Article FINSEQOP, MML version 4.99.1005
:: FINSEQOP:th 1
theorem
for b1 being Relation-like Function-like set holds
<:{},b1:> = {} & <:b1,{}:> = {};
:: FINSEQOP:th 2
theorem
for b1 being Relation-like Function-like set holds
[:{},b1:] = {} & [:b1,{}:] = {};
:: FINSEQOP:th 4
theorem
for b1, b2 being Relation-like Function-like set holds
b1 .:({},b2) = {} & b1 .:(b2,{}) = {};
:: FINSEQOP:th 5
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
b2 [:]({},b1) = {};
:: FINSEQOP:th 6
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
b2 [;](b1,{}) = {};
:: FINSEQOP:th 7
theorem
for b1, b2, b3 being set holds
<:b1 --> b2,b1 --> b3:> = b1 --> [b2,b3];
:: FINSEQOP:th 8
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4 being set
st [b3,b4] in proj1 b1
holds b1 .:(b2 --> b3,b2 --> b4) = b2 --> (b1 .(b3,b4));
:: FINSEQOP:funcnot 1 => FINSEQOP:func 1
definition
let a1, a2, a3 be non empty set;
let a4 be Function-like quasi_total Relation of [:a1,a2:],a3;
let a5 be FinSequence of a1;
let a6 be FinSequence of a2;
redefine func a4 .:(a5,a6) -> FinSequence of a3;
end;
:: FINSEQOP:funcnot 2 => FINSEQOP:func 2
definition
let a1, a2, a3 be non empty set;
let a4 be Function-like quasi_total Relation of [:a1,a2:],a3;
let a5 be FinSequence of a1;
let a6 be Element of a2;
redefine func a4 [:](a5,a6) -> FinSequence of a3;
end;
:: FINSEQOP:funcnot 3 => FINSEQOP:func 3
definition
let a1, a2, a3 be non empty set;
let a4 be Function-like quasi_total Relation of [:a1,a2:],a3;
let a5 be Element of a1;
let a6 be FinSequence of a2;
redefine func a4 [;](a5,a6) -> FinSequence of a3;
end;
:: FINSEQOP:funcnot 4 => FINSEQOP:func 4
definition
let a1 be non empty set;
let a2 be natural set;
let a3 be Element of a1;
redefine func a2 |-> a3 -> Element of a2 -tuples_on a1;
end;
:: FINSEQOP:funcnot 5 => FINSEQOP:func 5
definition
let a1, a2 be set;
let a3 be FinSequence of a1;
let a4 be Function-like quasi_total Relation of a1,a2;
redefine func a4 * a3 -> FinSequence of a2;
end;
:: FINSEQOP:th 9
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being FinSequence of b1
for b5 being Function-like quasi_total Relation of b1,b2 holds
b5 * (b4 ^ <*b3*>) = (b5 * b4) ^ <*b5 . b3*>;
:: FINSEQOP:th 10
theorem
for b1, b2 being non empty set
for b3, b4 being FinSequence of b1
for b5 being Function-like quasi_total Relation of b1,b2 holds
b5 * (b3 ^ b4) = (b5 * b3) ^ (b5 * b4);
:: FINSEQOP:th 11
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b2
for b5 being Element of b3
for b6 being natural set
for b7 being Function-like quasi_total Relation of [:b2,b3:],b1
for b8 being Element of b6 -tuples_on b2
for b9 being Element of b6 -tuples_on b3 holds
b7 .:(b8 ^ <*b4*>,b9 ^ <*b5*>) = (b7 .:(b8,b9)) ^ <*b7 .(b4,b5)*>;
:: FINSEQOP:th 12
theorem
for b1, b2, b3 being non empty set
for b4, b5 being natural set
for b6 being Function-like quasi_total Relation of [:b2,b3:],b1
for b7 being Element of b4 -tuples_on b2
for b8 being Element of b4 -tuples_on b3
for b9 being Element of b5 -tuples_on b2
for b10 being Element of b5 -tuples_on b3 holds
b6 .:(b7 ^ b9,b8 ^ b10) = (b6 .:(b7,b8)) ^ (b6 .:(b9,b10));
:: FINSEQOP:th 13
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b3
for b6 being Function-like quasi_total Relation of [:b1,b3:],b2
for b7 being FinSequence of b3 holds
b6 [;](b4,b7 ^ <*b5*>) = (b6 [;](b4,b7)) ^ <*b6 .(b4,b5)*>;
:: FINSEQOP:th 14
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function-like quasi_total Relation of [:b1,b3:],b2
for b6, b7 being FinSequence of b3 holds
b5 [;](b4,b6 ^ b7) = (b5 [;](b4,b6)) ^ (b5 [;](b4,b7));
:: FINSEQOP:th 15
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b3
for b5 being Element of b1
