Article ANPROJ_1, MML version 4.99.1005
:: ANPROJ_1:prednot 1 => RLVECT_1:attr 6
notation
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of the carrier of a1;
synonym a2 is_Prop_Vect for non-zero;
end;
:: ANPROJ_1:prednot 2 => ANPROJ_1:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
pred are_Prop A2,A3 means
ex b1, b2 being Element of REAL st
b1 * a2 = b2 * a3 & b1 <> 0 & b2 <> 0;
symmetry;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
:: for a2, a3 being Element of the carrier of a1
:: st are_Prop a2,a3
:: holds are_Prop a3,a2;
reflexivity;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
:: for a2 being Element of the carrier of a1 holds
:: are_Prop a2,a2;
end;
:: ANPROJ_1:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
To prove
are_Prop a2,a3
it is sufficient to prove
thus ex b1, b2 being Element of REAL st
b1 * a2 = b2 * a3 & b1 <> 0 & b2 <> 0;
:: ANPROJ_1:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
are_Prop b2,b3
iff
ex b4, b5 being Element of REAL st
b4 * b2 = b5 * b3 & b4 <> 0 & b5 <> 0;
:: ANPROJ_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
are_Prop b2,b3
iff
ex b4 being Element of REAL st
b4 <> 0 & b2 = b4 * b3;
:: ANPROJ_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st are_Prop b2,b3 & are_Prop b3,b4
holds are_Prop b2,b4;
:: ANPROJ_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
are_Prop b2,0. b1
iff
b2 = 0. b1;
:: ANPROJ_1:prednot 3 => ANPROJ_1:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
pred A2,A3,A4 are_LinDep means
ex b1, b2, b3 being Element of REAL st
((b1 * a2) + (b2 * a3)) + (b3 * a4) = 0. a1 &
(b1 = 0 & b2 = 0 implies b3 <> 0);
end;
:: ANPROJ_1:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
To prove
a2,a3,a4 are_LinDep
it is sufficient to prove
thus ex b1, b2, b3 being Element of REAL st
((b1 * a2) + (b2 * a3)) + (b3 * a4) = 0. a1 &
(b1 = 0 & b2 = 0 implies b3 <> 0);
:: ANPROJ_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3,b4 are_LinDep
iff
ex b5, b6, b7 being Element of REAL st
((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1 &
(b5 = 0 & b6 = 0 implies b7 <> 0);
:: ANPROJ_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st are_Prop b2,b3 & are_Prop b4,b5 & are_Prop b6,b7 & b2,b4,b6 are_LinDep
holds b3,b5,b7 are_LinDep;
:: ANPROJ_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 are_LinDep
holds b2,b4,b3 are_LinDep & b3,b2,b4 are_LinDep & b4,b3,b2 are_LinDep & b4,b2,b3 are_LinDep & b3,b4,b2 are_LinDep;
:: ANPROJ_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b2,b3
holds b2,b3,b4 are_LinDep
iff
ex b5, b6 being Element of REAL st
b4 = (b5 * b2) + (b6 * b3);
:: ANPROJ_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4, b5, b6, b7 being Element of REAL
st not are_Prop b2,b3 &
(b4 * b2) + (b5 * b3) = (b6 * b2) + (b7 * b3) &
b2 is non-zero(b1) &
b3 is non-zero(b1)
holds b4 = b6 & b5 = b7;
:: ANPROJ_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6, b7, b8, b9, b10 being Element of REAL
st not b2,b3,b4 are_LinDep &
((b5 * b2) + (b6 * b3)) + (b7 * b4) = ((b8 * b2) + (b9 * b3)) + (b10 * b4)
holds b5 = b8 & b6 = b9 & b7 = b10;
:: ANPROJ_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of REAL
st not are_Prop b2,b3 & b4 = (b6 * b2) + (b7 * b3) & b5 = (b8 * b2) + (b9 * b3) & (b6 * b9) - (b8 * b7) = 0 & b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b4,b5 & b4 <> 0. b1
holds b5 = 0. b1;
:: ANPROJ_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st (b2 <> 0. b1 & b3 <> 0. b1 implies b4 = 0. b1)
holds b2,b3,b4 are_LinDep;
:: ANPROJ_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st (not are_Prop b2,b3 & not are_Prop b4,b2 implies are_Prop b3,b4)
holds b4,b2,b3 are_LinDep;
:: ANPROJ_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st not b2,b3,b4 are_LinDep
