Theorem lgsfcl2 15734

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 15727). (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 9489	. . . . . . . . 9 |- 0 e. ZZ
2		0le1 8660	. . . . . . . . 9 |- 0 <_ 1
3		fveq2 5639	. . . . . . . . . . . 12 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 11618	. . . . . . . . . . . 12 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2280	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4098	. . . . . . . . . 10 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . . 10 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 2965	. . . . . . . . 9 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 950	. . . . . . . 8 |- 0 e. Z
10	9	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 0 e. Z )
11		1z 9504	. . . . . . . . . 10 |- 1 e. ZZ
12		1le1 8751	. . . . . . . . . 10 |- 1 <_ 1
13		fveq2 5639	. . . . . . . . . . . . 13 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
14		abs1 11632	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
15	13, 14	eqtrdi 2280	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = 1 )
16	15	breq1d 4098	. . . . . . . . . . 11 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
17	16, 7	elrab2 2965	. . . . . . . . . 10 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
18	11, 12, 17	mpbir2an 950	. . . . . . . . 9 |- 1 e. Z
19	18	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. Z )
20		neg1z 9510	. . . . . . . . . 10 |- -u 1 e. ZZ
21		fveq2 5639	. . . . . . . . . . . . 13 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
22		ax-1cn 8124	. . . . . . . . . . . . . . 15 |- 1 e. CC
23	22	absnegi 11707	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
24	23, 14	eqtri 2252	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
25	21, 24	eqtrdi 2280	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = 1 )
26	25	breq1d 4098	. . . . . . . . . . 11 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
27	26, 7	elrab2 2965	. . . . . . . . . 10 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
28	20, 12, 27	mpbir2an 950	. . . . . . . . 9 |- -u 1 e. Z
29	28	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. Z )
30		simp1 1023	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
31		8nn 9310	. . . . . . . . . . . . . 14 |- 8 e. NN
32	31	a1i 9	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 8 e. NN )
33	30, 32	zmodcld 10606	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. NN0 )
34	33	nn0zd 9599	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. ZZ )
35		zdceq 9554	. . . . . . . . . . 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 9309	. . . . . . . . . . . 12 |- 7 e. NN
38	37	nnzi 9499	. . . . . . . . . . 11 |- 7 e. ZZ
39		zdceq 9554	. . . . . . . . . . 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 943	. . . . . . . . . 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 3689	. . . . . . . . . . 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 845	. . . . . . . . 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 3643	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z )
48		2nn 9304	. . . . . . . . 9 |- 2 e. NN
49	48	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 2 e. NN )
50		dvdsdc 12358	. . . . . . . 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 3643	. . . . . 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 1026	. . . . . . 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 2409	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
59		eldifsn 3800	. . . . . . 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 15731	. . . . . 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 9600	. . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. ZZ )
65		2z 9506	. . . . . 6 |- 2 e. ZZ
66		zdceq 9554	. . . . . 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 3635	. . . 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 1063	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
71		simpll3 1064	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
72		pczcl 12870	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
73	69, 70, 71, 72	syl12anc 1271	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
74	7	ssrab3 3313	. . . . . 6 |- Z C_ ZZ
75		zsscn 9486	. . . . . 6 |- ZZ C_ CC
76	74, 75	sstri 3236	. . . . 5 |- Z C_ CC
77	7	lgslem3 15730	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
78	76, 77, 18	expcllem 10811	. . . 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 12701	. . . 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 3635	. 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 5801	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   {crab 2514   \ cdif 3197   ifcif 3605   {csn 3669   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   -->wf 5322   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   <_ cle 8214   - cmin 8349   -ucneg 8350   / cdiv 8851   NNcn 9142   2c2 9193   7c7 9198   8c8 9199   NN0cn0 9401   ZZcz 9478   mod cmo 10583   ^cexp 10799   abscabs 11557   || cdvds 12347   Primecprime 12678   pCnt cpc 12856

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-2o 6582  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111  df-dvds 12348  df-gcd 12524  df-prm 12679  df-phi 12782  df-pc 12857

This theorem is referenced by:  lgscllem  15735  lgsfcl  15736  lgsfle1  15737