Theorem fermltl 15489

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.)

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 15388	. . . 4 |- ( P e. Prime -> P e. NN )
2		dvdsval3 14987	. . . 4 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
3	1, 2	sylan 488	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
4		simp2 1062	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> A e. ZZ )
5		0zd 11389	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> 0 e. ZZ )
6	1	3ad2ant1 1082	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> P e. NN )
7	6	nnnn0d 11351	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> P e. NN0 )
8	6	nnrpd 11870	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> P e. RR+ )
9		simp3 1063	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( A mod P ) = 0 )
10		0mod 12701	. . . . . . . 8 |- ( P e. RR+ -> ( 0 mod P ) = 0 )
11	8, 10	syl 17	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( 0 mod P ) = 0 )
12	9, 11	eqtr4d 2659	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( A mod P ) = ( 0 mod P ) )
13		modexp 12999	. . . . . 6 |- ( ( ( A e. ZZ /\ 0 e. ZZ ) /\ ( P e. NN0 /\ P e. RR+ ) /\ ( A mod P ) = ( 0 mod P ) ) -> ( ( A ^ P ) mod P ) = ( ( 0 ^ P ) mod P ) )
14	4, 5, 7, 8, 12, 13	syl221anc 1337	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( ( A ^ P ) mod P ) = ( ( 0 ^ P ) mod P ) )
15	6	0expd 13024	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( 0 ^ P ) = 0 )
16	15	oveq1d 6665	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( ( 0 ^ P ) mod P ) = ( 0 mod P ) )
17	12, 16	eqtr4d 2659	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( A mod P ) = ( ( 0 ^ P ) mod P ) )
18	14, 17	eqtr4d 2659	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A mod P ) = 0 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
19	18	3expia 1267	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 0 -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
20	3, 19	sylbid 230	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
21		coprm 15423	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
22		prmz 15389	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
23		gcdcom 15235	. . . . . 6 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
24	22, 23	sylan 488	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
25	24	eqeq1d 2624	. . . 4 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
26	21, 25	bitrd 268	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( A gcd P ) = 1 ) )
27		simp2 1062	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. ZZ )
28	1	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. NN )
29	28	phicld 15477	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN )
30	29	nnnn0d 11351	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) e. NN0 )
31		zexpcl 12875	. . . . . . . 8 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
32	27, 30, 31	syl2anc 693	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
33	32	zred 11482	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) e. RR )
34		1red 10055	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> 1 e. RR )
35	28	nnrpd 11870	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> P e. RR+ )
36		eulerth 15488	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
37	1, 36	syl3an1 1359	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
38		modmul1 12723	. . . . . 6 |- ( ( ( ( A ^ ( phi ` P ) ) e. RR /\ 1 e. RR ) /\ ( A e. ZZ /\ P e. RR+ ) /\ ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( 1 x. A ) mod P ) )
39	33, 34, 27, 35, 37, 38	syl221anc 1337	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( 1 x. A ) mod P ) )
40		phiprm 15482	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
41	40	3ad2ant1 1082	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( phi ` P ) = ( P - 1 ) )
42	41	oveq2d 6666	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( P - 1 ) ) )
43	42	oveq1d 6665	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( ( A ^ ( P - 1 ) ) x. A ) )
44	27	zcnd 11483	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> A e. CC )
45		expm1t 12888	. . . . . . . 8 |- ( ( A e. CC /\ P e. NN ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
46	44, 28, 45	syl2anc 693	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( A ^ P ) = ( ( A ^ ( P - 1 ) ) x. A ) )
47	43, 46	eqtr4d 2659	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) x. A ) = ( A ^ P ) )
48	47	oveq1d 6665	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( ( A ^ ( phi ` P ) ) x. A ) mod P ) = ( ( A ^ P ) mod P ) )
49	44	mulid2d 10058	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( 1 x. A ) = A )
50	49	oveq1d 6665	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( 1 x. A ) mod P ) = ( A mod P ) )
51	39, 48, 50	3eqtr3d 2664	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )
52	51	3expia 1267	. . 3 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A gcd P ) = 1 -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
53	26, 52	sylbid 230	. 2 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A -> ( ( A ^ P ) mod P ) = ( A mod P ) ) )
54	20, 53	pm2.61d 170	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A ^ P ) mod P ) = ( A mod P ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   mod cmo 12668   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   Primecprime 15385   phicphi 15469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by: (None)