for b6 being Function-like quasi_total Relation of [:b3,b1:],b2
for b7 being FinSequence of b3 holds
b6 [:](b7 ^ <*b4*>,b5) = (b6 [:](b7,b5)) ^ <*b6 .(b4,b5)*>;
:: FINSEQOP:th 16
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function-like quasi_total Relation of [:b3,b1:],b2
for b6, b7 being FinSequence of b3 holds
b5 [:](b6 ^ b7,b4) = (b5 [:](b6,b4)) ^ (b5 [:](b7,b4));
:: FINSEQOP:th 17
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being natural set
for b5 being Function-like quasi_total Relation of b1,b2 holds
b5 * (b4 |-> b3) = b4 |-> (b5 . b3);
:: FINSEQOP:th 18
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b2
for b6 being natural set
for b7 being Function-like quasi_total Relation of [:b1,b2:],b3 holds
b7 .:(b6 |-> b4,b6 |-> b5) = b6 |-> (b7 .(b4,b5));
:: FINSEQOP:th 19
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b2
for b6 being natural set
for b7 being Function-like quasi_total Relation of [:b1,b2:],b3 holds
b7 [;](b4,b6 |-> b5) = b6 |-> (b7 .(b4,b5));
:: FINSEQOP:th 20
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b2
for b6 being natural set
for b7 being Function-like quasi_total Relation of [:b1,b2:],b3 holds
b7 [:](b6 |-> b4,b5) = b6 |-> (b7 .(b4,b5));
:: FINSEQOP:th 21
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being natural set
for b6 being Function-like quasi_total Relation of [:b1,b3:],b2
for b7 being Element of b5 -tuples_on b3 holds
b6 .:(b5 |-> b4,b7) = b6 [;](b4,b7);
:: FINSEQOP:th 22
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b3
for b5 being natural set
for b6 being Function-like quasi_total Relation of [:b3,b1:],b2
for b7 being Element of b5 -tuples_on b3 holds
b6 .:(b7,b5 |-> b4) = b6 [:](b7,b4);
:: FINSEQOP:th 23
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being natural set
for b6 being Function-like quasi_total Relation of [:b1,b3:],b2
for b7 being Element of b5 -tuples_on b3 holds
b6 [;](b4,b7) = b7 * (b6 [;](b4,id b3));
:: FINSEQOP:th 24
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b3
for b5 being natural set
for b6 being Function-like quasi_total Relation of [:b3,b1:],b2
for b7 being Element of b5 -tuples_on b3 holds
b6 [:](b7,b4) = b7 * (b6 [:](id b3,b4));
:: FINSEQOP:th 25
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4, b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of [:b2,b2:],b2
st b6 is associative(b2)
holds (b6 [;](b3,id b2)) * (b6 .:(b4,b5)) = b6 .:((b6 [;](b3,id b2)) * b4,b5);
:: FINSEQOP:th 26
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4, b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of [:b2,b2:],b2
st b6 is associative(b2)
holds (b6 [:](id b2,b3)) * (b6 .:(b4,b5)) = b6 .:(b4,(b6 [:](id b2,b3)) * b5);
:: FINSEQOP:th 27
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4, b5 being Element of b3 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds (b6 [;](b2,id b1)) * (b6 .:(b4,b5)) = b6 .:((b6 [;](b2,id b1)) * b4,b5);
:: FINSEQOP:th 28
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4, b5 being Element of b3 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds (b6 [:](id b1,b2)) * (b6 .:(b4,b5)) = b6 .:(b4,(b6 [:](id b1,b2)) * b5);
:: FINSEQOP:th 29
theorem
for b1 being non empty set
for b2 being natural set
for b3, b4, b5 being Element of b2 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds b6 .:(b6 .:(b3,b4),b5) = b6 .:(b3,b6 .:(b4,b5));
:: FINSEQOP:th 30
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being natural set
for b5 being Element of b4 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds b6 [:](b6 [;](b2,b5),b3) = b6 [;](b2,b6 [:](b5,b3));
:: FINSEQOP:th 31
