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:: Euler's {T}heorem and Small {F}ermat's Theorem
:: by Yoshinori Fujisawa , Yasushi Fuwa and Hidetaka Shimizu
::
:: Received June 10, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

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;

:: Very useful theorem

theorem :: EULER_2:1

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

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 Th3: :: EULER_2:3

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

proof end;

theorem Th4: :: EULER_2:4

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

proof end;

theorem Th5: :: EULER_2:5

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

proof end;

theorem :: EULER_2:6

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

proof end;

theorem Th7: :: EULER_2:7

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

proof end;

theorem :: EULER_2:8

for m, n, l being Nat holds (l * m) mod n = ((l mod n) * m) mod n by Th7;

theorem Th9: :: EULER_2:9

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

proof end;

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 Th10: :: EULER_2:10

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

proof end;

:: Function of (FinSequence of NAT) mod Element of NAT

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_2:11

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 Th12: :: EULER_2:12

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

proof end;

theorem :: EULER_2:13

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

proof end;

theorem :: EULER_2:14

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_2:15

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_2:16

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;

:: Function of (Element of NAT) |^ (Element of NAT)

theorem :: EULER_2:17

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

proof end;

:: Euler's Generalization of Fermat's Little Theorem

theorem Th18: :: EULER_2:18

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

proof end;

::End of Euler's Theorem
:: Small Fermat's Theorem

theorem :: EULER_2:19

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

proof end;