Theorem lgsfcl2 13507

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 13500). (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 9198	. . . . . . . . 9 |- 0 e. ZZ
2		0le1 8375	. . . . . . . . 9 |- 0 <_ 1
3		fveq2 5485	. . . . . . . . . . . 12 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 10996	. . . . . . . . . . . 12 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2214	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 3991	. . . . . . . . . 10 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . . 10 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 2884	. . . . . . . . 9 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 932	. . . . . . . 8 |- 0 e. Z
10	9	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 0 e. Z )
11		1z 9213	. . . . . . . . . 10 |- 1 e. ZZ
12		1le1 8466	. . . . . . . . . 10 |- 1 <_ 1
13		fveq2 5485	. . . . . . . . . . . . 13 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
14		abs1 11010	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
15	13, 14	eqtrdi 2214	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = 1 )
16	15	breq1d 3991	. . . . . . . . . . 11 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
17	16, 7	elrab2 2884	. . . . . . . . . 10 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
18	11, 12, 17	mpbir2an 932	. . . . . . . . 9 |- 1 e. Z
19	18	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. Z )
20		neg1z 9219	. . . . . . . . . 10 |- -u 1 e. ZZ
21		fveq2 5485	. . . . . . . . . . . . 13 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
22		ax-1cn 7842	. . . . . . . . . . . . . . 15 |- 1 e. CC
23	22	absnegi 11085	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
24	23, 14	eqtri 2186	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
25	21, 24	eqtrdi 2214	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = 1 )
26	25	breq1d 3991	. . . . . . . . . . 11 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
27	26, 7	elrab2 2884	. . . . . . . . . 10 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
28	20, 12, 27	mpbir2an 932	. . . . . . . . 9 |- -u 1 e. Z
29	28	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. Z )
30		simp1 987	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
31		8nn 9020	. . . . . . . . . . . . . 14 |- 8 e. NN
32	31	a1i 9	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 8 e. NN )
33	30, 32	zmodcld 10276	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. NN0 )
34	33	nn0zd 9307	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. ZZ )
35		zdceq 9262	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
36	34, 11, 35	sylancl 410	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 1 )
37		7nn 9019	. . . . . . . . . . . 12 |- 7 e. NN
38	37	nnzi 9208	. . . . . . . . . . 11 |- 7 e. ZZ
39		zdceq 9262	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
40	34, 38, 39	sylancl 410	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 7 )
41		dcor 925	. . . . . . . . . 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 3595	. . . . . . . . . . 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 828	. . . . . . . . 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 166	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
47	19, 29, 46	ifcldcd 3554	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z )
48		2nn 9014	. . . . . . . . 9 |- 2 e. NN
49	48	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 2 e. NN )
50		dvdsdc 11734	. . . . . . . 8 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
51	49, 30, 50	syl2anc 409	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID 2 || A )
52	10, 47, 51	ifcldcd 3554	. . . . . 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 484	. . . . 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 990	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> A e. ZZ )
55	54	ad2antrr 480	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> A e. ZZ )
56		simplr 520	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. Prime )
57		simpr 109	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> -. n = 2 )
58	57	neqned 2342	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
59		eldifsn 3702	. . . . . . 7 |- ( n e. ( Prime \ { 2 } ) <-> ( n e. Prime /\ n =/= 2 ) )
60	56, 58, 59	sylanbrc 414	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. ( Prime \ { 2 } ) )
61	7	lgslem4 13504	. . . . . 6 |- ( ( A e. ZZ /\ n e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
62	55, 60, 61	syl2anc 409	. . . . 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 520	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. NN )
64	63	nnzd 9308	. . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. ZZ )
65		2z 9215	. . . . . 6 |- 2 e. ZZ
66		zdceq 9262	. . . . . 6 |- ( ( n e. ZZ /\ 2 e. ZZ ) -> DECID n = 2 )
67	64, 65, 66	sylancl 410	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> DECID n = 2 )
68	53, 62, 67	ifcldadc 3548	. . . 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 109	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. Prime )
70		simpll2 1027	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
71		simpll3 1028	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
72		pczcl 12226	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
73	69, 70, 71, 72	syl12anc 1226	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
74	7	ssrab3 3227	. . . . . 6 |- Z C_ ZZ
75		zsscn 9195	. . . . . 6 |- ZZ C_ CC
76	74, 75	sstri 3150	. . . . 5 |- Z C_ CC
77	7	lgslem3 13503	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
78	76, 77, 18	expcllem 10462	. . . 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 409	. . 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 109	. . . 4 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> n e. NN )
82		prmdc 12058	. . . 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 3548	. 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 5638	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   {crab 2447   \ cdif 3112   ifcif 3519   {csn 3575   {cpr 3576   class class class wbr 3981   |-> cmpt 4042   -->wf 5183   ` cfv 5187  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   + caddc 7752   <_ cle 7930   - cmin 8065   -ucneg 8066   / cdiv 8564   NNcn 8853   2c2 8904   7c7 8909   8c8 8910   NN0cn0 9110   ZZcz 9187   mod cmo 10253   ^cexp 10450   abscabs 10935   || cdvds 11723   Primecprime 12035   pCnt cpc 12212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-2o 6381  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-5 8915  df-6 8916  df-7 8917  df-8 8918  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-proddc 11488  df-dvds 11724  df-gcd 11872  df-prm 12036  df-phi 12139  df-pc 12213

This theorem is referenced by:  lgscllem  13508  lgsfcl  13509  lgsfle1  13510