Theorem eulerth 12187

Description: Euler's theorem, a generalization of Fermat's little theorem. If A and N are coprime, then A ^ phi ( N ) == 1 (mod N). This is Metamath 100 proof #10. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (Contributed by Mario Carneiro, 28-Feb-2014.)

Assertion

Ref	Expression
eulerth	|- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )

Proof of Theorem eulerth

Dummy variables f y k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phicl 12169	. . . . . . . 8 |- ( N e. NN -> ( phi ` N ) e. NN )
2	1	nnnn0d 9188	. . . . . . 7 |- ( N e. NN -> ( phi ` N ) e. NN0 )
3		hashfz1 10717	. . . . . . 7 |- ( ( phi ` N ) e. NN0 -> (♯ ` ( 1 ... ( phi ` N ) ) ) = ( phi ` N ) )
4	2, 3	syl 14	. . . . . 6 |- ( N e. NN -> (♯ ` ( 1 ... ( phi ` N ) ) ) = ( phi ` N ) )
5		dfphi2 12174	. . . . . 6 |- ( N e. NN -> ( phi ` N ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
6	4, 5	eqtrd 2203	. . . . 5 |- ( N e. NN -> (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
7	6	3ad2ant1 1013	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
8		1zzd 9239	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
9	1	3ad2ant1 1013	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10	9	nnzd 9333	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. ZZ )
11	8, 10	fzfigd 10387	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( 1 ... ( phi ` N ) ) e. Fin )
12		id 19	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
13		oveq1 5860	. . . . . . . 8 |- ( k = y -> ( k gcd N ) = ( y gcd N ) )
14	13	eqeq1d 2179	. . . . . . 7 |- ( k = y -> ( ( k gcd N ) = 1 <-> ( y gcd N ) = 1 ) )
15	14	cbvrabv 2729	. . . . . 6 |- { k e. ( 0..^ N ) | ( k gcd N ) = 1 } = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
16	12, 15	eulerthlemfi 12182	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } e. Fin )
17		hashen 10718	. . . . 5 |- ( ( ( 1 ... ( phi ` N ) ) e. Fin /\ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } e. Fin ) -> ( (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) <-> ( 1 ... ( phi ` N ) ) ~~ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
18	11, 16, 17	syl2anc 409	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) <-> ( 1 ... ( phi ` N ) ) ~~ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
19	7, 18	mpbid 146	. . 3 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( 1 ... ( phi ` N ) ) ~~ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } )
20		bren 6725	. . 3 |- ( ( 1 ... ( phi ` N ) ) ~~ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } <-> E. f f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } )
21	19, 20	sylib 121	. 2 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> E. f f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } )
22		simpl 108	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
23		simpr 109	. . 3 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) -> f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } )
24	22, 15, 23	eulerthlemth 12186	. 2 |- ( ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) /\ f : ( 1 ... ( phi ` N ) ) -1-1-onto-> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
25	21, 24	exlimddv 1891	1 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   {crab 2452   class class class wbr 3989   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   ~~ cen 6716   Fincfn 6718   0cc0 7774   1c1 7775   NNcn 8878   NN0cn0 9135   ZZcz 9212   ...cfz 9965  ..^cfzo 10098   mod cmo 10278   ^cexp 10475  ♯chash 10709   gcd cgcd 11897   phicphi 12163

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514  df-dvds 11750  df-gcd 11898  df-phi 12165

This theorem is referenced by:  fermltl  12188  prmdiv  12189  odzcllem  12196  odzphi  12200  vfermltl  12205  lgslem1  13695