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4, b5 being Element of b3 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds b6 .:(b6 [:](b4,b2),b5) = b6 .:(b4,b6 [;](b2,b5));
:: FINSEQOP:th 32
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being natural set
for b5 being Element of b4 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds b6 [;](b6 .(b2,b3),b5) = b6 [;](b2,b6 [;](b3,b5));
:: FINSEQOP:th 33
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being natural set
for b5 being Element of b4 -tuples_on b1
for b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is associative(b1)
holds b6 [:](b5,b6 .(b2,b3)) = b6 [:](b6 [:](b5,b2),b3);
:: FINSEQOP:th 34
theorem
for b1 being non empty set
for b2 being natural set
for b3, b4 being Element of b2 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is commutative(b1)
holds b5 .:(b3,b4) = b5 .:(b4,b3);
:: FINSEQOP:th 35
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4 being Element of b3 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is commutative(b1)
holds b5 [;](b2,b4) = b5 [:](b4,b2);
:: FINSEQOP:th 36
theorem
for b1, b2 being non empty set
for b3, b4 being Element of b2
for b5 being Function-like quasi_total Relation of b1,b2
for b6, b7 being Function-like quasi_total Relation of [:b2,b2:],b2
st b6 is_distributive_wrt b7
holds b6 [;](b7 .(b3,b4),b5) = b7 .:(b6 [;](b3,b5),b6 [;](b4,b5));
:: FINSEQOP:th 37
theorem
for b1, b2 being non empty set
for b3, b4 being Element of b2
for b5 being Function-like quasi_total Relation of b1,b2
for b6, b7 being Function-like quasi_total Relation of [:b2,b2:],b2
st b6 is_distributive_wrt b7
holds b6 [:](b5,b7 .(b3,b4)) = b7 .:(b6 [:](b5,b3),b6 [:](b5,b4));
:: FINSEQOP:th 38
theorem
for b1, b2, b3 being non empty set
for b4, b5 being Function-like quasi_total Relation of b1,b3
for b6 being Function-like quasi_total Relation of b3,b2
for b7 being Function-like quasi_total Relation of [:b3,b3:],b3
for b8 being Function-like quasi_total Relation of [:b2,b2:],b2
st for b9, b10 being Element of b3 holds
b6 . (b7 .(b9,b10)) = b8 .(b6 . b9,b6 . b10)
holds b6 * (b7 .:(b4,b5)) = b8 .:(b6 * b4,b6 * b5);
:: FINSEQOP:th 39
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of b1,b3
for b6 being Function-like quasi_total Relation of b3,b2
for b7 being Function-like quasi_total Relation of [:b3,b3:],b3
for b8 being Function-like quasi_total Relation of [:b2,b2:],b2
st for b9, b10 being Element of b3 holds
b6 . (b7 .(b9,b10)) = b8 .(b6 . b9,b6 . b10)
holds b6 * (b7 [;](b4,b5)) = b8 [;](b6 . b4,b6 * b5);
:: FINSEQOP:th 40
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of b1,b3
for b6 being Function-like quasi_total Relation of b3,b2
for b7 being Function-like quasi_total Relation of [:b3,b3:],b3
for b8 being Function-like quasi_total Relation of [:b2,b2:],b2
st for b9, b10 being Element of b3 holds
b6 . (b7 .(b9,b10)) = b8 .(b6 . b9,b6 . b10)
holds b6 * (b7 [:](b5,b4)) = b8 [:](b6 * b5,b6 . b4);
:: FINSEQOP:th 41
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b2,b2
st b6 is_distributive_wrt b5
holds b6 * (b5 .:(b3,b4)) = b5 .:(b6 * b3,b6 * b4);
:: FINSEQOP:th 42
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b2,b2
st b6 is_distributive_wrt b5
holds b6 * (b5 [;](b3,b4)) = b5 [;](b6 . b3,b6 * b4);
:: FINSEQOP:th 43
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b2,b2
st b6 is_distributive_wrt b5
holds b6 * (b5 [:](b4,b3)) = b5 [:](b6 * b4,b6 . b3);
:: FINSEQOP:th 44
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1)
holds b4 .:(b2 --> the_unity_wrt b4,b3) = b3 & b4 .:(b3,b2 --> the_unity_wrt b4) = b3;
:: FINSEQOP:th 45
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
st b4 is having_a_unity(b2)
