:: Euler's {T}heorem and Small {F}ermat's Theorem
:: by Yoshinori Fujisawa , Yasushi Fuwa and Hidetaka Shimizu
::
:: Received June 10, 1998
:: Copyright (c) 1998-2011 Association of Mizar Users

begin

Lm1: for t being Integer holds
( t < 1 iff t <= 0 )

proof end;

Lm2: for a being Nat st a <> 0 holds
a - 1 >= 0

proof end;

Lm3: for z being Integer holds 1 gcd z = 1

proof end;

theorem :: EULER_2:1

for a, b being Nat holds
( a,b are_relative_prime iff a,b are_relative_prime ) ;

theorem Th2: :: EULER_2:2

for m being Nat
for t being Integer st m * t >= 1 holds
t >= 1

proof end;

Lm4: for m being Nat
for z being Integer st 1 - m <= z & z <= m - 1 & m divides z holds
z = 0

proof end;

theorem :: EULER_2:3

canceled;

theorem :: EULER_2:4

canceled;

theorem Th5: :: EULER_2:5

for a, b, m being Nat st a <> 0 & b <> 0 & m <> 0 & a,m are_relative_prime & b,m are_relative_prime holds
m,(a * b) mod m are_relative_prime

proof end;

theorem Th6: :: EULER_2:6

for m, n, a, b being Nat st m > 1 & m,n are_relative_prime & n = (a * b) mod m holds
m,b are_relative_prime

proof end;

theorem Th7: :: EULER_2:7

for m, n being Nat holds (m mod n) mod n = m mod n

proof end;

theorem :: EULER_2:8

for l, m, n being Nat holds (l + m) mod n = ((l mod n) + (m mod n)) mod n

proof end;

theorem Th9: :: EULER_2:9

for l, m, n being Nat holds (l * m) mod n = (l * (m mod n)) mod n

proof end;

theorem :: EULER_2:10

for l, m, n being Nat holds (l * m) mod n = ((l mod n) * m) mod n by Th9;

theorem Th11: :: EULER_2:11

for l, m, n being Nat holds (l * m) mod n = ((l mod n) * (m mod n)) mod n

proof end;

begin

definition

let f be FinSequence of NAT ;
let a be Nat;
:: original: *
redefine func a * f -> FinSequence of NAT ;
coherence
a * f is FinSequence of NAT

proof end;

end;

theorem :: EULER_2:12

canceled;

theorem :: EULER_2:13

canceled;

theorem :: EULER_2:14

canceled;

theorem :: EULER_2:15

canceled;

theorem :: EULER_2:16

canceled;

theorem :: EULER_2:17

canceled;

theorem :: EULER_2:18

canceled;

theorem :: EULER_2:19

canceled;

theorem :: EULER_2:20

canceled;

theorem :: EULER_2:21

canceled;

theorem :: EULER_2:22

canceled;

theorem :: EULER_2:23

canceled;

theorem :: EULER_2:24

canceled;

theorem Th25: :: EULER_2:25

for R1, R2 being FinSequence of NAT st R1,R2 are_fiberwise_equipotent holds
Product R1 = Product R2

proof end;

begin

definition

let f be FinSequence of NAT ;
let m be Nat;
func f mod m -> FinSequence of NAT means :Def1: :: EULER_2:def 1
( len it = len f & ( for i being Nat st i in dom f holds
it . i = (f . i) mod m ) );
existence
ex b₁ being FinSequence of NAT st
( len b₁ = len f & ( for i being Nat st i in dom f holds
b₁ . i = (f . i) mod m ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of NAT st len b₁ = len f & ( for i being Nat st i in dom f holds
b₁ . i = (f . i) mod m ) & len b₂ = len f & ( for i being Nat st i in dom f holds
b₂ . i = (f . i) mod m ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines mod EULER_2:def 1 :
for f being FinSequence of NAT
for m being Nat
for b₃ being FinSequence of NAT holds
( b₃ = f mod m iff ( len b₃ = len f & ( for i being Nat st i in dom f holds
b₃ . i = (f . i) mod m ) ) );

theorem :: EULER_2:26

for m being Nat
for f being FinSequence of NAT st m <> 0 holds
(Product (f mod m)) mod m = (Product f) mod m

proof end;

theorem Th27: :: EULER_2:27

for a, m, n being Nat st a <> 0 & m > 1 & (a * n) mod m = n mod m & m,n are_relative_prime holds
a mod m = 1

proof end;

theorem :: EULER_2:28

for m being Nat
for F being FinSequence of NAT holds (F mod m) mod m = F mod m

proof end;

theorem :: EULER_2:29

for a, m being Nat
for F being FinSequence of NAT holds (a * (F mod m)) mod m = (a * F) mod m

proof end;

theorem :: EULER_2:30

for m being Nat
for F, G being FinSequence of NAT holds (F ^ G) mod m = (F mod m) ^ (G mod m)

proof end;

theorem :: EULER_2:31

for a, m being Nat
for F, G being FinSequence of NAT holds (a * (F ^ G)) mod m = ((a * F) mod m) ^ ((a * G) mod m)

proof end;

theorem :: EULER_2:32

for a, m being Nat st a <> 0 & m <> 0 & a,m are_relative_prime holds
for b being Nat holds a |^ b,m are_relative_prime

proof end;

begin

theorem Th33: :: EULER_2:33

for a, m being Nat st a <> 0 & m > 1 & a,m are_relative_prime holds
(a |^ (Euler m)) mod m = 1

proof end;

theorem :: EULER_2:34

for a, m being Nat st a <> 0 & m is prime & a,m are_relative_prime holds
(a |^ m) mod m = a mod m

proof end;