Article ORTSP_1, MML version 4.99.1005
:: ORTSP_1:attrnot 1 => ORTSP_1:attr 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed SymStr over a1;
attr a2 is OrtSp-like means
for b1, b2, b3, b4, b5 being Element of the carrier of a2
for b6 being Element of the carrier of a1 holds
(b1 <> 0. a2 & b2 <> 0. a2 & b3 <> 0. a2 & b4 <> 0. a2 implies ex b7 being Element of the carrier of a2 st
not b7 <= b1 & not b7 <= b2 & not b7 <= b3 & not b7 <= b4) &
(b1 <= b2 implies b6 * b1 <= b2) &
(b2 <= b1 & b3 <= b1 implies b2 + b3 <= b1) &
(not b2 <= b1 implies ex b7 being Element of the carrier of a1 st
b5 - (b7 * b2) <= b1) &
(b1 <= b2 - b3 & b2 <= b3 - b1 implies b3 <= b1 - b2);
end;
:: ORTSP_1:dfs 1
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed SymStr over a1;
To prove
a2 is OrtSp-like
it is sufficient to prove
thus for b1, b2, b3, b4, b5 being Element of the carrier of a2
for b6 being Element of the carrier of a1 holds
(b1 <> 0. a2 & b2 <> 0. a2 & b3 <> 0. a2 & b4 <> 0. a2 implies ex b7 being Element of the carrier of a2 st
not b7 <= b1 & not b7 <= b2 & not b7 <= b3 & not b7 <= b4) &
(b1 <= b2 implies b6 * b1 <= b2) &
(b2 <= b1 & b3 <= b1 implies b2 + b3 <= b1) &
(not b2 <= b1 implies ex b7 being Element of the carrier of a1 st
b5 - (b7 * b2) <= b1) &
(b1 <= b2 - b3 & b2 <= b3 - b1 implies b3 <= b1 - b2);
:: ORTSP_1:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed SymStr over b1 holds
b2 is OrtSp-like(b1)
iff
for b3, b4, b5, b6, b7 being Element of the carrier of b2
for b8 being Element of the carrier of b1 holds
(b3 <> 0. b2 & b4 <> 0. b2 & b5 <> 0. b2 & b6 <> 0. b2 implies ex b9 being Element of the carrier of b2 st
not b9 <= b3 & not b9 <= b4 & not b9 <= b5 & not b9 <= b6) &
(b3 <= b4 implies b8 * b3 <= b4) &
(b4 <= b3 & b5 <= b3 implies b4 + b5 <= b3) &
(not b4 <= b3 implies ex b9 being Element of the carrier of b1 st
b7 - (b9 * b4) <= b3) &
(b3 <= b4 - b5 & b4 <= b5 - b3 implies b5 <= b3 - b4);
:: ORTSP_1:exreg 1
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
cluster non empty right_complementable Abelian add-associative right_zeroed VectSp-like strict OrtSp-like SymStr over a1;
end;
:: ORTSP_1:modenot 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
mode OrtSp of a1 is non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
end;
:: ORTSP_1:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3 being Element of the carrier of b2 holds
0. b2 <= b3;
:: ORTSP_1:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
st b3 <= b4
holds b4 <= b3;
:: ORTSP_1:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & b5 + b3 <= b4
holds not b5 <= b4;
:: ORTSP_1:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & b5 <= b4
holds not b3 + b5 <= b4;
:: ORTSP_1:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Element of the carrier of b1
st not b3 <= b4 & b5 <> 0. b1
holds not b5 * b3 <= b4 & not b3 <= b5 * b4;
:: ORTSP_1:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
st b3 <= b4
holds - b3 <= b4;
:: ORTSP_1:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st b3 - b4 <= b5 & b3 - b6 <= b5
holds b4 - b6 <= b5;
:: ORTSP_1:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b1
st not b3 <= b4 & b5 - (b6 * b3) <= b4 & b5 - (b7 * b3) <= b4
holds b6 = b7;
:: ORTSP_1:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
st b3 <= b3 & b4 <= b4
holds b3 + b4 <= b3 - b4;
:: ORTSP_1:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
st (1_ b1) + 1_ b1 <> 0. b1 &
(ex b3 being Element of the carrier of b2 st
b3 <> 0. b2)
holds ex b3 being Element of the carrier of b2 st
not b3 <= b3;
:: ORTSP_1:funcnot 1 => ORTSP_1:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
let a3, a4, a5 be Element of the carrier of a2;
assume not a4 <= a3;
func ProJ(A3,A4,A5) -> Element of the carrier of a1 means
for b1 being Element of the carrier of a1
st a5 - (b1 * a4) <= a3
holds it = b1;
end;
:: ORTSP_1:def 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b4 <= b3
for b6 being Element of the carrier of b1 holds
b6 = ProJ(b3,b4,b5)
iff
for b7 being Element of the carrier of b1
st b5 - (b7 * b4) <= b3
holds b6 = b7;
:: ORTSP_1:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4
