Theorem eulerth 12810

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 12792	. . . . . . . 8 |- ( N e. NN -> ( phi ` N ) e. NN )
2	1	nnnn0d 9455	. . . . . . 7 |- ( N e. NN -> ( phi ` N ) e. NN0 )
3		hashfz1 11046	. . . . . . 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 12797	. . . . . 6 |- ( N e. NN -> ( phi ` N ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
6	4, 5	eqtrd 2264	. . . . 5 |- ( N e. NN -> (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
7	6	3ad2ant1 1044	. . . 4 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> (♯ ` ( 1 ... ( phi ` N ) ) ) = (♯ ` { k e. ( 0..^ N ) | ( k gcd N ) = 1 } ) )
8		1zzd 9506	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> 1 e. ZZ )
9	1	3ad2ant1 1044	. . . . . . 7 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. NN )
10	9	nnzd 9601	. . . . . 6 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( phi ` N ) e. ZZ )
11	8, 10	fzfigd 10694	. . . . 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 6025	. . . . . . . 8 |- ( k = y -> ( k gcd N ) = ( y gcd N ) )
14	13	eqeq1d 2240	. . . . . . 7 |- ( k = y -> ( ( k gcd N ) = 1 <-> ( y gcd N ) = 1 ) )
15	14	cbvrabv 2801	. . . . . 6 |- { k e. ( 0..^ N ) | ( k gcd N ) = 1 } = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
16	12, 15	eulerthlemfi 12805	. . . . 5 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> { k e. ( 0..^ N ) | ( k gcd N ) = 1 } e. Fin )
17		hashen 11047	. . . . 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 411	. . . 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 147	. . 3 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( 1 ... ( phi ` N ) ) ~~ { k e. ( 0..^ N ) | ( k gcd N ) = 1 } )
20		bren 6917	. . 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 122	. 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 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 } ) -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
23		simpr 110	. . 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 12809	. 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 1947	1 |- ( ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   {crab 2514   class class class wbr 4088   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6018   ~~ cen 6907   Fincfn 6909   0cc0 8032   1c1 8033   NNcn 9143   NN0cn0 9402   ZZcz 9479   ...cfz 10243  ..^cfzo 10377   mod cmo 10585   ^cexp 10801  ♯chash 11038   gcd cgcd 12529   phicphi 12786

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-clim 11844  df-proddc 12117  df-dvds 12354  df-gcd 12530  df-phi 12788

This theorem is referenced by:  fermltl  12811  prmdiv  12812  odzcllem  12820  odzphi  12824  vfermltl  12829  lgslem1  15735