Theorem lgslem4 24306

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 768	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> A e. ZZ )
2		oddprm 14844	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
3	2	ad2antlr 741	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
4	3	nnnn0d 10949	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
5		zexpcl 12325	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
6	1, 4, 5	syl2anc 673	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
7	6	zred 11063	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
8		0red 9662	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. RR )
9		1red 9676	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 e. RR )
10		eldifi 3544	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11	10	ad2antlr 741	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. Prime )
12		prmuz2 14721	. . . . . . . . . . 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 11254	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
15	13, 14	sylib 201	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P e. NN /\ 1 < P ) )
16	15	simpld 466	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. NN )
17	16	nnrpd 11362	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR+ )
18		0zd 10973	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 0 e. ZZ )
19		simpr 468	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P || A )
20		dvdsval3 14386	. . . . . . . . . . . 12 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
21	16, 1, 20	syl2anc 673	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( P || A <-> ( A mod P ) = 0 ) )
22	19, 21	mpbid 215	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = 0 )
23		0mod 12161	. . . . . . . . . . 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 2508	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( A mod P ) = ( 0 mod P ) )
26		modexp 12445	. . . . . . . . 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 1303	. . . . . . . 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 12470	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 0 ^ ( ( P - 1 ) / 2 ) ) = 0 )
29	28	oveq1d 6323	. . . . . . . 8 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( 0 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
30	27, 29	eqtrd 2505	. . . . . . 7 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 0 mod P ) )
31		modadd1 12167	. . . . . . 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 1303	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( ( 0 + 1 ) mod P ) )
33		0p1e1 10743	. . . . . . 7 |- ( 0 + 1 ) = 1
34	33	oveq1i 6318	. . . . . 6 |- ( ( 0 + 1 ) mod P ) = ( 1 mod P )
35	32, 34	syl6eq 2521	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 1 mod P ) )
36	16	nnred 10646	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> P e. RR )
37	15	simprd 470	. . . . . 6 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> 1 < P )
38		1mod 12162	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
39	36, 37, 38	syl2anc 673	. . . . 5 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( 1 mod P ) = 1 )
40	35, 39	eqtrd 2505	. . . 4 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 1 )
41	40	oveq1d 6323	. . 3 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 1 - 1 ) )
42		1m1e0 10700	. . . 4 |- ( 1 - 1 ) = 0
43		lgslem2.z	. . . . . 6 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
44	43	lgslem2 24304	. . . . 5 |- ( -u 1 e. Z /\ 0 e. Z /\ 1 e. Z )
45	44	simp2i 1040	. . . 4 |- 0 e. Z
46	42, 45	eqeltri 2545	. . 3 |- ( 1 - 1 ) e. Z
47	41, 46	syl6eqel 2557	. 2 |- ( ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) /\ P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
48		lgslem1 24303	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )
49		elpri 3976	. . . 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 6315	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 0 - 1 ) )
51		df-neg 9883	. . . . . . 7 |- -u 1 = ( 0 - 1 )
52	44	simp1i 1039	. . . . . . 7 |- -u 1 e. Z
53	51, 52	eqeltrri 2546	. . . . . 6 |- ( 0 - 1 ) e. Z
54	50, 53	syl6eqel 2557	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
55		oveq1 6315	. . . . . 6 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( 2 - 1 ) )
56		2m1e1 10746	. . . . . . 7 |- ( 2 - 1 ) = 1
57	44	simp3i 1041	. . . . . . 7 |- 1 e. Z
58	56, 57	eqeltri 2545	. . . . . 6 |- ( 2 - 1 ) e. Z
59	55, 58	syl6eqel 2557	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) e. Z )
60	54, 59	jaoi 386	. . . 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 1231	. 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 808	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 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {crab 2760   \ cdif 3387   {csn 3959   {cpr 3961   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   < clt 9693   <_ cle 9694   - cmin 9880   -ucneg 9881   / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   mod cmo 12129   ^cexp 12310   abscabs 13374   || cdvds 14382   Primecprime 14701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793

This theorem is referenced by:  lgsfcl2  24309