Article ALGSTR_2, MML version 4.99.1005
:: ALGSTR_2:funcreg 1
registration
cluster F_Real -> strict multLoop_0-like;
end;
:: ALGSTR_2:funcnot 1 => ALGSTR_2:func 1
definition
let a1 be non empty left_add-cancelable add-right-invertible addLoopStr;
let a2 be Element of the carrier of a1;
func - A2 -> Element of the carrier of a1 means
a2 + it = 0. a1;
end;
:: ALGSTR_2:def 7
theorem
for b1 being non empty left_add-cancelable add-right-invertible addLoopStr
for b2, b3 being Element of the carrier of b1 holds
b3 = - b2
iff
b2 + b3 = 0. b1;
:: ALGSTR_2:funcnot 2 => ALGSTR_2:func 2
definition
let a1 be non empty left_add-cancelable add-right-invertible addLoopStr;
let a2, a3 be Element of the carrier of a1;
func A2 - A3 -> Element of the carrier of a1 equals
a2 + - a3;
end;
:: ALGSTR_2:def 8
theorem
for b1 being non empty left_add-cancelable add-right-invertible addLoopStr
for b2, b3 being Element of the carrier of b1 holds
b2 - b3 = b2 + - b3;
:: ALGSTR_2:exreg 1
registration
cluster non empty non degenerated strict left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
end;
:: ALGSTR_2:modenot 1
definition
mode doubleLoop is non empty left_zeroed Loop-like multLoop_0-like right_zeroed well-unital doubleLoopStr;
end;
:: ALGSTR_2:modenot 2
definition
mode leftQuasi-Field is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed right-distributive well-unital doubleLoopStr;
end;
:: ALGSTR_2:th 12
theorem
for b1 being non empty doubleLoopStr holds
b1 is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed right-distributive well-unital doubleLoopStr
iff
(for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 + b3 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2) &
0. b1 <> 1. b1 &
(for b2 being Element of the carrier of b1 holds
b2 * 1. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
(1. b1) * b2 = b2) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b2 * b4 = b3) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b4 * b2 = b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b2 * b3 = b2 * b4
holds b3 = b4) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b3 * b2 = b4 * b2
holds b3 = b4) &
(for b2 being Element of the carrier of b1 holds
b2 * 0. b1 = 0. b1) &
(for b2 being Element of the carrier of b1 holds
(0. b1) * b2 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4));
:: ALGSTR_2:th 14
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like Abelian right_zeroed right-distributive well-unital doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
b2 * - b3 = - (b2 * b3);
:: ALGSTR_2:th 15
theorem
for b1 being non empty left_add-cancelable add-right-invertible Abelian addLoopStr
for b2 being Element of the carrier of b1 holds
- - b2 = b2;
:: ALGSTR_2:th 16
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like Abelian right_zeroed right-distributive well-unital doubleLoopStr holds
(- 1. b1) * - 1. b1 = 1. b1;
:: ALGSTR_2:th 17
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like Abelian right_zeroed right-distributive well-unital doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 - b4) = (b2 * b3) - (b2 * b4);
:: ALGSTR_2:modenot 3
definition
mode rightQuasi-Field is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed left-distributive well-unital doubleLoopStr;
end;
:: ALGSTR_2:th 19
theorem
for b1 being non empty doubleLoopStr holds
b1 is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed left-distributive well-unital doubleLoopStr
iff
(for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 + b3 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2) &
0. b1 <> 1. b1 &
(for b2 being Element of the carrier of b1 holds
b2 * 1. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
(1. b1) * b2 = b2) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b2 * b4 = b3) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b4 * b2 = b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b2 * b3 = b2 * b4
holds b3 = b4) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b3 * b2 = b4 * b2
holds b3 = b4) &
(for b2 being Element of the carrier of b1 holds
b2 * 0. b1 = 0. b1) &
(for b2 being Element of the carrier of b1 holds
(0. b1) * b2 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2));
:: ALGSTR_2:th 21
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like right_zeroed left-distributive well-unital doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
(- b2) * b3 = - (b2 * b3);
:: ALGSTR_2:th 23
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like Abelian right_zeroed left-distributive well-unital doubleLoopStr holds
(- 1. b1) * - 1. b1 = 1. b1;
:: ALGSTR_2:th 24
theorem
for b1 being non empty left_zeroed Loop-like multLoop_0-like right_zeroed left-distributive well-unital doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 - b3) * b4 = (b2 * b4) - (b3 * b4);
:: ALGSTR_2:modenot 4
definition
mode doublesidedQuasi-Field is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
end;
:: ALGSTR_2:th 26
theorem
for b1 being non empty doubleLoopStr holds
b1 is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
iff
(for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 + b3 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2) &
0. b1 <> 1. b1 &
(for b2 being Element of the carrier of b1 holds
b2 * 1. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
(1. b1) * b2 = b2) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b2 * b4 = b3) &
(for b2, b3 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b4 being Element of the carrier of b1 st
b4 * b2 = b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b2 * b3 = b2 * b4
holds b3 = b4) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> 0. b1 & b3 * b2 = b4 * b2
holds b3 = b4) &
(for b2 being Element of the carrier of b1 holds
b2 * 0. b1 = 0. b1) &
(for b2 being Element of the carrier of b1 holds
(0. b1) * b2 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2));
:: ALGSTR_2:modenot 5
definition
mode _Skew-Field is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
end;
:: ALGSTR_2:th 32
theorem
for b1 being non empty doubleLoopStr holds
b1 is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
iff
(for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 + b3 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2) &
0. b1 <> 1. b1 &
(for b2 being Element of the carrier of b1 holds
b2 * 1. b1 = b2) &
(for b2 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b3 being Element of the carrier of b1 st
b2 * b3 = 1. b1) &
(for b2 being Element of the carrier of b1 holds
b2 * 0. b1 = 0. b1) &
(for b2 being Element of the carrier of b1 holds
(0. b1) * b2 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4)) &
(for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2));
:: ALGSTR_2:modenot 6
definition
mode _Field is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
end;
:: ALGSTR_2:th 34
theorem
for b1 being non empty doubleLoopStr holds
b1 is non empty non degenerated left_zeroed Loop-like multLoop_0-like Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
iff
(for b2 being Element of the carrier of b1 holds
b2 + 0. b1 = b2) &
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 + b3 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 + b3 = b3 + b2) &
0. b1 <> 1. b1 &
(for b2 being Element of the carrier of b1 holds
b2 * 1. b1 = b2) &
(for b2 being Element of the carrier of b1
st b2 <> 0. b1
holds ex b3 being Element of the carrier of b1 st
b2 * b3 = 1. b1) &
(for b2 being Element of the carrier of b1 holds
b2 * 0. b1 = 0. b1) &
(for b2, b3, b4 being Element of the carrier of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4)) &
(for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
(for b2, b3 being Element of the carrier of b1 holds
b2 * b3 = b3 * b2);