Article MULTOP_1, MML version 4.99.1005
:: MULTOP_1:funcnot 1 => MULTOP_1:func 1
definition
let a1 be Relation-like Function-like set;
let a2, a3, a4 be set;
func A1 .(A2,A3,A4) -> set equals
a1 . [a2,a3,a4];
end;
:: MULTOP_1:def 1
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4 being set holds
b1 .(b2,b3,b4) = b1 . [b2,b3,b4];
:: MULTOP_1:funcnot 2 => MULTOP_1:func 2
definition
let a1, a2, a3, a4 be non empty set;
let a5 be Function-like quasi_total Relation of [:a1,a2,a3:],a4;
let a6 be Element of a1;
let a7 be Element of a2;
let a8 be Element of a3;
redefine func a5 .(a6,a7,a8) -> Element of a4;
end;
:: MULTOP_1:th 2
theorem
for b1 being non empty set
for b2, b3, b4 being set
for b5, b6 being Function-like quasi_total Relation of [:b2,b3,b4:],b1
st for b7, b8, b9 being set
st b7 in b2 & b8 in b3 & b9 in b4
holds b5 . [b7,b8,b9] = b6 . [b7,b8,b9]
holds b5 = b6;
:: MULTOP_1:th 3
theorem
for b1, b2, b3, b4 being non empty set
for b5, b6 being Function-like quasi_total Relation of [:b1,b2,b3:],b4
st for b7 being Element of b1
for b8 being Element of b2
for b9 being Element of b3 holds
b5 . [b7,b8,b9] = b6 . [b7,b8,b9]
holds b5 = b6;
:: MULTOP_1:th 4
theorem
for b1, b2, b3, b4 being non empty set
for b5, b6 being Function-like quasi_total Relation of [:b1,b2,b3:],b4
st for b7 being Element of b1
for b8 being Element of b2
for b9 being Element of b3 holds
b5 .(b7,b8,b9) = b6 .(b7,b8,b9)
holds b5 = b6;
:: MULTOP_1:modenot 1
definition
let a1 be set;
mode TriOp of a1 is Function-like quasi_total Relation of [:a1,a1,a1:],a1;
end;
:: MULTOP_1:sch 1
scheme MULTOP_1:sch 1
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> non empty set}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2(),F3():],F4() st
for b2 being Element of F1()
for b3 being Element of F2()
for b4 being Element of F3() holds
P1[b2, b3, b4, b1 . [b2,b3,b4]]
provided
for b1 being Element of F1()
for b2 being Element of F2()
for b3 being Element of F3() holds
ex b4 being Element of F4() st
P1[b1, b2, b3, b4];
:: MULTOP_1:sch 2
scheme MULTOP_1:sch 2
{F1 -> non empty set}:
ex b1 being Function-like quasi_total Relation of [:F1(),F1(),F1():],F1() st
for b2, b3, b4 being Element of F1() holds
P1[b2, b3, b4, b1 .(b2,b3,b4)]
provided
for b1, b2, b3 being Element of F1() holds
ex b4 being Element of F1() st
P1[b1, b2, b3, b4];
:: MULTOP_1:sch 3
scheme MULTOP_1:sch 3
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> non empty set,
F5 -> Element of F4()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2(),F3():],F4() st
for b2 being Element of F1()
for b3 being Element of F2()
for b4 being Element of F3() holds
b1 . [b2,b3,b4] = F5(b2, b3, b4)
:: MULTOP_1:sch 4
scheme MULTOP_1:sch 4
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> non empty set,
F5 -> Element of F4()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2(),F3():],F4() st
for b2 being Element of F1()
for b3 being Element of F2()
for b4 being Element of F3() holds
b1 .(b2,b3,b4) = F5(b2, b3, b4)
:: MULTOP_1:funcnot 3 => MULTOP_1:func 3
definition
let a1 be Relation-like Function-like set;
let a2, a3, a4, a5 be set;
