:: EULER_1 semantic presentation
Lemma27:
for k, n being Element of NAT holds k gcd n = k hcf n
Lemma28:
for a, b being Element of NAT holds
( a,b are_relative_prime iff a,b are_relative_prime )
Lemma29:
for a, k being Element of NAT
for i being Integer st i > 0 & a = i & k > 0 holds
i mod k = a mod k
Lemma35:
for i being Integer
for a, m being Nat holds
( a = i & m > 0 & m divides i iff ( a = i & m > 0 & m divides a ) )
Lemma36:
for k being Element of NAT
for j being Integer st k <> 0 & [\(j / k)/] + 1 >= j / k holds
k >= j - ([\(j / k)/] * k)
theorem Th1: :: EULER_1:1
Lemma38:
not 1 is prime
by INT_2:def 5;
theorem Th2: :: EULER_1:2
theorem Th3: :: EULER_1:3
theorem Th4: :: EULER_1:4
theorem Th5: :: EULER_1:5
theorem Th6: :: EULER_1:6
theorem Th7: :: EULER_1:7
theorem Th8: :: EULER_1:8
theorem Th9: :: EULER_1:9
theorem Th10: :: EULER_1:10
theorem Th11: :: EULER_1:11
theorem Th12: :: EULER_1:12
theorem Th13: :: EULER_1:13
theorem Th14: :: EULER_1:14
theorem Th15: :: EULER_1:15
theorem Th16: :: EULER_1:16
theorem Th17: :: EULER_1:17
:: deftheorem Def1 defines Euler EULER_1:def 1 :
set X = { k where k is Element of NAT : ( 1,k are_relative_prime & k >= 1 & k <= 1 ) } ;
for x being set holds
( x in { k where k is Element of NAT : ( 1,k are_relative_prime & k >= 1 & k <= 1 ) } iff x = 1 )
then Lemma94:
Card {1} = Euler 1
by TARSKI:def 1;
theorem Th18: :: EULER_1:18
set X = { k where k is Element of NAT : ( 2,k are_relative_prime & k >= 1 & k <= 2 ) } ;
for x being set holds
( x in { k where k is Element of NAT : ( 2,k are_relative_prime & k >= 1 & k <= 2 ) } iff x = 1 )
then Lemma95:
Card {1} = Euler 2
by TARSKI:def 1;
theorem Th19: :: EULER_1:19
theorem Th20: :: EULER_1:20
theorem Th21: :: EULER_1:21
theorem Th22: :: EULER_1:22