holds b4 [;](the_unity_wrt b4,b3) = b3;
:: FINSEQOP:th 46
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
st b4 is having_a_unity(b2)
holds b4 [:](b3,the_unity_wrt b4) = b3;
:: FINSEQOP:th 47
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being natural set
for b5 being Element of b4 -tuples_on b1
for b6, b7 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is_distributive_wrt b7
holds b6 [;](b7 .(b2,b3),b5) = b7 .:(b6 [;](b2,b5),b6 [;](b3,b5));
:: FINSEQOP:th 48
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being natural set
for b5 being Element of b4 -tuples_on b1
for b6, b7 being Function-like quasi_total Relation of [:b1,b1:],b1
st b6 is_distributive_wrt b7
holds b6 [:](b5,b7 .(b2,b3)) = b7 .:(b6 [:](b5,b2),b6 [:](b5,b3));
:: FINSEQOP:th 49
theorem
for b1, b2 being non empty set
for b3 being natural set
for b4 being Function-like quasi_total Relation of b2,b1
for b5, b6 being Element of b3 -tuples_on b2
for b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of [:b1,b1:],b1
st for b9, b10 being Element of b2 holds
b4 . (b7 .(b9,b10)) = b8 .(b4 . b9,b4 . b10)
holds b4 * (b7 .:(b5,b6)) = b8 .:(b4 * b5,b4 * b6);
:: FINSEQOP:th 50
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being natural set
for b5 being Function-like quasi_total Relation of b2,b1
for b6 being Element of b4 -tuples_on b2
for b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of [:b1,b1:],b1
st for b9, b10 being Element of b2 holds
b5 . (b7 .(b9,b10)) = b8 .(b5 . b9,b5 . b10)
holds b5 * (b7 [;](b3,b6)) = b8 [;](b5 . b3,b5 * b6);
:: FINSEQOP:th 51
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being natural set
for b5 being Function-like quasi_total Relation of b2,b1
for b6 being Element of b4 -tuples_on b2
for b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of [:b1,b1:],b1
st for b9, b10 being Element of b2 holds
b5 . (b7 .(b9,b10)) = b8 .(b5 . b9,b5 . b10)
holds b5 * (b7 [:](b6,b3)) = b8 [:](b5 * b6,b5 . b3);
:: FINSEQOP:th 52
theorem
for b1 being non empty set
for b2 being natural set
for b3, b4 being Element of b2 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being Function-like quasi_total Relation of b1,b1
st b6 is_distributive_wrt b5
holds b6 * (b5 .:(b3,b4)) = b5 .:(b6 * b3,b6 * b4);
:: FINSEQOP:th 53
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4 being Element of b3 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being Function-like quasi_total Relation of b1,b1
st b6 is_distributive_wrt b5
holds b6 * (b5 [;](b2,b4)) = b5 [;](b6 . b2,b6 * b4);
:: FINSEQOP:th 54
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4 being Element of b3 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being Function-like quasi_total Relation of b1,b1
st b6 is_distributive_wrt b5
holds b6 * (b5 [:](b4,b2)) = b5 [:](b6 * b4,b6 . b2);
:: FINSEQOP:th 55
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being Function-like quasi_total Relation of b1,b1
st b3 is_distributive_wrt b4 & b5 = b3 [;](b2,id b1)
holds b5 is_distributive_wrt b4;
:: FINSEQOP:th 56
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being Function-like quasi_total Relation of b1,b1
st b3 is_distributive_wrt b4 & b5 = b3 [:](id b1,b2)
holds b5 is_distributive_wrt b4;
:: FINSEQOP:th 57
theorem
for b1 being non empty set
for b2 being natural set
for b3 being Element of b2 -tuples_on b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1)
holds b4 .:(b2 |-> the_unity_wrt b4,b3) = b3 & b4 .:(b3,b2 |-> the_unity_wrt b4) = b3;
:: FINSEQOP:th 58
theorem
for b1 being non empty set
for b2 being natural set
for b3 being Element of b2 -tuples_on b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1)
