Article VECTSP_8, MML version 4.99.1005
:: VECTSP_8:funcnot 1 => VECTSP_8:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func lattice A2 -> non empty strict Lattice-like bounded LattStr equals
LattStr(#Subspaces a2,SubJoin a2,SubMeet a2#);
end;
:: VECTSP_8:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
lattice b2 = LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#);
:: VECTSP_8:modenot 1 => VECTSP_8:mode 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode SubVS-Family of A2 means
for b1 being set
st b1 in it
holds b1 is Subspace of a2;
end;
:: VECTSP_8:dfs 2
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be set;
To prove
a3 is SubVS-Family of a2
it is sufficient to prove
thus for b1 being set
st b1 in a3
holds b1 is Subspace of a2;
:: VECTSP_8:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being set holds
b3 is SubVS-Family of b2
iff
for b4 being set
st b4 in b3
holds b4 is Subspace of b2;
:: VECTSP_8:exreg 1
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
cluster non empty SubVS-Family of a2;
end;
:: VECTSP_8:funcnot 2 => VECTSP_8:func 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
redefine func Subspaces a2 -> non empty SubVS-Family of a2;
end;
:: VECTSP_8:modenot 2 => VECTSP_8:mode 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be non empty SubVS-Family of a2;
redefine mode Element of a3 -> Subspace of a2;
end;
:: VECTSP_8:funcnot 3 => VECTSP_8:func 3
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Element of Subspaces a2;
func carr A3 -> Element of bool the carrier of a2 means
ex b1 being Subspace of a2 st
a3 = b1 & it = the carrier of b1;
end;
:: VECTSP_8:def 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of Subspaces b2
for b4 being Element of bool the carrier of b2 holds
b4 = carr b3
iff
ex b5 being Subspace of b2 st
b3 = b5 & b4 = the carrier of b5;
:: VECTSP_8:funcnot 4 => VECTSP_8:func 4
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func carr A2 -> Function-like quasi_total Relation of Subspaces a2,bool the carrier of a2 means
for b1 being Element of Subspaces a2
for b2 being Subspace of a2
st b1 = b2
holds it . b1 = the carrier of b2;
end;
:: VECTSP_8:def 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Function-like quasi_total Relation of Subspaces b2,bool the carrier of b2 holds
b3 = carr b2
iff
for b4 being Element of Subspaces b2
for b5 being Subspace of b2
st b4 = b5
holds b3 . b4 = the carrier of b5;
:: VECTSP_8:th 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being non empty Element of bool Subspaces b2 holds
(carr b2) .: b3 is not empty;
:: VECTSP_8:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
0. b2 in (carr b2) . b3;
:: VECTSP_8:funcnot 5 => VECTSP_8:func 5
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be non empty Element of bool Subspaces a2;
func meet A3 -> strict Subspace of a2 means
the carrier of it = meet ((carr a2) .: a3);
end;
:: VECTSP_8:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being non empty Element of bool Subspaces b2
for b4 being strict Subspace of b2 holds
b4 = meet b3
iff
the carrier of b4 = meet ((carr b2) .: b3);
:: VECTSP_8:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
Subspaces b2 = the carrier of lattice b2;
:: VECTSP_8:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
the L_meet of lattice b2 = SubMeet b2;
:: VECTSP_8:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
the L_join of lattice b2 = SubJoin b2;
:: VECTSP_8:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of lattice b2
for b5, b6 being strict Subspace of b2
st b3 = b5 & b4 = b6
holds b3 [= b4
iff
the carrier of b5 c= the carrier of b6;
:: VECTSP_8:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of lattice b2
for b5, b6 being Subspace of b2
st b3 = b5 & b4 = b6
holds b3 "\/" b4 = b5 + b6;
:: VECTSP_8:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of lattice b2
for b5, b6 being Subspace of b2
st b3 = b5 & b4 = b6
