Theorem lgslem4 24227

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 760	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> A e. ZZ )
2		oddprm 14765	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
3	2	ad2antlr 733	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
4	3	nnnn0d 10925	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
5		zexpcl 12287	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
6	1, 4, 5	syl2anc 667	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
7	6	zred 11040	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
8		0red 9644	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. RR )
9		1red 9658	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 e. RR )
10		eldifi 3555	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11	10	ad2antlr 733	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. Prime )
12		prmuz2 14642	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. ( ZZ>= ` 2 ) )
14		eluz2b2 11231	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
15	13, 14	sylib 200	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P e. NN /\ 1 < P ) )
16	15	simpld 461	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. NN )
17	16	nnrpd 11339	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR+ )
18		0zd 10949	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. ZZ )
19		simpr 463	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P || A )
20		dvdsval3 14309	. . . . . . . . . . . 12 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
21	16, 1, 20	syl2anc 667	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P || A <-> ( A mod P ) = 0 ) )
22	19, 21	mpbid 214	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = 0 )
23		0mod 12128	. . . . . . . . . . 11 |- ( P e. RR+ -> ( 0 mod P ) = 0 )
24	17, 23	syl 17	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 mod P ) = 0 )
25	22, 24	eqtr4d 2488	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = ( 0 mod P ) )
26		modexp 12407	. . . . . . . . 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 1279	. . . . . . . 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 12432	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 ^ ( ( P - 1 ) / 2 ) ) = 0 )
29	28	oveq1d 6305	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
30	27, 29	eqtrd 2485	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
31		modadd1 12134	. . . . . . 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 1279	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
33		0p1e1 10721	. . . . . . 7 |- ( 0 + 1 ) = 1
34	33	oveq1i 6300	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
35	32, 34	syl6eq 2501	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 1 mod P ) )
36	16	nnred 10624	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR )
37	15	simprd 465	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
38		1mod 12129	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
39	36, 37, 38	syl2anc 667	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 1 mod P ) = 1 )
40	35, 39	eqtrd 2485	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
41	40	oveq1d 6305	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
42		1m1e0 10678	. . . 4 |- ( 1 - 1 ) = 0
43		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
44	43	lgslem2 24225	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
45	44	simp2i 1018	. . . 4 |- 0 e. Z
46	42, 45	eqeltri 2525	. . 3 |- ( 1 - 1 ) e. Z
47	41, 46	syl6eqel 2537	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
48		lgslem1 24224	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
49		elpri 3985	. . . 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 6297	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
51		df-neg 9863	. . . . . . 7 |- -u 1 = ( 0 - 1 )
52	44	simp1i 1017	. . . . . . 7 |- -u 1 e. Z
53	51, 52	eqeltrri 2526	. . . . . 6 |- ( 0 - 1 ) e. Z
54	50, 53	syl6eqel 2537	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
55		oveq1 6297	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
56		2m1e1 10724	. . . . . . 7 |- ( 2 - 1 ) = 1
57	44	simp3i 1019	. . . . . . 7 |- 1 e. Z
58	56, 57	eqeltri 2525	. . . . . 6 |- ( 2 - 1 ) e. Z
59	55, 58	syl6eqel 2537	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
60	54, 59	jaoi 381	. . . 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 18	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
62	61	3expa 1208	. 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 800	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 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   {crab 2741   \ cdif 3401   {csn 3968   {cpr 3970   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   -ucneg 9861   / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   mod cmo 12096   ^cexp 12272   abscabs 13297   || cdvds 14305   Primecprime 14622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-gcd 14469  df-prm 14623  df-phi 14714

This theorem is referenced by:  lgsfcl2  24230