:: BINOP_1 semantic presentation
:: deftheorem Def1 defines . BINOP_1:def 1 :
theorem Th1: :: BINOP_1:1
for
b1,
b2,
b3 being
set for
b4,
b5 being
Function of
[:b1,b2:],
b3 st ( for
b6,
b7 being
set st
b6 in b1 &
b7 in b2 holds
b4 . b6,
b7 = b5 . b6,
b7 ) holds
b4 = b5
theorem Th2: :: BINOP_1:2
for
b1,
b2,
b3 being
set for
b4,
b5 being
Function of
[:b1,b2:],
b3 st ( for
b6 being
Element of
b1for
b7 being
Element of
b2 holds
b4 . b6,
b7 = b5 . b6,
b7 ) holds
b4 = b5
scheme :: BINOP_1:sch 1
s1{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
P1[
set ,
set ,
set ] } :
ex
b1 being
Function of
[:F1(),F2():],
F3() st
for
b2,
b3 being
set st
b2 in F1() &
b3 in F2() holds
P1[
b2,
b3,
b1 . b2,
b3]
provided
E2:
for
b1,
b2 being
set st
b1 in F1() &
b2 in F2() holds
ex
b3 being
set st
(
b3 in F3() &
P1[
b1,
b2,
b3] )
scheme :: BINOP_1:sch 2
s2{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4(
set ,
set )
-> set } :
ex
b1 being
Function of
[:F1(),F2():],
F3() st
for
b2,
b3 being
set st
b2 in F1() &
b3 in F2() holds
b1 . b2,
b3 = F4(
b2,
b3)
provided
E2:
for
b1,
b2 being
set st
b1 in F1() &
b2 in F2() holds
F4(
b1,
b2)
in F3()
definition
let c1 be
set ;
let c2 be
BinOp of
c1;
attr a2 is
commutative means :
Def2:
:: BINOP_1:def 2
for
b1,
b2 being
Element of
a1 holds
a2 . b1,
b2 = a2 . b2,
b1;
attr a2 is
associative means :
Def3:
:: BINOP_1:def 3
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . b1,
(a2 . b2,b3) = a2 . (a2 . b1,b2),
b3;
attr a2 is
idempotent means :
Def4:
:: BINOP_1:def 4
for
b1 being
Element of
a1 holds
a2 . b1,
b1 = b1;
end;
:: deftheorem Def2 defines commutative BINOP_1:def 2 :
:: deftheorem Def3 defines associative BINOP_1:def 3 :
for
b1 being
set for
b2 being
BinOp of
b1 holds
(
b2 is
associative iff for
b3,
b4,
b5 being
Element of
b1 holds
b2 . b3,
(b2 . b4,b5) = b2 . (b2 . b3,b4),
b5 );
:: deftheorem Def4 defines idempotent BINOP_1:def 4 :
:: deftheorem Def5 defines is_a_left_unity_wrt BINOP_1:def 5 :
:: deftheorem Def6 defines is_a_right_unity_wrt BINOP_1:def 6 :
:: deftheorem Def7 defines is_a_unity_wrt BINOP_1:def 7 :
theorem Th3: :: BINOP_1:3
canceled;
theorem Th4: :: BINOP_1:4
canceled;
theorem Th5: :: BINOP_1:5
canceled;
theorem Th6: :: BINOP_1:6
canceled;
theorem Th7: :: BINOP_1:7
canceled;
theorem Th8: :: BINOP_1:8
canceled;
theorem Th9: :: BINOP_1:9
canceled;
theorem Th10: :: BINOP_1:10
canceled;
theorem Th11: :: BINOP_1:11
theorem Th12: :: BINOP_1:12
theorem Th13: :: BINOP_1:13
theorem Th14: :: BINOP_1:14
theorem Th15: :: BINOP_1:15
theorem Th16: :: BINOP_1:16
theorem Th17: :: BINOP_1:17
theorem Th18: :: BINOP_1:18
:: deftheorem Def8 defines the_unity_wrt BINOP_1:def 8 :
definition
let c1 be
set ;
let c2,
c3 be
BinOp of
c1;
pred c2 is_left_distributive_wrt c3 means :
Def9:
:: BINOP_1:def 9
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . b1,
(a3 . b2,b3) = a3 . (a2 . b1,b2),
(a2 . b1,b3);
pred c2 is_right_distributive_wrt c3 means :
Def10:
:: BINOP_1:def 10
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . (a3 . b1,b2),
b3 = a3 . (a2 . b1,b3),
