Theorem powm2modprm 12494

Description: If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)

Assertion

Ref	Expression
powm2modprm	|- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 ) )

Proof of Theorem powm2modprm

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. Prime )
2		simpr 110	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. ZZ )
3	2	adantr 276	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> A e. ZZ )
4		m1dvdsndvds 12490	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> -. P || A ) )
5	4	imp 124	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> -. P || A )
6		eqid 2204	. . . . . 6 |- ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P )
7	6	modprminv 12491	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 ) )
8		simpr 110	. . . . . 6 |- ( ( ( ( A ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 ) -> ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 )
9	8	eqcomd 2210	. . . . 5 |- ( ( ( ( A ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 ) -> 1 = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
10	7, 9	syl 14	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
11	1, 3, 5, 10	syl3anc 1249	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 1 = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
12		modprm1div 12489	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 1 <-> P || ( A - 1 ) ) )
13	12	biimpar 297	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A mod P ) = 1 )
14	13	oveq1d 5949	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A mod P ) x. ( ( A ^ ( P - 2 ) ) mod P ) ) = ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) )
15	14	oveq1d 5949	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( ( A mod P ) x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
16		zq 9729	. . . . . 6 |- ( A e. ZZ -> A e. QQ )
17	3, 16	syl 14	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> A e. QQ )
18		prmm2nn0 12374	. . . . . . . . . 10 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
19	18	anim1ci 341	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A e. ZZ /\ ( P - 2 ) e. NN0 ) )
20	19	adantr 276	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A e. ZZ /\ ( P - 2 ) e. NN0 ) )
21		zexpcl 10680	. . . . . . . 8 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
22	20, 21	syl 14	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A ^ ( P - 2 ) ) e. ZZ )
23		prmnn 12351	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
24	23	adantr 276	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> P e. NN )
25	24	adantr 276	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. NN )
26	22, 25	zmodcld 10471	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
27	26	nn0zd 9475	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ZZ )
28	25	nnzd 9476	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. ZZ )
29		zq 9729	. . . . . 6 |- ( P e. ZZ -> P e. QQ )
30	28, 29	syl 14	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. QQ )
31	25	nngt0d 9062	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 0 < P )
32		modqmulmod 10515	. . . . 5 |- ( ( ( A e. QQ /\ ( ( A ^ ( P - 2 ) ) mod P ) e. ZZ ) /\ ( P e. QQ /\ 0 < P ) ) -> ( ( ( A mod P ) x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
33	17, 27, 30, 31, 32	syl22anc 1250	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( ( A mod P ) x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
34	19, 21	syl 14	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A ^ ( P - 2 ) ) e. ZZ )
35	34, 24	zmodcld 10471	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
36	35	nn0cnd 9332	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. CC )
37	36	mulid2d 8073	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) = ( ( A ^ ( P - 2 ) ) mod P ) )
38	37	oveq1d 5949	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) )
39	38	adantr 276	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) )
40		zq 9729	. . . . . . 7 |- ( ( A ^ ( P - 2 ) ) e. ZZ -> ( A ^ ( P - 2 ) ) e. QQ )
41	22, 40	syl 14	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A ^ ( P - 2 ) ) e. QQ )
42		modqabs2 10484	. . . . . 6 |- ( ( ( A ^ ( P - 2 ) ) e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
43	41, 30, 31, 42	syl3anc 1249	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
44	39, 43	eqtrd 2237	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
45	15, 33, 44	3eqtr3d 2245	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
46	11, 45	eqtr2d 2238	. 2 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 )
47	46	ex 115	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5934   0cc0 7907   1c1 7908   x. cmul 7912   < clt 8089   - cmin 8225   NNcn 9018   2c2 9069   NN0cn0 9277   ZZcz 9354   QQcq 9722   ...cfz 10112   mod cmo 10448   ^cexp 10664   || cdvds 12017   Primecprime 12348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-isom 5277  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-frec 6467  df-1o 6492  df-2o 6493  df-oadd 6496  df-er 6610  df-en 6818  df-dom 6819  df-fin 6820  df-sup 7068  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-q 9723  df-rp 9758  df-fz 10113  df-fzo 10247  df-fl 10394  df-mod 10449  df-seqfrec 10574  df-exp 10665  df-ihash 10902  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229  df-clim 11509  df-proddc 11781  df-dvds 12018  df-gcd 12194  df-prm 12349  df-phi 12452

This theorem is referenced by: (None)