Theorem powm2modprm 12736

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 12732	. . . . 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 2207	. . . . . 6 |- ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P )
7	6	modprminv 12733	. . . . 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 2213	. . . . 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 1250	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 1 = ( ( A x. ( ( A ^ ( P - 2 ) ) mod P ) ) mod P ) )
12		modprm1div 12731	. . . . . . 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 5984	. . . . 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 5984	. . . 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 9784	. . . . . 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 12616	. . . . . . . . . 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 10738	. . . . . . . 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 12593	. . . . . . . . 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 10529	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
27	26	nn0zd 9530	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ZZ )
28	25	nnzd 9531	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> P e. ZZ )
29		zq 9784	. . . . . 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 9117	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ ) /\ P || ( A - 1 ) ) -> 0 < P )
32		modqmulmod 10573	. . . . 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 1251	. . . 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 10529	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. NN0 )
36	35	nn0cnd 9387	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. CC )
37	36	mulid2d 8128	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( 1 x. ( ( A ^ ( P - 2 ) ) mod P ) ) = ( ( A ^ ( P - 2 ) ) mod P ) )
38	37	oveq1d 5984	. . . . . 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 9784	. . . . . . 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 10542	. . . . . 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 1250	. . . . 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 2240	. . . 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 2248	. . 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 2241	. 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 981   = wceq 1373   e. wcel 2178   class class class wbr 4060  (class class class)co 5969   0cc0 7962   1c1 7963   x. cmul 7967   < clt 8144   - cmin 8280   NNcn 9073   2c2 9124   NN0cn0 9332   ZZcz 9409   QQcq 9777   ...cfz 10167   mod cmo 10506   ^cexp 10722   || cdvds 12259   Primecprime 12590

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-mulrcl 8061  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-precex 8072  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-apti 8077  ax-pre-ltadd 8078  ax-pre-mulgt0 8079  ax-pre-mulext 8080  ax-arch 8081  ax-caucvg 8082

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-po 4362  df-iso 4363  df-iord 4432  df-on 4434  df-ilim 4435  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-isom 5300  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-recs 6416  df-irdg 6481  df-frec 6502  df-1o 6527  df-2o 6528  df-oadd 6531  df-er 6645  df-en 6853  df-dom 6854  df-fin 6855  df-sup 7114  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-reap 8685  df-ap 8692  df-div 8783  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-n0 9333  df-z 9410  df-uz 9686  df-q 9778  df-rp 9813  df-fz 10168  df-fzo 10302  df-fl 10452  df-mod 10507  df-seqfrec 10632  df-exp 10723  df-ihash 10960  df-cj 11314  df-re 11315  df-im 11316  df-rsqrt 11470  df-abs 11471  df-clim 11751  df-proddc 12023  df-dvds 12260  df-gcd 12436  df-prm 12591  df-phi 12694

This theorem is referenced by: (None)