Theorem powm2modprm 12517

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 12513	. . . . 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 12514	. . . . 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 12512	. . . . . . 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 5958	. . . . 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 5958	. . . 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 9746	. . . . . 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 12397	. . . . . . . . . 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 10697	. . . . . . . 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 12374	. . . . . . . . 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 10488	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
27	26	nn0zd 9492	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ZZ )
28	25	nnzd 9493	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. ZZ )
29		zq 9746	. . . . . 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 9079	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 0 < P )
32		modqmulmod 10532	. . . . 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 10488	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
36	35	nn0cnd 9349	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. CC )
37	36	mulid2d 8090	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) = ( ( A ^ ( P - 2 ) ) mod P ) )
38	37	oveq1d 5958	. . . . . 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 9746	. . . . . . 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 10501	. . . . . 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 5943   0cc0 7924   1c1 7925   x. cmul 7929   < clt 8106   - cmin 8242   NNcn 9035   2c2 9086   NN0cn0 9294   ZZcz 9371   QQcq 9739   ...cfz 10129   mod cmo 10465   ^cexp 10681   || cdvds 12040   Primecprime 12371

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 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

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 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-2o 6502  df-oadd 6505  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-sup 7085  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-ihash 10919  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-clim 11532  df-proddc 11804  df-dvds 12041  df-gcd 12217  df-prm 12372  df-phi 12475

This theorem is referenced by: (None)