holds b4 [;](the_unity_wrt b4,b3) = b3;
:: FINSEQOP:th 59
theorem
for b1 being non empty set
for b2 being natural set
for b3 being Element of b2 -tuples_on b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1)
holds b4 [:](b3,the_unity_wrt b4) = b3;
:: FINSEQOP:prednot 1 => FINSEQOP:pred 1
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of a1,a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
pred A2 is_an_inverseOp_wrt A3 means
for b1 being Element of a1 holds
a3 .(b1,a2 . b1) = the_unity_wrt a3 & a3 .(a2 . b1,b1) = the_unity_wrt a3;
end;
:: FINSEQOP:dfs 1
definiens
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of a1,a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
To prove
a2 is_an_inverseOp_wrt a3
it is sufficient to prove
thus for b1 being Element of a1 holds
a3 .(b1,a2 . b1) = the_unity_wrt a3 & a3 .(a2 . b1,b1) = the_unity_wrt a3;
:: FINSEQOP:def 1
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
b2 is_an_inverseOp_wrt b3
iff
for b4 being Element of b1 holds
b3 .(b4,b2 . b4) = the_unity_wrt b3 & b3 .(b2 . b4,b4) = the_unity_wrt b3;
:: FINSEQOP:attrnot 1 => FINSEQOP:attr 1
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
attr a2 is having_an_inverseOp means
ex b1 being Function-like quasi_total Relation of a1,a1 st
b1 is_an_inverseOp_wrt a2;
end;
:: FINSEQOP:dfs 2
definiens
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
To prove
a2 is having_an_inverseOp
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of a1,a1 st
b1 is_an_inverseOp_wrt a2;
:: FINSEQOP:def 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
b2 is having_an_inverseOp(b1)
iff
ex b3 being Function-like quasi_total Relation of b1,b1 st
b3 is_an_inverseOp_wrt b2;
:: FINSEQOP:prednot 2 => FINSEQOP:attr 1
notation
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
synonym a2 has_an_inverseOp for having_an_inverseOp;
end;
:: FINSEQOP:funcnot 6 => FINSEQOP:func 6
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
assume a2 is having_a_unity(a1) & a2 is associative(a1) & a2 is having_an_inverseOp(a1);
func the_inverseOp_wrt A2 -> Function-like quasi_total Relation of a1,a1 means
it is_an_inverseOp_wrt a2;
end;
:: FINSEQOP:def 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
st b2 is having_a_unity(b1) & b2 is associative(b1) & b2 is having_an_inverseOp(b1)
for b3 being Function-like quasi_total Relation of b1,b1 holds
b3 = the_inverseOp_wrt b2
iff
b3 is_an_inverseOp_wrt b2;
:: FINSEQOP:th 63
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is having_a_unity(b1) & b3 is associative(b1) & b3 is having_an_inverseOp(b1)
holds b3 .((the_inverseOp_wrt b3) . b2,b2) = the_unity_wrt b3 &
b3 .(b2,(the_inverseOp_wrt b3) . b2) = the_unity_wrt b3;
:: FINSEQOP:th 64
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1) & b4 is associative(b1) & b4 is having_an_inverseOp(b1) & b4 .(b2,b3) = the_unity_wrt b4
holds b2 = (the_inverseOp_wrt b4) . b3 & (the_inverseOp_wrt b4) . b2 = b3;
:: FINSEQOP:th 65
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
st b2 is having_a_unity(b1) & b2 is associative(b1) & b2 is having_an_inverseOp(b1)
holds (the_inverseOp_wrt b2) . the_unity_wrt b2 = the_unity_wrt b2;
:: FINSEQOP:th 66
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is having_a_unity(b1) & b3 is associative(b1) & b3 is having_an_inverseOp(b1)
holds (the_inverseOp_wrt b3) . ((the_inverseOp_wrt b3) . b2) = b2;
:: FINSEQOP:th 67
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
st b2 is having_a_unity(b1) & b2 is associative(b1) & b2 is commutative(b1) & b2 is having_an_inverseOp(b1)
holds the_inverseOp_wrt b2 is_distributive_wrt b2;
:: FINSEQOP:th 68
