Theorem lgslem4 25025

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.)

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 790	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> A e. ZZ )
2		oddprm 15515	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
3	2	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
4	3	nnnn0d 11351	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
5		zexpcl 12875	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
6	1, 4, 5	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
7	6	zred 11482	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
8		0red 10041	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. RR )
9		1red 10055	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 e. RR )
10		eldifi 3732	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11	10	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. Prime )
12		prmuz2 15408	. . . . . . . . . . 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 11761	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
15	13, 14	sylib 208	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P e. NN /\ 1 < P ) )
16	15	simpld 475	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. NN )
17	16	nnrpd 11870	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR+ )
18		0zd 11389	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. ZZ )
19		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P || A )
20		dvdsval3 14987	. . . . . . . . . . . 12 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
21	16, 1, 20	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P || A <-> ( A mod P ) = 0 ) )
22	19, 21	mpbid 222	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = 0 )
23		0mod 12701	. . . . . . . . . . 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 2659	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = ( 0 mod P ) )
26		modexp 12999	. . . . . . . . 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 1337	. . . . . . . 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 13024	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 ^ ( ( P - 1 ) / 2 ) ) = 0 )
29	28	oveq1d 6665	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
30	27, 29	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
31		modadd1 12707	. . . . . . 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 1337	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
33		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
34	33	oveq1i 6660	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
35	32, 34	syl6eq 2672	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 1 mod P ) )
36	16	nnred 11035	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR )
37	15	simprd 479	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
38		1mod 12702	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
39	36, 37, 38	syl2anc 693	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 1 mod P ) = 1 )
40	35, 39	eqtrd 2656	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
41	40	oveq1d 6665	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
42		1m1e0 11089	. . . 4 |- ( 1 - 1 ) = 0
43		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
44	43	lgslem2 25023	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
45	44	simp2i 1071	. . . 4 |- 0 e. Z
46	42, 45	eqeltri 2697	. . 3 |- ( 1 - 1 ) e. Z
47	41, 46	syl6eqel 2709	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
48		lgslem1 25022	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
49		elpri 4197	. . . 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 6657	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
51		df-neg 10269	. . . . . . 7 |- -u 1 = ( 0 - 1 )
52	44	simp1i 1070	. . . . . . 7 |- -u 1 e. Z
53	51, 52	eqeltrri 2698	. . . . . 6 |- ( 0 - 1 ) e. Z
54	50, 53	syl6eqel 2709	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
55		oveq1 6657	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
56		2m1e1 11135	. . . . . . 7 |- ( 2 - 1 ) = 1
57	44	simp3i 1072	. . . . . . 7 |- 1 e. Z
58	56, 57	eqeltri 2697	. . . . . 6 |- ( 2 - 1 ) e. Z
59	55, 58	syl6eqel 2709	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
60	54, 59	jaoi 394	. . . 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 1265	. 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 832	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 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   {csn 4177   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   mod cmo 12668   ^cexp 12860   abscabs 13974   || cdvds 14983   Primecprime 15385

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-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

This theorem is referenced by:  lgsfcl2  25028