func A1 .(A2,A3,A4,A5) -> set equals
a1 . [a2,a3,a4,a5];
end;
:: MULTOP_1:def 2
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4, b5 being set holds
b1 .(b2,b3,b4,b5) = b1 . [b2,b3,b4,b5];
:: MULTOP_1:funcnot 4 => MULTOP_1:func 4
definition
let a1, a2, a3, a4, a5 be non empty set;
let a6 be Function-like quasi_total Relation of [:a1,a2,a3,a4:],a5;
let a7 be Element of a1;
let a8 be Element of a2;
let a9 be Element of a3;
let a10 be Element of a4;
redefine func a6 .(a7,a8,a9,a10) -> Element of a5;
end;
:: MULTOP_1:th 6
theorem
for b1 being non empty set
for b2, b3, b4, b5 being set
for b6, b7 being Function-like quasi_total Relation of [:b2,b3,b4,b5:],b1
st for b8, b9, b10, b11 being set
st b8 in b2 & b9 in b3 & b10 in b4 & b11 in b5
holds b6 . [b8,b9,b10,b11] = b7 . [b8,b9,b10,b11]
holds b6 = b7;
:: MULTOP_1:th 7
theorem
for b1, b2, b3, b4, b5 being non empty set
for b6, b7 being Function-like quasi_total Relation of [:b1,b2,b3,b4:],b5
st for b8 being Element of b1
for b9 being Element of b2
for b10 being Element of b3
for b11 being Element of b4 holds
b6 . [b8,b9,b10,b11] = b7 . [b8,b9,b10,b11]
holds b6 = b7;
:: MULTOP_1:th 8
theorem
for b1, b2, b3, b4, b5 being non empty set
for b6, b7 being Function-like quasi_total Relation of [:b1,b2,b3,b4:],b5
st for b8 being Element of b1
for b9 being Element of b2
for b10 being Element of b3
for b11 being Element of b4 holds
b6 .(b8,b9,b10,b11) = b7 .(b8,b9,b10,b11)
holds b6 = b7;
:: MULTOP_1:modenot 2
definition
let a1 be non empty set;
mode QuaOp of a1 is Function-like quasi_total Relation of [:a1,a1,a1,a1:],a1;
end;
:: MULTOP_1:sch 5
scheme MULTOP_1:sch 5
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> non empty set,
F5 -> non empty set}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2(),F3(),F4():],F5() st
for b2 being Element of F1()
for b3 being Element of F2()
for b4 being Element of F3()
for b5 being Element of F4() holds
P1[b2, b3, b4, b5, b1 . [b2,b3,b4,b5]]
provided
for b1 being Element of F1()
for b2 being Element of F2()
for b3 being Element of F3()
for b4 being Element of F4() holds
ex b5 being Element of F5() st
P1[b1, b2, b3, b4, b5];
:: MULTOP_1:sch 6
scheme MULTOP_1:sch 6
{F1 -> non empty set}:
ex b1 being Function-like quasi_total Relation of [:F1(),F1(),F1(),F1():],F1() st
for b2, b3, b4, b5 being Element of F1() holds
P1[b2, b3, b4, b5, b1 .(b2,b3,b4,b5)]
provided
for b1, b2, b3, b4 being Element of F1() holds
ex b5 being Element of F1() st
P1[b1, b2, b3, b4, b5];
:: MULTOP_1:sch 7
scheme MULTOP_1:sch 7
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> non empty set,
F5 -> non empty set,
F6 -> Element of F5()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2(),F3(),F4():],F5() st
for b2 being Element of F1()
for b3 being Element of F2()
for b4 being Element of F3()
for b5 being Element of F4() holds
b1 . [b2,b3,b4,b5] = F6(b2, b3, b4, b5)
:: MULTOP_1:sch 8
scheme MULTOP_1:sch 8
{F1 -> non empty set,
F2 -> Element of F1()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F1(),F1(),F1():],F1() st
for b2, b3, b4, b5 being Element of F1() holds
b1 .(b2,b3,b4,b5) = F2(b2, b3, b4, b5)