Theorem lgsfcl2 24228

Description: The function F is closed in integers with absolute value less than 1 (namely { -u 1 , 0 , 1 } although this representation is less useful to us). (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 10955	. . . . . . . 8 |- 0 e. ZZ
2		0le1 10144	. . . . . . . 8 |- 0 <_ 1
3		fveq2 5881	. . . . . . . . . . 11 |- ( x = 0 -> ( abs ` x ) = ( abs ` 0 ) )
4		abs0 13348	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
5	3, 4	syl6eq 2479	. . . . . . . . . 10 |- ( x = 0 -> ( abs ` x ) = 0 )
6	5	breq1d 4433	. . . . . . . . 9 |- ( x = 0 -> ( ( abs ` x ) <_ 1 <-> 0 <_ 1 ) )
7		lgsfcl2.z	. . . . . . . . 9 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
8	6, 7	elrab2 3230	. . . . . . . 8 |- ( 0 e. Z <-> ( 0 e. ZZ /\ 0 <_ 1 ) )
9	1, 2, 8	mpbir2an 928	. . . . . . 7 |- 0 e. Z
10		1z 10974	. . . . . . . . 9 |- 1 e. ZZ
11		1le1 10247	. . . . . . . . 9 |- 1 <_ 1
12		fveq2 5881	. . . . . . . . . . . 12 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
13		abs1 13360	. . . . . . . . . . . 12 |- ( abs ` 1 ) = 1
14	12, 13	syl6eq 2479	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = 1 )
15	14	breq1d 4433	. . . . . . . . . 10 |- ( x = 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
16	15, 7	elrab2 3230	. . . . . . . . 9 |- ( 1 e. Z <-> ( 1 e. ZZ /\ 1 <_ 1 ) )
17	10, 11, 16	mpbir2an 928	. . . . . . . 8 |- 1 e. Z
18		neg1z 10980	. . . . . . . . 9 |- -u 1 e. ZZ
19		fveq2 5881	. . . . . . . . . . . 12 |- ( x = -u 1 -> ( abs ` x ) = ( abs ` -u 1 ) )
20		ax-1cn 9604	. . . . . . . . . . . . . 14 |- 1 e. CC
21	20	absnegi 13462	. . . . . . . . . . . . 13 |- ( abs ` -u 1 ) = ( abs ` 1 )
22	21, 13	eqtri 2451	. . . . . . . . . . . 12 |- ( abs ` -u 1 ) = 1
23	19, 22	syl6eq 2479	. . . . . . . . . . 11 |- ( x = -u 1 -> ( abs ` x ) = 1 )
24	23	breq1d 4433	. . . . . . . . . 10 |- ( x = -u 1 -> ( ( abs ` x ) <_ 1 <-> 1 <_ 1 ) )
25	24, 7	elrab2 3230	. . . . . . . . 9 |- ( -u 1 e. Z <-> ( -u 1 e. ZZ /\ 1 <_ 1 ) )
26	18, 11, 25	mpbir2an 928	. . . . . . . 8 |- -u 1 e. Z
27	17, 26	keepel 3978	. . . . . . 7 |- if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. Z
28	9, 27	keepel 3978	. . . . . 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 1008	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> A e. ZZ )
31	30	ad2antrr 730	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> A e. ZZ )
32		simplr 760	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. Prime )
33		simpr 462	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> -. n = 2 )
34	33	neqned 2623	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n =/= 2 )
35		eldifsn 4125	. . . . . . 7 |- ( n e. ( Prime \ { 2 } ) <-> ( n e. Prime /\ n =/= 2 ) )
36	32, 34, 35	sylanbrc 668	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) /\ -. n = 2 ) -> n e. ( Prime \ { 2 } ) )
37	7	lgslem4 24225	. . . . . 6 |- ( ( A e. ZZ /\ n e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) e. Z )
38	31, 36, 37	syl2anc 665	. . . . 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 3943	. . . 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 462	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> n e. Prime )
41		simpll2 1045	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N e. ZZ )
42		simpll3 1046	. . . . 5 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> N =/= 0 )
43		pczcl 14797	. . . . 5 |- ( ( n e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( n pCnt N ) e. NN0 )
44	40, 41, 42, 43	syl12anc 1262	. . . 4 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
45		ssrab2 3546	. . . . . . 7 |- { x e. ZZ | ( abs ` x ) <_ 1 } C_ ZZ
46	7, 45	eqsstri 3494	. . . . . 6 |- Z C_ ZZ
47		zsscn 10952	. . . . . 6 |- ZZ C_ CC
48	46, 47	sstri 3473	. . . . 5 |- Z C_ CC
49	7	lgslem3 24224	. . . . 5 |- ( ( a e. Z /\ b e. Z ) -> ( a x. b ) e. Z )
50	48, 49, 17	expcllem 12289	. . . 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 665	. . 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 3943	. 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 6061	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F : NN --> Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2614   {crab 2775   \ cdif 3433   ifcif 3911   {csn 3998   {cpr 4000   class class class wbr 4423   |-> cmpt 4482   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9544   0cc0 9546   1c1 9547   + caddc 9549   <_ cle 9683   - cmin 9867   -ucneg 9868   / cdiv 10276   NNcn 10616   2c2 10666   7c7 10671   8c8 10672   NN0cn0 10876   ZZcz 10944   mod cmo 12102   ^cexp 12278   abscabs 13297   || cdvds 14304   Primecprime 14621   pCnt cpc 14785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-fz 11792  df-fzo 11923  df-fl 12034  df-mod 12103  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14305  df-gcd 14468  df-prm 14622  df-phi 14713  df-pc 14786

This theorem is referenced by:  lgscllem  24229  lgsfcl  24230  lgsfle1  24231