(a2 . b2,b3);
end;
:: deftheorem Def9 defines is_left_distributive_wrt BINOP_1:def 9 :
:: deftheorem Def10 defines is_right_distributive_wrt BINOP_1:def 10 :
:: deftheorem Def11 defines is_distributive_wrt BINOP_1:def 11 :
theorem Th19: :: BINOP_1:19
canceled;
theorem Th20: :: BINOP_1:20
canceled;
theorem Th21: :: BINOP_1:21
canceled;
theorem Th22: :: BINOP_1:22
canceled;
theorem Th23: :: BINOP_1:23
for
b1 being
set for
b2,
b3 being
BinOp of
b1 holds
(
b2 is_distributive_wrt b3 iff for
b4,
b5,
b6 being
Element of
b1 holds
(
b2 . b4,
(b3 . b5,b6) = b3 . (b2 . b4,b5),
(b2 . b4,b6) &
b2 . (b3 . b4,b5),
b6 = b3 . (b2 . b4,b6),
(b2 . b5,b6) ) )
theorem Th24: :: BINOP_1:24
theorem Th25: :: BINOP_1:25
theorem Th26: :: BINOP_1:26
theorem Th27: :: BINOP_1:27
theorem Th28: :: BINOP_1:28
:: deftheorem Def12 defines is_distributive_wrt BINOP_1:def 12 :
definition
let c1 be non
empty set ;
let c2 be
BinOp of
c1;
redefine attr a2 is
commutative means :: BINOP_1:def 13
for
b1,
b2 being
Element of
a1 holds
a2 . b1,
b2 = a2 . b2,
b1;
correctness
compatibility
( c2 is commutative iff for b1, b2 being Element of c1 holds c2 . b1,b2 = c2 . b2,b1 );
by Def2;
redefine attr a2 is
associative means :: BINOP_1:def 14
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . b1,
(a2 . b2,b3) = a2 . (a2 . b1,b2),
b3;
correctness
compatibility
( c2 is associative iff for b1, b2, b3 being Element of c1 holds c2 . b1,(c2 . b2,b3) = c2 . (c2 . b1,b2),b3 );
by Def3;
redefine attr a2 is
idempotent means :: BINOP_1:def 15
for
b1 being
Element of
a1 holds
a2 . b1,
b1 = b1;
correctness
compatibility
( c2 is idempotent iff for b1 being Element of c1 holds c2 . b1,b1 = b1 );
by Def4;
end;
:: deftheorem Def13 defines commutative BINOP_1:def 13 :
:: deftheorem Def14 defines associative BINOP_1:def 14 :
:: deftheorem Def15 defines idempotent BINOP_1:def 15 :
:: deftheorem Def16 defines is_a_left_unity_wrt BINOP_1:def 16 :
:: deftheorem Def17 defines is_a_right_unity_wrt BINOP_1:def 17 :
definition
let c1 be non
empty set ;
let c2,
c3 be
BinOp of
c1;
redefine pred c2 is_left_distributive_wrt c3 means :: BINOP_1:def 18
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . b1,
(a3 . b2,b3) = a3 . (a2 . b1,b2),
(a2 . b1,b3);
correctness
compatibility
( c2 is_left_distributive_wrt c3 iff for b1, b2, b3 being Element of c1 holds c2 . b1,(c3 . b2,b3) = c3 . (c2 . b1,b2),(c2 . b1,b3) );
by Def9;
redefine pred c2 is_right_distributive_wrt c3 means :: BINOP_1:def 19
for
b1,
b2,
b3 being
Element of
a1 holds
a2 . (a3 . b1,b2),
b3 = a3 . (a2 . b1,b3),
(a2 . b2,b3);
correctness
compatibility
( c2 is_right_distributive_wrt c3 iff for b1, b2, b3 being Element of c1 holds c2 . (c3 . b1,b2),b3 = c3 . (c2 . b1,b3),(c2 . b2,b3) );
by Def10;
end;
:: deftheorem Def18 defines is_left_distributive_wrt BINOP_1:def 18 :
:: deftheorem Def19 defines is_right_distributive_wrt BINOP_1:def 19 :
:: deftheorem Def20 defines is_distributive_wrt BINOP_1:def 20 :
theorem Th29: :: BINOP_1:29
theorem Th30: :: BINOP_1:30
theorem Th31: :: BINOP_1:31