:: EULER_1 semantic presentation
Lemma1:
for b1, b2 being Nat holds b1 gcd b2 = b1 hcf b2
Lemma2:
for b1, b2 being Nat holds
( b1,b2 are_relative_prime iff b1,b2 are_relative_prime )
Lemma3:
for b1, b2 being Nat
for b3 being Integer holds
( b3 > 0 & b1 = b3 & b2 > 0 implies b3 mod b2 = b1 mod b2 )
Lemma4:
for b1, b2 being Nat
for b3 being Integer holds
( ( b1 = b3 & b2 > 0 & b2 divides b3 implies ( b1 = b3 & b2 > 0 & b2 divides b1 ) ) & ( b1 = b3 & b2 > 0 & b2 divides b1 implies ( b1 = b3 & b2 > 0 & b2 divides b3 ) ) )
Lemma5:
for b1 being Nat
for b2 being Integer holds
( b1 <> 0 & [\(b2 / b1)/] + 1 >= b2 / b1 implies b1 >= b2 - ([\(b2 / b1)/] * b1) )
theorem Th1: :: EULER_1:1
Lemma7:
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
for b
1, b
2 being
Nat holds
( b
1 hcf b
2 = 1 implies for b
3 being
Nat holds
(b1 * b3) hcf (b2 * b3) = b
3 )
theorem Th7: :: EULER_1:7
theorem Th8: :: EULER_1:8
for b
1, b
2 being
Nat holds
( b
1 hcf b
2 = 1 implies
(b1 + b2) hcf b
2 = 1 )
theorem Th9: :: EULER_1:9
for b
1, b
2, b
3 being
Nat holds
(b1 + (b2 * b3)) hcf b
2 = b
1 hcf b
2
theorem Th10: :: EULER_1:10
for b
1, b
2 being
Nat holds
not ( b
1,b
2 are_relative_prime & ( for b
3 being
Nat holds
not ( ex b
4, b
5 being
Integer st
( b
3 = (b4 * b1) + (b5 * b2) & b
3 > 0 ) & ( for b
4 being
Nat holds
( ex b
5, b
6 being
Integer st
( b
4 = (b5 * b1) + (b6 * b2) & b
4 > 0 ) implies b
3 <= b
4 ) ) ) ) )
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
for b
1, b
2, b
3 being
Integer holds
( b
1 <> 0 & b
2 <> 0 & b
3 > 0 implies
(b3 * b1) gcd (b3 * b2) = b
3 * (b1 gcd b2) )
theorem Th17: :: EULER_1:17
:: deftheorem Def1 defines Euler EULER_1:def 1 :
set c1 = { b1 where B is Nat : ( 1,b1 are_relative_prime & b1 >= 1 & b1 <= 1 ) } ;
for b1 being set holds
( b1 in { b2 where B is Nat : ( 1,b2 are_relative_prime & b2 >= 1 & b2 <= 1 ) } iff b1 = 1 )
then E24: Card {1} =
Card { b1 where B is Nat : ( 1,b1 are_relative_prime & b1 >= 1 & b1 <= 1 ) }
by TARSKI:def 1
.=
Euler 1
;
theorem Th18: :: EULER_1:18
set c2 = { b1 where B is Nat : ( 2,b1 are_relative_prime & b1 >= 1 & b1 <= 2 ) } ;
for b1 being set holds
( b1 in { b2 where B is Nat : ( 2,b2 are_relative_prime & b2 >= 1 & b2 <= 2 ) } iff b1 = 1 )
then E25: Card {1} =
Card { b1 where B is Nat : ( 2,b1 are_relative_prime & b1 >= 1 & b1 <= 2 ) }
by TARSKI:def 1
.=
Euler 2
;
theorem Th19: :: EULER_1:19
theorem Th20: :: EULER_1:20
for b
1 being
Nat holds
( b
1 > 1 implies
Euler b
1 <= b
1 - 1 )
theorem Th21: :: EULER_1:21
theorem Th22: :: EULER_1:22