Theorem powm2modprm 12206

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 524	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. Prime )
2		simpr 109	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> A e. ZZ )
3	2	adantr 274	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> A e. ZZ )
4		m1dvdsndvds 12202	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> -. P || A ) )
5	4	imp 123	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> -. P || A )
6		eqid 2170	. . . . . 6 |- ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P )
7	6	modprminv 12203	. . . . 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 109	. . . . . 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 2176	. . . . 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 1233	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 1 = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
12		modprm1div 12201	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 1 <-> P || ( A - 1 ) ) )
13	12	biimpar 295	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A mod P ) = 1 )
14	13	oveq1d 5868	. . . . 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 5868	. . . 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 9585	. . . . . 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 12087	. . . . . . . . . 10 |- ( P e. Prime -> ( P - 2 ) e. NN0 )
19	18	anim1ci 339	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( A e. ZZ /\ ( P - 2 ) e. NN0 ) )
20	19	adantr 274	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( A e. ZZ /\ ( P - 2 ) e. NN0 ) )
21		zexpcl 10491	. . . . . . . 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 12064	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
24	23	adantr 274	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> P e. NN )
25	24	adantr 274	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. NN )
26	22, 25	zmodcld 10301	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
27	26	nn0zd 9332	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ZZ )
28	25	nnzd 9333	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. ZZ )
29		zq 9585	. . . . . 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 8922	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 0 < P )
32		modqmulmod 10345	. . . . 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 1234	. . . 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 10301	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
36	35	nn0cnd 9190	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. CC )
37	36	mulid2d 7938	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) = ( ( A ^ ( P - 2 ) ) mod P ) )
38	37	oveq1d 5868	. . . . . 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 274	. . . . 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 9585	. . . . . . 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 10314	. . . . . 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 1233	. . . . 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 2203	. . . 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 2211	. . 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 2204	. 2 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 )
47	46	ex 114	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || ( A - 1 ) -> ( ( A ^ ( P - 2 ) ) mod P ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   < clt 7954   - cmin 8090   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   QQcq 9578   ...cfz 9965   mod cmo 10278   ^cexp 10475   || cdvds 11749   Primecprime 12061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-2o 6396  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514  df-dvds 11750  df-gcd 11898  df-prm 12062  df-phi 12165

This theorem is referenced by: (None)