Theorem lgsfcl2 14885

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 14878). (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 9295	. . . . . . . . 9 |- 0 e. ZZ
2		0le1 8469	. . . . . . . . 9 |- 0 <_ 1
3		fveq2 5534	. . . . . . . . . . . 12 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 11102	. . . . . . . . . . . 12 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2238	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4028	. . . . . . . . . 10 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . . 10 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 2911	. . . . . . . . 9 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 944	. . . . . . . 8 |- 0 e. Z
10	9	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 0 e. Z )
11		1z 9310	. . . . . . . . . 10 |- 1 e. ZZ
12		1le1 8560	. . . . . . . . . 10 |- 1 <_ 1
13		fveq2 5534	. . . . . . . . . . . . 13 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
14		abs1 11116	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
15	13, 14	eqtrdi 2238	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = 1 )
16	15	breq1d 4028	. . . . . . . . . . 11 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
17	16, 7	elrab2 2911	. . . . . . . . . 10 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
18	11, 12, 17	mpbir2an 944	. . . . . . . . 9 |- 1 e. Z
19	18	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. Z )
20		neg1z 9316	. . . . . . . . . 10 |- -u 1 e. ZZ
21		fveq2 5534	. . . . . . . . . . . . 13 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
22		ax-1cn 7935	. . . . . . . . . . . . . . 15 |- 1 e. CC
23	22	absnegi 11191	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
24	23, 14	eqtri 2210	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
25	21, 24	eqtrdi 2238	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = 1 )
26	25	breq1d 4028	. . . . . . . . . . 11 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
27	26, 7	elrab2 2911	. . . . . . . . . 10 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
28	20, 12, 27	mpbir2an 944	. . . . . . . . 9 |- -u 1 e. Z
29	28	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. Z )
30		simp1 999	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
31		8nn 9117	. . . . . . . . . . . . . 14 |- 8 e. NN
32	31	a1i 9	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 8 e. NN )
33	30, 32	zmodcld 10378	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. NN0 )
34	33	nn0zd 9404	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. ZZ )
35		zdceq 9359	. . . . . . . . . . 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 9116	. . . . . . . . . . . 12 |- 7 e. NN
38	37	nnzi 9305	. . . . . . . . . . 11 |- 7 e. ZZ
39		zdceq 9359	. . . . . . . . . . 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 937	. . . . . . . . . 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 3627	. . . . . . . . . . 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 839	. . . . . . . . 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 3585	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z )
48		2nn 9111	. . . . . . . . 9 |- 2 e. NN
49	48	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 2 e. NN )
50		dvdsdc 11840	. . . . . . . 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 3585	. . . . . 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 1002	. . . . . . 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 528	. . . . . . 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 2367	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
59		eldifsn 3734	. . . . . . 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 14882	. . . . . 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 528	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. NN )
64	63	nnzd 9405	. . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. ZZ )
65		2z 9312	. . . . . 6 |- 2 e. ZZ
66		zdceq 9359	. . . . . 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 3578	. . . 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 1039	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
71		simpll3 1040	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
72		pczcl 12333	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
73	69, 70, 71, 72	syl12anc 1247	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
74	7	ssrab3 3256	. . . . . 6 |- Z C_ ZZ
75		zsscn 9292	. . . . . 6 |- ZZ C_ CC
76	74, 75	sstri 3179	. . . . 5 |- Z C_ CC
77	7	lgslem3 14881	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
78	76, 77, 18	expcllem 10565	. . . 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 12165	. . . 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 3578	. 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 5691	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2160   =/= wne 2360   {crab 2472   \ cdif 3141   ifcif 3549   {csn 3607   {cpr 3608   class class class wbr 4018   |-> cmpt 4079   -->wf 5231   ` cfv 5235  (class class class)co 5897   CCcc 7840   0cc0 7842   1c1 7843   + caddc 7845   <_ cle 8024   - cmin 8159   -ucneg 8160   / cdiv 8660   NNcn 8950   2c2 9001   7c7 9006   8c8 9007   NN0cn0 9207   ZZcz 9284   mod cmo 10355   ^cexp 10553   abscabs 11041   || cdvds 11829   Primecprime 12142   pCnt cpc 12319

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-2o 6443  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-5 9012  df-6 9013  df-7 9014  df-8 9015  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322  df-proddc 11594  df-dvds 11830  df-gcd 11979  df-prm 12143  df-phi 12246  df-pc 12320

This theorem is referenced by:  lgscllem  14886  lgsfcl  14887  lgsfle1  14888