Theorem fermltl 12756

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 12632	. . . 4 |- ( P e. Prime -> P e. NN )
2		dvdsmodexp 12306	. . . . 5 |- ( ( P e. NN /\ P e. NN /\ P || A ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
3	2	3exp 1226	. . . 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 12666	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
7		prmz 12633	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
8		gcdcom 12494	. . . . . 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 2238	. . . 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 1022	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. ZZ )
13	1	3ad2ant1 1042	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. NN )
14	13	phicld 12740	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN )
15	14	nnnn0d 9422	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN0 )
16		zexpcl 10776	. . . . . . . 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 9821	. . . . . . 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 9472	. . . . . . 7 |- 1 e. ZZ
21		zq 9821	. . . . . . 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 9828	. . . . . . 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 9154	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 0 < P )
26		eulerth 12755	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
27	1, 26	syl3an1 1304	. . . . . 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 10599	. . . . 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 12745	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
30	29	3ad2ant1 1042	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) = ( P - 1 ) )
31	30	oveq2d 6017	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( P - 1 ) ) )
32	31	oveq1d 6016	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( ( A ^ ( P - 1 ) ) x. A ) )
33	12	zcnd 9570	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. CC )
34		expm1t 10789	. . . . . . . 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 2265	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( A ^ P ) )
37	36	oveq1d 6016	. . . . 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 8165	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( 1 x. A ) = A )
39	38	oveq1d 6016	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( 1 x. A ) mod P ) = ( A mod P ) )
40	28, 37, 39	3eqtr3d 2270	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
41	40	3expia 1229	. . 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 12309	. . . 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 841	. . 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 723	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   1c1 8000   x. cmul 8004   - cmin 8317   NNcn 9110   NN0cn0 9369   ZZcz 9446   QQcq 9814   mod cmo 10544   ^cexp 10760   || cdvds 12298   gcd cgcd 12474   Primecprime 12629   phicphi 12731

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-2o 6563  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-proddc 12062  df-dvds 12299  df-gcd 12475  df-prm 12630  df-phi 12733

This theorem is referenced by: (None)