Theorem lgsfcl2 16039

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 16032). (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 9634	. . . . . . . . 9 |- 0 e. ZZ
2		0le1 8799	. . . . . . . . 9 |- 0 <_ 1
3		fveq2 5690	. . . . . . . . . . . 12 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 11802	. . . . . . . . . . . 12 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2287	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4135	. . . . . . . . . 10 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . . 10 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 2985	. . . . . . . . 9 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 955	. . . . . . . 8 |- 0 e. Z
10	9	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 0 e. Z )
11		1z 9649	. . . . . . . . . 10 |- 1 e. ZZ
12		1le1 8890	. . . . . . . . . 10 |- 1 <_ 1
13		fveq2 5690	. . . . . . . . . . . . 13 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
14		abs1 11816	. . . . . . . . . . . . 13 |- ( abs ` 1 ) = 1
15	13, 14	eqtrdi 2287	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = 1 )
16	15	breq1d 4135	. . . . . . . . . . 11 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
17	16, 7	elrab2 2985	. . . . . . . . . 10 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
18	11, 12, 17	mpbir2an 955	. . . . . . . . 9 |- 1 e. Z
19	18	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. Z )
20		neg1z 9655	. . . . . . . . . 10 |- -u 1 e. ZZ
21		fveq2 5690	. . . . . . . . . . . . 13 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
22		ax-1cn 8262	. . . . . . . . . . . . . . 15 |- 1 e. CC
23	22	absnegi 11891	. . . . . . . . . . . . . 14 |- ( abs ` -u 1 ) = ( abs ` 1 )
24	23, 14	eqtri 2259	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = 1
25	21, 24	eqtrdi 2287	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = 1 )
26	25	breq1d 4135	. . . . . . . . . . 11 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
27	26, 7	elrab2 2985	. . . . . . . . . 10 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
28	20, 12, 27	mpbir2an 955	. . . . . . . . 9 |- -u 1 e. Z
29	28	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. Z )
30		simp1 1028	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
31		8nn 9451	. . . . . . . . . . . . . 14 |- 8 e. NN
32	31	a1i 9	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 8 e. NN )
33	30, 32	zmodcld 10760	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. NN0 )
34	33	nn0zd 9745	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A mod 8 ) e. ZZ )
35		zdceq 9699	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
36	34, 11, 35	sylancl 417	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 1 )
37		7nn 9450	. . . . . . . . . . . 12 |- 7 e. NN
38	37	nnzi 9644	. . . . . . . . . . 11 |- 7 e. ZZ
39		zdceq 9699	. . . . . . . . . . 11 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
40	34, 38, 39	sylancl 417	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( A mod 8 ) = 7 )
41		dcor 948	. . . . . . . . . 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 3725	. . . . . . . . . . 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 850	. . . . . . . . 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 3675	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z )
48		2nn 9445	. . . . . . . . 9 |- 2 e. NN
49	48	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 2 e. NN )
50		dvdsdc 12543	. . . . . . . 8 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
51	49, 30, 50	syl2anc 415	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID 2 || A )
52	10, 47, 51	ifcldcd 3675	. . . . . 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 496	. . . . 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 1031	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> A e. ZZ )
55	54	ad2antrr 492	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> A e. ZZ )
56		simplr 533	. . . . . . 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 2427	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
59		eldifsn 3836	. . . . . . 7 |- ( n e. ( Prime \ { 2 } ) <-> ( n e. Prime /\ n =/= 2 ) )
60	56, 58, 59	sylanbrc 421	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. ( Prime \ { 2 } ) )
61	7	lgslem4 16036	. . . . . 6 |- ( ( A e. ZZ /\ n e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
62	55, 60, 61	syl2anc 415	. . . . 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 533	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. NN )
64	63	nnzd 9746	. . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. ZZ )
65		2z 9651	. . . . . 6 |- 2 e. ZZ
66		zdceq 9699	. . . . . 6 |- ( ( n e. ZZ /\ 2 e. ZZ ) -> DECID n = 2 )
67	64, 65, 66	sylancl 417	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> DECID n = 2 )
68	53, 62, 67	ifcldadc 3667	. . . 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 1068	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
71		simpll3 1069	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
72		pczcl 13055	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
73	69, 70, 71, 72	syl12anc 1276	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
74	7	ssrab3 3334	. . . . . 6 |- Z C_ ZZ
75		zsscn 9631	. . . . . 6 |- ZZ C_ CC
76	74, 75	sstri 3257	. . . . 5 |- Z C_ CC
77	7	lgslem3 16035	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
78	76, 77, 18	expcllem 10965	. . . 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 415	. . 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 12886	. . . 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 3667	. 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 5853	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   \ cdif 3217   ifcif 3635   {csn 3705   {cpr 3706   class class class wbr 4125   |-> cmpt 4187   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   <_ cle 8351   - cmin 8487   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   7c7 9339   8c8 9340   NN0cn0 9542   ZZcz 9623   mod cmo 10737   ^cexp 10953   abscabs 11741   || cdvds 12532   Primecprime 12863   pCnt cpc 13041

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  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  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-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  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-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  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  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296  df-dvds 12533  df-gcd 12709  df-prm 12864  df-phi 12967  df-pc 13042

This theorem is referenced by:  lgscllem  16040  lgsfcl  16041  lgsfle1  16042