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is having_a_unity(b1) &
b5 is associative(b1) &
b5 is having_an_inverseOp(b1) &
(b5 .(b2,b3) = b5 .(b2,b4) or b5 .(b3,b2) = b5 .(b4,b2))
holds b3 = b4;
:: FINSEQOP:th 69
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1) & b4 is associative(b1) & b4 is having_an_inverseOp(b1) & (b4 .(b2,b3) = b3 or b4 .(b3,b2) = b3)
holds b2 = the_unity_wrt b4;
:: FINSEQOP:th 70
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3 & b2 = the_unity_wrt b3
for b5 being Element of b1 holds
b4 .(b2,b5) = b2 & b4 .(b5,b2) = b2;
:: FINSEQOP:th 71
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4, b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being Function-like quasi_total Relation of b1,b1
st b4 is having_a_unity(b1) & b4 is associative(b1) & b4 is having_an_inverseOp(b1) & b6 = the_inverseOp_wrt b4 & b5 is_distributive_wrt b4
holds b6 . (b5 .(b2,b3)) = b5 .(b6 . b2,b3) &
b6 . (b5 .(b2,b3)) = b5 .(b2,b6 . b3);
:: FINSEQOP:th 72
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of [:b1,b1:],b1
for b4 being Function-like quasi_total Relation of b1,b1
st b2 is having_a_unity(b1) & b2 is associative(b1) & b2 is having_an_inverseOp(b1) & b4 = the_inverseOp_wrt b2 & b3 is_distributive_wrt b2 & b3 is having_a_unity(b1)
holds b3 [;](b4 . the_unity_wrt b3,id b1) = b4;
:: FINSEQOP:th 73
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3
holds (b4 [;](b2,id b1)) . the_unity_wrt b3 = the_unity_wrt b3;
:: FINSEQOP:th 74
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3
holds (b4 [:](id b1,b2)) . the_unity_wrt b3 = the_unity_wrt b3;
:: FINSEQOP:th 75
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1) & b4 is associative(b1) & b4 is having_an_inverseOp(b1)
holds b4 .:(b3,(the_inverseOp_wrt b4) * b3) = b2 --> the_unity_wrt b4 &
b4 .:((the_inverseOp_wrt b4) * b3,b3) = b2 --> the_unity_wrt b4;
:: FINSEQOP:th 76
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like quasi_total Relation of b2,b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is associative(b1) & b5 is having_an_inverseOp(b1) & b5 is having_a_unity(b1) & b5 .:(b3,b4) = b2 --> the_unity_wrt b5
holds b3 = (the_inverseOp_wrt b5) * b4 & (the_inverseOp_wrt b5) * b3 = b4;
:: FINSEQOP:th 77
theorem
for b1 being non empty set
for b2 being natural set
for b3 being Element of b2 -tuples_on b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
st b4 is having_a_unity(b1) & b4 is associative(b1) & b4 is having_an_inverseOp(b1)
holds b4 .:(b3,(the_inverseOp_wrt b4) * b3) = b2 |-> the_unity_wrt b4 &
b4 .:((the_inverseOp_wrt b4) * b3,b3) = b2 |-> the_unity_wrt b4;
:: FINSEQOP:th 78
theorem
for b1 being non empty set
for b2 being natural set
for b3, b4 being Element of b2 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is associative(b1) & b5 is having_an_inverseOp(b1) & b5 is having_a_unity(b1) & b5 .:(b3,b4) = b2 |-> the_unity_wrt b5
holds b3 = (the_inverseOp_wrt b5) * b4 & (the_inverseOp_wrt b5) * b3 = b4;
:: FINSEQOP:th 79
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function-like quasi_total Relation of b2,b1
for b5, b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is associative(b1) & b5 is having_a_unity(b1) & b3 = the_unity_wrt b5 & b5 is having_an_inverseOp(b1) & b6 is_distributive_wrt b5
holds b6 [;](b3,b4) = b2 --> b3;
:: FINSEQOP:th 80
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4 being Element of b3 -tuples_on b1
for b5, b6 being Function-like quasi_total Relation of [:b1,b1:],b1
st b5 is associative(b1) & b5 is having_a_unity(b1) & b2 = the_unity_wrt b5 & b5 is having_an_inverseOp(b1) & b6 is_distributive_wrt b5
holds b6 [;](b2,b4) = b3 |-> b2;
:: FINSEQOP:funcnot 7 => FINSEQOP:func 7
