Theorem lgslem4 23440

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.)

Hypothesis

Ref	Expression
lgslem2.z	|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }

Assertion

Ref	Expression
lgslem4	|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )

Distinct variable group:   x, A

Allowed substitution hints:   P( x)   Z( x)

Proof of Theorem lgslem4

Step	Hyp	Ref	Expression
1		simpll 753	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> A e. ZZ )
2		oddprm 14215	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
3	2	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
4	3	nnnn0d 10864	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
5		zexpcl 12161	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
6	1, 4, 5	syl2anc 661	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
7	6	zred 10978	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
8		0red 9609	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. RR )
9		1red 9623	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 e. RR )
10		eldifi 3631	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11	10	ad2antlr 726	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. Prime )
12		prmuz2 14111	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
13	11, 12	syl 16	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. ( ZZ>= ` 2 ) )
14		eluz2b2 11166	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
15	13, 14	sylib 196	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P e. NN /\ 1 < P ) )
16	15	simpld 459	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. NN )
17	16	nnrpd 11267	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR+ )
18		0zd 10888	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. ZZ )
19		simpr 461	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P || A )
20		dvdsval3 13868	. . . . . . . . . . . 12 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
21	16, 1, 20	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P || A <-> ( A mod P ) = 0 ) )
22	19, 21	mpbid 210	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = 0 )
23		0mod 12007	. . . . . . . . . . 11 |- ( P e. RR+ -> ( 0 mod P ) = 0 )
24	17, 23	syl 16	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 mod P ) = 0 )
25	22, 24	eqtr4d 2511	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = ( 0 mod P ) )
26		modexp 12281	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ 0 e. ZZ ) /\ ( ( ( P - 1 ) / 2 ) e. NN0 /\ P e. RR+ ) /\ ( A mod P ) = ( 0 mod P ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
27	1, 18, 4, 17, 25, 26	syl221anc 1239	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
28	3	0expd 12306	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 ^ ( ( P - 1 ) / 2 ) ) = 0 )
29	28	oveq1d 6310	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
30	27, 29	eqtrd 2508	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
31		modadd1 12013	. . . . . . 7 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) e. RR /\ 0 e. RR ) /\ ( 1 e. RR /\ P e. RR+ ) /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
32	7, 8, 9, 17, 30, 31	syl221anc 1239	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
33		0p1e1 10659	. . . . . . 7 |- ( 0 + 1 ) = 1
34	33	oveq1i 6305	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
35	32, 34	syl6eq 2524	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 1 mod P ) )
36	16	nnred 10563	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR )
37	15	simprd 463	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
38		1mod 12008	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
39	36, 37, 38	syl2anc 661	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 1 mod P ) = 1 )
40	35, 39	eqtrd 2508	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
41	40	oveq1d 6310	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
42		1m1e0 10616	. . . 4 |- ( 1 - 1 ) = 0
43		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
44	43	lgslem2 23438	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
45	44	simp2i 1006	. . . 4 |- 0 e. Z
46	42, 45	eqeltri 2551	. . 3 |- ( 1 - 1 ) e. Z
47	41, 46	syl6eqel 2563	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
48		lgslem1 23437	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
49		elpri 4053	. . . 4 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) )
50		oveq1 6302	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
51		df-neg 9820	. . . . . . 7 |- -u 1 = ( 0 - 1 )
52	44	simp1i 1005	. . . . . . 7 |- -u 1 e. Z
53	51, 52	eqeltrri 2552	. . . . . 6 |- ( 0 - 1 ) e. Z
54	50, 53	syl6eqel 2563	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
55		oveq1 6302	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
56		2m1e1 10662	. . . . . . 7 |- ( 2 - 1 ) = 1
57	44	simp3i 1007	. . . . . . 7 |- 1 e. Z
58	56, 57	eqeltri 2551	. . . . . 6 |- ( 2 - 1 ) e. Z
59	55, 58	syl6eqel 2563	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
60	54, 59	jaoi 379	. . . 4 |- ( ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
61	48, 49, 60	3syl 20	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
62	61	3expa 1196	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
63	47, 62	pm2.61dan 789	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   {crab 2821   \ cdif 3478   {csn 4033   {cpr 4035   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505   + caddc 9507   < clt 9640   <_ cle 9641   - cmin 9817   -ucneg 9818   / cdiv 10218   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   RR+crp 11232   mod cmo 11976   ^cexp 12146   abscabs 13047   || cdivides 13864   Primecprime 14093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172

This theorem is referenced by:  lgsfcl2  23443