holds b5 - ((ProJ(b4,b3,b5)) * b3) <= b4;
:: ORTSP_1:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st not b3 <= b4
holds ProJ(b4,b3,b6 * b5) = b6 * ProJ(b4,b3,b5);
:: ORTSP_1:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4
holds ProJ(b4,b3,b5 + b6) = (ProJ(b4,b3,b5)) + ProJ(b4,b3,b6);
:: ORTSP_1:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st not b3 <= b4 & b6 <> 0. b1
holds ProJ(b4,b6 * b3,b5) = b6 " * ProJ(b4,b3,b5);
:: ORTSP_1:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
st not b3 <= b4 & b6 <> 0. b1
holds ProJ(b6 * b4,b3,b5) = ProJ(b4,b3,b5);
:: ORTSP_1:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & b5 <= b4
holds ProJ(b4,b3 + b5,b6) = ProJ(b4,b3,b6) & ProJ(b4,b3,b6 + b5) = ProJ(b4,b3,b6);
:: ORTSP_1:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & b5 <= b3 & b5 <= b6
holds ProJ(b4 + b5,b3,b6) = ProJ(b4,b3,b6);
:: ORTSP_1:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & b5 - b3 <= b4
holds ProJ(b4,b3,b5) = 1_ b1;
:: ORTSP_1:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
st not b3 <= b4
holds ProJ(b4,b3,b3) = 1_ b1;
:: ORTSP_1:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4
holds b5 <= b4
iff
ProJ(b4,b3,b5) = 0. b1;
:: ORTSP_1:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & not b5 <= b4
holds (ProJ(b4,b3,b6)) * ((ProJ(b4,b3,b5)) ") = ProJ(b4,b5,b6);
:: ORTSP_1:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & not b5 <= b4
holds ProJ(b4,b3,b5) = (ProJ(b4,b5,b3)) ";
:: ORTSP_1:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & b3 <= b5 + b4
holds ProJ(b4,b3,b5) = - ProJ(b5,b3,b4);
:: ORTSP_1:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
st not b3 <= b4 & not b5 <= b4
holds ProJ(b5,b4,b3) = (ProJ(b4,b3,b5)) " * ProJ(b3,b4,b5);
:: ORTSP_1:th 38
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & not b3 <= b5 & not b6 <= b4 & not b6 <= b5
holds (ProJ(b4,b6,b3)) * ProJ(b3,b4,b5) = (ProJ(b6,b4,b5)) * ProJ(b5,b6,b3);
:: ORTSP_1:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of b2
st not b3 <= b4 & not b3 <= b5 & not b6 <= b4 & not b6 <= b5 & not b7 <= b4
holds ((ProJ(b4,b7,b3)) * ProJ(b3,b4,b5)) * ProJ(b5,b3,b8) = ((ProJ(b4,b7,b6)) * ProJ(b6,b4,b5)) * ProJ(b5,b6,b8);
:: ORTSP_1:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & not b5 <= b4 & not b6 <= b4
holds (ProJ(b4,b3,b5)) * ProJ(b5,b4,b6) = (ProJ(b4,b3,b6)) * ProJ(b6,b4,b5);
:: ORTSP_1:funcnot 2 => ORTSP_1:func 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
let a3, a4, a5, a6 be Element of the carrier of a2;
assume not a6 <= a5;
func PProJ(A5,A6,A3,A4) -> Element of the carrier of a1 means
for b1 being Element of the carrier of a2
st not b1 <= a5 & not b1 <= a3
holds it = ((ProJ(a5,a6,b1)) * ProJ(b1,a5,a3)) * ProJ(a3,b1,a4)
if ex b1 being Element of the carrier of a2 st
not b1 <= a5 & not b1 <= a3
otherwise case for b1 being Element of the carrier of a2
st not b1 <= a5
holds b1 <= a3;
thus it = 0. a1;
end;
;
end;
:: ORTSP_1:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b6 <= b5
for b7 being Element of the carrier of b1 holds
(for b8 being Element of the carrier of b2
st not b8 <= b5
holds b8 <= b3 or (b7 = PProJ(b5,b6,b3,b4)
iff
for b8 being Element of the carrier of b2
st not b8 <= b5 & not b8 <= b3
holds b7 = ((ProJ(b5,b6,b8)) * ProJ(b8,b5,b3)) * ProJ(b3,b8,b4))) &
(for b8 being Element of the carrier of b2
st not b8 <= b5
holds b8 <= b3 implies (b7 = PProJ(b5,b6,b3,b4)
iff
b7 = 0. b1));
:: ORTSP_1:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4 & b5 = 0. b2
holds PProJ(b4,b3,b5,b6) = 0. b1;
:: ORTSP_1:th 44
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4
holds PProJ(b4,b3,b5,b6) = 0. b1
iff
b6 <= b5;
:: ORTSP_1:th 45
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
st not b3 <= b4
holds PProJ(b4,b3,b5,b6) = PProJ(b4,b3,b6,b5);
:: ORTSP_1:th 46
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
for b7 being Element of the carrier of b1
st not b3 <= b4
holds PProJ(b4,b3,b5,b7 * b6) = b7 * PProJ(b4,b3,b5,b6);
:: ORTSP_1:th 47
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6, b7 being Element of the carrier of b2
st not b3 <= b4
holds PProJ(b4,b3,b5,b6 + b7) = (PProJ(b4,b3,b5,b6)) + PProJ(b4,b3,b5,b7);