Theorem fermltl 12931

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 12807	. . . 4 |- ( P e. Prime -> P e. NN )
2		dvdsmodexp 12481	. . . . 5 |- ( ( P e. NN /\ P e. NN /\ P || A ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
3	2	3exp 1229	. . . 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 12841	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
7		prmz 12808	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
8		gcdcom 12669	. . . . . 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 2241	. . . 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 1025	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. ZZ )
13	1	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. NN )
14	13	phicld 12915	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN )
15	14	nnnn0d 9553	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN0 )
16		zexpcl 10916	. . . . . . . 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 9958	. . . . . . 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 9603	. . . . . . 7 |- 1 e. ZZ
21		zq 9958	. . . . . . 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 9965	. . . . . . 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 9281	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 0 < P )
26		eulerth 12930	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
27	1, 26	syl3an1 1307	. . . . . 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 10739	. . . . 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 12920	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
30	29	3ad2ant1 1045	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) = ( P - 1 ) )
31	30	oveq2d 6066	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( P - 1 ) ) )
32	31	oveq1d 6065	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( ( A ^ ( P - 1 ) ) x. A ) )
33	12	zcnd 9701	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. CC )
34		expm1t 10929	. . . . . . . 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 2268	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( A ^ P ) )
37	36	oveq1d 6065	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( A ^ P ) mod P ) )
38	33	mullidd 8292	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( 1 x. A ) = A )
39	38	oveq1d 6065	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( 1 x. A ) mod P ) = ( A mod P ) )
40	28, 37, 39	3eqtr3d 2273	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
41	40	3expia 1232	. . 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 12484	. . . 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 844	. . 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 726	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   1c1 8128   x. cmul 8132   - cmin 8444   NNcn 9237   NN0cn0 9496   ZZcz 9577   QQcq 9951   mod cmo 10684   ^cexp 10900   || cdvds 12473   gcd cgcd 12649   Primecprime 12804   phicphi 12906

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-prm 12805  df-phi 12908

This theorem is referenced by: (None)