definition
let a1, a2, a3 be Relation-like Function-like set;
func A1 *(A2,A3) -> Relation-like Function-like set equals
[:a2,a3:] * a1;
end;
:: FINSEQOP:def 4
theorem
for b1, b2, b3 being Relation-like Function-like set holds
b1 *(b2,b3) = [:b2,b3:] * b1;
:: FINSEQOP:th 82
theorem
for b1, b2 being set
for b3, b4, b5 being Relation-like Function-like set
st [b1,b2] in proj1 (b3 *(b4,b5))
holds (b3 *(b4,b5)) .(b1,b2) = b3 .(b4 . b1,b5 . b2);
:: FINSEQOP:th 83
theorem
for b1, b2 being set
for b3, b4, b5 being Relation-like Function-like set
st [b1,b2] in proj1 (b3 *(b4,b5))
holds (b3 *(b4,b5)) .(b1,b2) = b3 .(b4 . b1,b5 . b2);
:: FINSEQOP:th 84
theorem
for b1, b2, b3, b4, b5 being non empty set
for b6 being Function-like quasi_total Relation of [:b1,b2:],b3
for b7 being Function-like quasi_total Relation of b4,b1
for b8 being Function-like quasi_total Relation of b5,b2 holds
b6 *(b7,b8) is Function-like quasi_total Relation of [:b4,b5:],b3;
:: FINSEQOP:th 85
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3, b4 being Function-like quasi_total Relation of b1,b1 holds
b2 *(b3,b4) is Function-like quasi_total Relation of [:b1,b1:],b1;
:: FINSEQOP:funcnot 8 => FINSEQOP:func 8
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a3, a4 be Function-like quasi_total Relation of a1,a1;
redefine func a2 *(a3,a4) -> Function-like quasi_total Relation of [:a1,a1:],a1;
end;
:: FINSEQOP:th 86
theorem
for b1, b2, b3, b4, b5 being non empty set
for b6 being Element of b4
for b7 being Element of b5
for b8 being Function-like quasi_total Relation of [:b1,b2:],b3
for b9 being Function-like quasi_total Relation of b4,b1
for b10 being Function-like quasi_total Relation of b5,b2 holds
(b8 *(b9,b10)) .(b6,b7) = b8 .(b9 . b6,b10 . b7);
:: FINSEQOP:th 87
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being Function-like quasi_total Relation of b1,b1 holds
(b4 *(id b1,b5)) .(b2,b3) = b4 .(b2,b5 . b3) &
(b4 *(b5,id b1)) .(b2,b3) = b4 .(b5 . b2,b3);
:: FINSEQOP:th 88
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b2,b2 holds
(b5 *(id b2,b6)) .:(b3,b4) = b5 .:(b3,b6 * b4);
:: FINSEQOP:th 89
theorem
for b1 being non empty set
for b2 being natural set
for b3, b4 being Element of b2 -tuples_on b1
for b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being Function-like quasi_total Relation of b1,b1 holds
(b5 *(id b1,b6)) .:(b3,b4) = b5 .:(b3,b6 * b4);
:: FINSEQOP:th 90
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being Function-like quasi_total Relation of b1,b1
st b4 is associative(b1) & b4 is having_a_unity(b1) & b4 is commutative(b1) & b4 is having_an_inverseOp(b1) & b5 = the_inverseOp_wrt b4
holds b5 . ((b4 *(id b1,b5)) .(b2,b3)) = (b4 *(b5,id b1)) .(b2,b3) &
(b4 *(id b1,b5)) .(b2,b3) = b5 . ((b4 *(b5,id b1)) .(b2,b3));
:: FINSEQOP:th 91
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1)
holds (b3 *(id b1,the_inverseOp_wrt b3)) .(b2,b2) = the_unity_wrt b3;
:: FINSEQOP:th 92
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1)
holds (b3 *(id b1,the_inverseOp_wrt b3)) .(b2,the_unity_wrt b3) = b2;
:: FINSEQOP:th 93
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
for b4 being Function-like quasi_total Relation of b1,b1
st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 = the_inverseOp_wrt b3
holds (b3 *(id b1,b4)) .(the_unity_wrt b3,b2) = b4 . b2;
:: FINSEQOP:th 94
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of [:b1,b1:],b1
st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1) & b2 is having_an_inverseOp(b1) & b3 = b2 *(id b1,the_inverseOp_wrt b2)
for b4, b5, b6, b7 being Element of b1 holds
b2 .(b3 .(b4,b5),b3 .(b6,b7)) = b3 .(b2 .(b4,b6),b2 .(b5,b7));