Theorem lgsfcl2 25028

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 }, see zabsle1 25021). (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:   x, n, A   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 11388	. . . . . . . 8 |- 0 e. ZZ
2		0le1 10551	. . . . . . . 8 |- 0 <_ 1
3		fveq2 6191	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 14025	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
5	3, 4	syl6eq 2672	. . . . . . . . . 10 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4663	. . . . . . . . 9 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . 9 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 3366	. . . . . . . 8 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 955	. . . . . . 7 |- 0 e. Z
10		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
11		1le1 10655	. . . . . . . . 9 |- 1 <_ 1
12		fveq2 6191	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
13		abs1 14037	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
14	12, 13	syl6eq 2672	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = 1 )
15	14	breq1d 4663	. . . . . . . . . 10 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
16	15, 7	elrab2 3366	. . . . . . . . 9 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
17	10, 11, 16	mpbir2an 955	. . . . . . . 8 |- 1 e. Z
18		neg1z 11413	. . . . . . . . 9 |- -u 1 e. ZZ
19		fveq2 6191	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
20		ax-1cn 9994	. . . . . . . . . . . . . 14 |- 1 e. CC
21	20	absnegi 14139	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = ( abs ` 1 )
22	21, 13	eqtri 2644	. . . . . . . . . . . 12 |- ( abs ` -u 1 ) = 1
23	19, 22	syl6eq 2672	. . . . . . . . . . 11 |- ( x = -u 1 -> ( abs ` x ) = 1 )
24	23	breq1d 4663	. . . . . . . . . 10 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
25	24, 7	elrab2 3366	. . . . . . . . 9 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
26	18, 11, 25	mpbir2an 955	. . . . . . . 8 |- -u 1 e. Z
27	17, 26	keepel 4155	. . . . . . 7 |- if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z
28	9, 27	keepel 4155	. . . . . 6 |- if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) e. Z
29	28	a1i 11	. . . . 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 )
30		simpl1 1064	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> A e. ZZ )
31	30	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> A e. ZZ )
32		simplr 792	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. Prime )
33		simpr 477	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> -. n = 2 )
34	33	neqned 2801	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
35		eldifsn 4317	. . . . . . 7 |- ( n e. ( Prime \ { 2 } ) <-> ( n e. Prime /\ n =/= 2 ) )
36	32, 34, 35	sylanbrc 698	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. ( Prime \ { 2 } ) )
37	7	lgslem4 25025	. . . . . 6 |- ( ( A e. ZZ /\ n e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
38	31, 36, 37	syl2anc 693	. . . . 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 )
39	29, 38	ifclda 4120	. . . 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 )
40		simpr 477	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. Prime )
41		simpll2 1101	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
42		simpll3 1102	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
43		pczcl 15553	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
44	40, 41, 42, 43	syl12anc 1324	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
45		ssrab2 3687	. . . . . . 7 |- { x e. ZZ | ( abs ` x ) <_ 1 } C_ ZZ
46	7, 45	eqsstri 3635	. . . . . 6 |- Z C_ ZZ
47		zsscn 11385	. . . . . 6 |- ZZ C_ CC
48	46, 47	sstri 3612	. . . . 5 |- Z C_ CC
49	7	lgslem3 25024	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
50	48, 49, 17	expcllem 12871	. . . 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 )
51	39, 44, 50	syl2anc 693	. . 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 )
52	17	a1i 11	. . 3 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ -. n e. Prime ) -> 1 e. Z )
53	51, 52	ifclda 4120	. 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 )
54		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 ) )
55	53, 54	fmptd 6385	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   ifcif 4086   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   7c7 11075   8c8 11076   NN0cn0 11292   ZZcz 11377   mod cmo 12668   ^cexp 12860   abscabs 13974   || cdvds 14983   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542

This theorem is referenced by:  lgscllem  25029  lgsfcl  25030  lgsfle1  25031