holds b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1) & not are_Prop b2,b3 & not are_Prop b3,b4 & not are_Prop b4,b2;
:: ANPROJ_1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st b2 + b3 = 0. b1
holds are_Prop b2,b3;
:: ANPROJ_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not are_Prop b2,b3 & b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b2,b3,b6 are_LinDep & b2 is non-zero(b1) & b3 is non-zero(b1)
holds b4,b5,b6 are_LinDep;
:: ANPROJ_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b3,b4,b6 are_LinDep
holds ex b7 being Element of the carrier of b1 st
b2,b4,b7 are_LinDep & b5,b6,b7 are_LinDep & b7 is non-zero(b1);
:: ANPROJ_1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st not are_Prop b2,b3 & b2 is non-zero(b1) & b3 is non-zero(b1)
for b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
b6 is non-zero(b1) & b4,b5,b6 are_LinDep & not are_Prop b4,b6 & not are_Prop b5,b6;
:: ANPROJ_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st not b2,b3,b4 are_LinDep
for b5, b6 being Element of the carrier of b1
st b5 is non-zero(b1) & b6 is non-zero(b1) & not are_Prop b5,b6
holds ex b7 being Element of the carrier of b1 st
b7 is non-zero(b1) & not b5,b6,b7 are_LinDep;
:: ANPROJ_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st b2,b3,b4 are_LinDep & b5,b6,b4 are_LinDep & b2,b5,b7 are_LinDep & b3,b6,b7 are_LinDep & b2,b6,b8 are_LinDep & b3,b5,b8 are_LinDep & b7,b4,b8 are_LinDep & b7 is non-zero(b1) & b4 is non-zero(b1) & b8 is non-zero(b1) & not b2,b3,b6 are_LinDep & not b2,b3,b5 are_LinDep & not b2,b5,b6 are_LinDep
holds b3,b5,b6 are_LinDep;
:: ANPROJ_1:funcnot 1 => ANPROJ_1:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func Proper_Vectors_of A1 -> set means
for b1 being set holds
b1 in it
iff
b1 <> 0. a1 & b1 is Element of the carrier of a1;
end;
:: ANPROJ_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
b2 = Proper_Vectors_of b1
iff
for b3 being set holds
b3 in b2
iff
b3 <> 0. b1 & b3 is Element of the carrier of b1;
:: ANPROJ_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
b2 in Proper_Vectors_of b1
iff
b2 is non-zero(b1);
:: ANPROJ_1:funcnot 2 => ANPROJ_1:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func Proportionality_as_EqRel_of A1 -> symmetric transitive total Relation of Proper_Vectors_of a1,Proper_Vectors_of a1 means
for b1, b2 being set holds
[b1,b2] in it
iff
b1 in Proper_Vectors_of a1 &
b2 in Proper_Vectors_of a1 &
(ex b3, b4 being Element of the carrier of a1 st
b1 = b3 & b2 = b4 & are_Prop b3,b4);
end;
:: ANPROJ_1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being symmetric transitive total Relation of Proper_Vectors_of b1,Proper_Vectors_of b1 holds
b2 = Proportionality_as_EqRel_of b1
iff
for b3, b4 being set holds
[b3,b4] in b2
iff
b3 in Proper_Vectors_of b1 &
b4 in Proper_Vectors_of b1 &
(ex b5, b6 being Element of the carrier of b1 st
b3 = b5 & b4 = b6 & are_Prop b5,b6);
:: ANPROJ_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being set
st [b2,b3] in Proportionality_as_EqRel_of b1
holds b2 is Element of the carrier of b1 & b3 is Element of the carrier of b1;
:: ANPROJ_1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
[b2,b3] in Proportionality_as_EqRel_of b1
iff
b2 is non-zero(b1) & b3 is non-zero(b1) & are_Prop b2,b3;
:: ANPROJ_1:funcnot 3 => ANPROJ_1:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of the carrier of a1;
func Dir A2 -> Element of bool Proper_Vectors_of a1 equals
Class(Proportionality_as_EqRel_of a1,a2);
end;
:: ANPROJ_1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
Dir b2 = Class(Proportionality_as_EqRel_of b1,b2);
:: ANPROJ_1:funcnot 4 => ANPROJ_1:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func ProjectivePoints A1 -> set means
ex b1 being Element of bool bool Proper_Vectors_of a1 st
b1 = Class Proportionality_as_EqRel_of a1 & it = b1;
