Theorem fermltl 12097

Description: Fermat's little theorem. When P is prime, A ^ P == A (mod P) for any A, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.)

Assertion

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

Proof of Theorem fermltl

Step	Hyp	Ref	Expression
1		prmnn 11978	. . . 4 |- ( P e. Prime -> P e. NN )
2		dvdsmodexp 11684	. . . . 5 |- ( ( P e. NN /\ P e. NN /\ P || A ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
3	2	3exp 1184	. . . 4 |- ( P e. NN -> ( P e. NN -> ( P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) ) )
4	1, 1, 3	sylc 62	. . 3 |- ( P e. Prime -> ( P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
5	4	adantr 274	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
6		coprm 12009	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
7		prmz 11979	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
8		gcdcom 11848	. . . . . 6 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
9	7, 8	sylan 281	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
10	9	eqeq1d 2166	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
11	6, 10	bitrd 187	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( A gcd P ) = 1 ) )
12		simp2 983	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. ZZ )
13	1	3ad2ant1 1003	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. NN )
14	13	phicld 12081	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN )
15	14	nnnn0d 9137	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN0 )
16		zexpcl 10427	. . . . . . . 8 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
17	12, 15, 16	syl2anc 409	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
18		zq 9528	. . . . . . 7 |- ( ( A ^ ( phi ` P ) ) e. ZZ -> ( A ^ ( phi ` P ) ) e. QQ )
19	17, 18	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) e. QQ )
20		1z 9187	. . . . . . 7 |- 1 e. ZZ
21		zq 9528	. . . . . . 7 |- ( 1 e. ZZ -> 1 e. QQ )
22	20, 21	mp1i 10	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 1 e. QQ )
23		nnq 9535	. . . . . . 7 |- ( P e. NN -> P e. QQ )
24	13, 23	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. QQ )
25	13	nngt0d 8871	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 0 < P )
26		eulerth 12096	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
27	1, 26	syl3an1 1253	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
28	19, 22, 12, 24, 25, 27	modqmul1 10269	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( 1 x. A ) mod P ) )
29		phiprm 12086	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
30	29	3ad2ant1 1003	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) = ( P - 1 ) )
31	30	oveq2d 5837	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( P - 1 ) ) )
32	31	oveq1d 5836	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( ( A ^ ( P - 1 ) ) x. A ) )
33	12	zcnd 9281	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. CC )
34		expm1t 10440	. . . . . . . 8 |- ( ( A e. CC /\ P e. NN ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
35	33, 13, 34	syl2anc 409	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
36	32, 35	eqtr4d 2193	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( A ^ P ) )
37	36	oveq1d 5836	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( A ^ P ) mod P ) )
38	33	mulid2d 7890	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( 1 x. A ) = A )
39	38	oveq1d 5836	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( 1 x. A ) mod P ) = ( A mod P ) )
40	28, 37, 39	3eqtr3d 2198	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
41	40	3expia 1187	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A gcd P ) = 1 -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
42	11, 41	sylbid 149	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
43		dvdsdc 11687	. . . 4 |- ( ( P e. NN /\ A e. ZZ ) -> DECID P || A )
44	1, 43	sylan 281	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> DECID P || A )
45		exmiddc 822	. . 3 |- (DECID P || A -> ( P || A \/ -. P || A ) )
46	44, 45	syl 14	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A \/ -. P || A ) )
47	5, 42, 46	mpjaod 708	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965   ` cfv 5169  (class class class)co 5821   CCcc 7724   1c1 7727   x. cmul 7731   - cmin 8040   NNcn 8827   NN0cn0 9084   ZZcz 9161   QQcq 9521   mod cmo 10214   ^cexp 10411   || cdvds 11676   gcd cgcd 11821   Primecprime 11975   phicphi 12073

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-2o 6361  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-sup 6924  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-fz 9906  df-fzo 10035  df-fl 10162  df-mod 10215  df-seqfrec 10338  df-exp 10412  df-ihash 10643  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-clim 11169  df-proddc 11441  df-dvds 11677  df-gcd 11822  df-prm 11976  df-phi 12074

This theorem is referenced by: (None)