holds b3 "/\" b4 = b5 /\ b6;
:: VECTSP_8:attrnot 1 => LATTICE3:attr 4
definition
let a1 be non empty LattStr;
attr a1 is complete means
for b1 being Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_less_than b1 &
(for b3 being Element of the carrier of a1
st b3 is_less_than b1
holds b3 [= b2);
end;
:: VECTSP_8:dfs 6
definiens
let a1 be non empty LattStr;
To prove
a1 is complete
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 is_less_than b1 &
(for b3 being Element of the carrier of a1
st b3 is_less_than b1
holds b3 [= b2);
:: VECTSP_8:def 6
theorem
for b1 being non empty LattStr holds
b1 is complete
iff
for b2 being Element of bool the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b3 is_less_than b2 &
(for b4 being Element of the carrier of b1
st b4 is_less_than b2
holds b4 [= b3);
:: VECTSP_8:th 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
lattice b2 is complete;
:: VECTSP_8:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being set
for b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Element of bool the carrier of b3
st b4 is not empty & b4 is linearly-closed(b1, b3) & b2 in Lin b4
holds b2 in b4;
:: VECTSP_8:funcnot 6 => VECTSP_8:func 6
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a3;
func FuncLatt A4 -> Function-like quasi_total Relation of the carrier of lattice a2,the carrier of lattice a3 means
for b1 being strict Subspace of a2
for b2 being Element of bool the carrier of a3
st b2 = a4 .: the carrier of b1
holds it . b1 = Lin b2;
end;
:: VECTSP_8:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of lattice b2,the carrier of lattice b3 holds
b5 = FuncLatt b4
iff
for b6 being strict Subspace of b2
for b7 being Element of bool the carrier of b3
st b7 = b4 .: the carrier of b6
holds b5 . b6 = Lin b7;
:: VECTSP_8:modenot 3 => VECTSP_8:mode 3
definition
let a1, a2 be non empty Lattice-like LattStr;
mode Semilattice-Homomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
for b1, b2 being Element of the carrier of a1 holds
it . (b1 "/\" b2) = (it . b1) "/\" (it . b2);
end;
:: VECTSP_8:dfs 8
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is Semilattice-Homomorphism of a1,a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 "/\" b2) = (a3 . b1) "/\" (a3 . b2);
:: VECTSP_8:def 8
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is Semilattice-Homomorphism of b1,b2
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 "/\" b5) = (b3 . b4) "/\" (b3 . b5);
:: VECTSP_8:modenot 4 => VECTSP_8:mode 4
definition
let a1, a2 be non empty Lattice-like LattStr;
mode sup-Semilattice-Homomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
for b1, b2 being Element of the carrier of a1 holds
it . (b1 "\/" b2) = (it . b1) "\/" (it . b2);
end;
:: VECTSP_8:dfs 9
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is sup-Semilattice-Homomorphism of a1,a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 "\/" b2) = (a3 . b1) "\/" (a3 . b2);
:: VECTSP_8:def 9
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is sup-Semilattice-Homomorphism of b1,b2
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 "\/" b5) = (b3 . b4) "\/" (b3 . b5);
:: VECTSP_8:th 11
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is Homomorphism of b1,b2
iff
b3 is sup-Semilattice-Homomorphism of b1,b2 & b3 is Semilattice-Homomorphism of b1,b2;
:: VECTSP_8:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st b4 is linear(b1, b2, b3)
holds FuncLatt b4 is sup-Semilattice-Homomorphism of lattice b2,lattice b3;
:: VECTSP_8:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st b4 is one-to-one & b4 is linear(b1, b2, b3)
holds FuncLatt b4 is Homomorphism of lattice b2,lattice b3;
:: VECTSP_8:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st b4 is linear(b1, b2, b3) & b4 is one-to-one
holds FuncLatt b4 is one-to-one;
:: VECTSP_8:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
FuncLatt id the carrier of b2 = id the carrier of lattice b2;