end;
:: ANPROJ_1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
b2 = ProjectivePoints b1
iff
ex b3 being Element of bool bool Proper_Vectors_of b1 st
b3 = Class Proportionality_as_EqRel_of b1 & b2 = b3;
:: ANPROJ_1:attrnot 1 => STRUCT_0:attr 7
definition
let a1 be 1-sorted;
attr a1 is trivial means
for b1 being Element of the carrier of a1 holds
b1 = 0. a1;
end;
:: ANPROJ_1:dfs 7
definiens
let a1 be non empty ZeroStr;
To prove
a1 is trivial
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 = 0. a1;
:: ANPROJ_1:def 8
theorem
for b1 being non empty ZeroStr holds
b1 is trivial
iff
for b2 being Element of the carrier of b1 holds
b2 = 0. b1;
:: ANPROJ_1:exreg 1
registration
cluster non empty non trivial left_complementable right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like zeroed RLSStruct;
end;
:: ANPROJ_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
b1 is non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
iff
ex b2 being Element of the carrier of b1 st
b2 in Proper_Vectors_of b1;
:: ANPROJ_1:funcreg 1
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster Proper_Vectors_of a1 -> non empty;
end;
:: ANPROJ_1:funcreg 2
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster ProjectivePoints a1 -> non empty;
end;
:: ANPROJ_1:th 34
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
st b2 is non-zero(b1)
holds Dir b2 is Element of ProjectivePoints b1;
:: ANPROJ_1:th 35
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st b2 is non-zero(b1) & b3 is non-zero(b1)
holds Dir b2 = Dir b3
iff
are_Prop b2,b3;
:: ANPROJ_1:funcnot 5 => ANPROJ_1:func 5
definition
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func ProjectiveCollinearity A1 -> Relation3 of ProjectivePoints a1 means
for b1, b2, b3 being set holds
[b1,b2,b3] in it
iff
ex b4, b5, b6 being Element of the carrier of a1 st
b1 = Dir b4 & b2 = Dir b5 & b3 = Dir b6 & b4 is non-zero(a1) & b5 is non-zero(a1) & b6 is non-zero(a1) & b4,b5,b6 are_LinDep;
end;
:: ANPROJ_1:def 9
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Relation3 of ProjectivePoints b1 holds
b2 = ProjectiveCollinearity b1
iff
for b3, b4, b5 being set holds
[b3,b4,b5] in b2
iff
ex b6, b7, b8 being Element of the carrier of b1 st
b3 = Dir b6 & b4 = Dir b7 & b5 = Dir b8 & b6 is non-zero(b1) & b7 is non-zero(b1) & b8 is non-zero(b1) & b6,b7,b8 are_LinDep;
:: ANPROJ_1:funcnot 6 => ANPROJ_1:func 6
definition
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func ProjectiveSpace A1 -> strict CollStr equals
CollStr(#ProjectivePoints a1,ProjectiveCollinearity a1#);
end;
:: ANPROJ_1:def 10
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
ProjectiveSpace b1 = CollStr(#ProjectivePoints b1,ProjectiveCollinearity b1#);
:: ANPROJ_1:funcreg 3
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster ProjectiveSpace a1 -> non empty strict;
end;
:: ANPROJ_1:th 39
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the carrier of ProjectiveSpace b1 = ProjectivePoints b1 & the Collinearity of ProjectiveSpace b1 = ProjectiveCollinearity b1;
:: ANPROJ_1:th 40
theorem
for b1, b2, b3 being set
for b4 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st [b1,b2,b3] in the Collinearity of ProjectiveSpace b4
holds ex b5, b6, b7 being Element of the carrier of b4 st
b1 = Dir b5 & b2 = Dir b6 & b3 = Dir b7 & b5 is non-zero(b4) & b6 is non-zero(b4) & b7 is non-zero(b4) & b5,b6,b7 are_LinDep;
:: ANPROJ_1:th 41
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1)
holds [Dir b2,Dir b3,Dir b4] in the Collinearity of ProjectiveSpace b1
iff
b2,b3,b4 are_LinDep;
:: ANPROJ_1:th 42
theorem
for b1 being set
for b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
b1 is Element of the carrier of ProjectiveSpace b2
iff
ex b3 being Element of the carrier of b2 st
b3 is non-zero(b2) & b1 = Dir b3;