 8460   -ucneg 8461   / cdiv 8963   NNcn 9254   2c2 9305   7c7 9310   8c8 9311   NN0cn0 9513   ZZcz 9594   mod cmo 10708   ^cexp 10924   abscabs 11707   || cdvds 12498   Primecprime 12829   pCnt cpc 13007

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-phi 12933  df-pc 13008

This theorem is referenced by:  lgscllem  16006  lgsfcl  16007  lgsfle1  16008