Theorem lgslem4 22638

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 13882	. . . . . . . . . . 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 10636	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
5		zexpcl 11880	. . . . . . . . 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 10747	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
8		0red 9387	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. RR )
9		1red 9401	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 e. RR )
10		eldifi 3478	. . . . . . . . . . . 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 13781	. . . . . . . . . . 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 10927	. . . . . . . . . 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 11026	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR+ )
18		0zd 10658	. . . . . . . . 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 13539	. . . . . . . . . . . 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 11739	. . . . . . . . . . 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 2478	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = ( 0 mod P ) )
26		modexp 11999	. . . . . . . . 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 1229	. . . . . . . 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 12024	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 ^ ( ( P - 1 ) / 2 ) ) = 0 )
29	28	oveq1d 6106	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
30	27, 29	eqtrd 2475	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
31		modadd1 11745	. . . . . . 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 1229	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
33		0p1e1 10433	. . . . . . 7 |- ( 0 + 1 ) = 1
34	33	oveq1i 6101	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
35	32, 34	syl6eq 2491	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 1 mod P ) )
36	16	nnred 10337	. . . . . 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 11740	. . . . . 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 2475	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
41	40	oveq1d 6106	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
42		1m1e0 10390	. . . 4 |- ( 1 - 1 ) = 0
43		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
44	43	lgslem2 22636	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
45	44	simp2i 998	. . . 4 |- 0 e. Z
46	42, 45	eqeltri 2513	. . 3 |- ( 1 - 1 ) e. Z
47	41, 46	syl6eqel 2531	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
48		lgslem1 22635	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
49		elpri 3897	. . . 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 6098	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
51		df-neg 9598	. . . . . . 7 |- -u 1 = ( 0 - 1 )
52	44	simp1i 997	. . . . . . 7 |- -u 1 e. Z
53	51, 52	eqeltrri 2514	. . . . . 6 |- ( 0 - 1 ) e. Z
54	50, 53	syl6eqel 2531	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
55		oveq1 6098	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
56		2m1e1 10436	. . . . . . 7 |- ( 2 - 1 ) = 1
57	44	simp3i 999	. . . . . . 7 |- 1 e. Z
58	56, 57	eqeltri 2513	. . . . . 6 |- ( 2 - 1 ) e. Z
59	55, 58	syl6eqel 2531	. . . . 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 1187	. 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 965   = wceq 1369   e. wcel 1756   {crab 2719   \ cdif 3325   {csn 3877   {cpr 3879   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283   + caddc 9285   < clt 9418   <_ cle 9419   - cmin 9595   -ucneg 9596   / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   RR+crp 10991   mod cmo 11708   ^cexp 11865   abscabs 12723   || cdivides 13535   Primecprime 13763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-phi 13841

This theorem is referenced by:  lgsfcl2  22641