Theorem fermltl 12372

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 12248	. . . 4 |- ( P e. Prime -> P e. NN )
2		dvdsmodexp 11938	. . . . 5 |- ( ( P e. NN /\ P e. NN /\ P || A ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
3	2	3exp 1204	. . . 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 276	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
6		coprm 12282	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
7		prmz 12249	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
8		gcdcom 12110	. . . . . 6 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
9	7, 8	sylan 283	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
10	9	eqeq1d 2202	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
11	6, 10	bitrd 188	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( A gcd P ) = 1 ) )
12		simp2 1000	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. ZZ )
13	1	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. NN )
14	13	phicld 12356	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN )
15	14	nnnn0d 9293	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN0 )
16		zexpcl 10625	. . . . . . . 8 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
17	12, 15, 16	syl2anc 411	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
18		zq 9691	. . . . . . 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 9343	. . . . . . 7 |- 1 e. ZZ
21		zq 9691	. . . . . . 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 9698	. . . . . . 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 9026	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 0 < P )
26		eulerth 12371	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
27	1, 26	syl3an1 1282	. . . . . 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 10448	. . . . 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 12361	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
30	29	3ad2ant1 1020	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) = ( P - 1 ) )
31	30	oveq2d 5934	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( P - 1 ) ) )
32	31	oveq1d 5933	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( ( A ^ ( P - 1 ) ) x. A ) )
33	12	zcnd 9440	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. CC )
34		expm1t 10638	. . . . . . . 8 |- ( ( A e. CC /\ P e. NN ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
35	33, 13, 34	syl2anc 411	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
36	32, 35	eqtr4d 2229	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( A ^ P ) )
37	36	oveq1d 5933	. . . . 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 8038	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( 1 x. A ) = A )
39	38	oveq1d 5933	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( 1 x. A ) mod P ) = ( A mod P ) )
40	28, 37, 39	3eqtr3d 2234	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
41	40	3expia 1207	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A gcd P ) = 1 -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
42	11, 41	sylbid 150	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
43		dvdsdc 11941	. . . 4 |- ( ( P e. NN /\ A e. ZZ ) -> DECID P || A )
44	1, 43	sylan 283	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> DECID P || A )
45		exmiddc 837	. . 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 719	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   1c1 7873   x. cmul 7877   - cmin 8190   NNcn 8982   NN0cn0 9240   ZZcz 9317   QQcq 9684   mod cmo 10393   ^cexp 10609   || cdvds 11930   gcd cgcd 12079   Primecprime 12245   phicphi 12347

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694  df-dvds 11931  df-gcd 12080  df-prm 12246  df-phi 12349

This theorem is referenced by: (None)