let X be non empty set ; :: thesis: for V being ComplexAlgebra
for V1 being non empty Subset of V
for d1, d2 being Element of X
for A being BinOp of X
for M being Function of [:X,X:],X
for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let V be ComplexAlgebra; :: thesis: for V1 being non empty Subset of V
for d1, d2 being Element of X
for A being BinOp of X
for M being Function of [:X,X:],X
for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let V1 be non empty Subset of V; :: thesis: for d1, d2 being Element of X
for A being BinOp of X
for M being Function of [:X,X:],X
for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let d1, d2 be Element of X; :: thesis: for A being BinOp of X
for M being Function of [:X,X:],X
for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let A be BinOp of X; :: thesis: for M being Function of [:X,X:],X
for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let M be Function of [:X,X:],X; :: thesis: for MR being Function of ,X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse holds
ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V

let MR be Function of ,X; :: thesis: ( V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:COMPLEX,V1:] & V1 is having-inverse implies ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V )
assume that
A1: V1 = X and
A2: d1 = 0. V and
A3: d2 = 1. V and
A4: A = the addF of V || V1 and
A5: M = the multF of V || V1 and
A6: MR = the Mult of V | [:COMPLEX,V1:] and
A7: for v being Element of V st v in V1 holds
- v in V1 ; :: according to C0SP1:def 1 :: thesis: ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V
reconsider W = ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) as non empty ComplexAlgebraStr ;
A8: now :: thesis: for x, y being Element of W holds x + y = the addF of V . (x,y)
let x, y be Element of W; :: thesis: x + y = the addF of V . (x,y)
reconsider x1 = x, y1 = y as Element of V by ;
x + y = the addF of V . [x,y] by ;
hence x + y = the addF of V . (x,y) ; :: thesis: verum
end;
A9: now :: thesis: for a, x being VECTOR of W holds a * x = the multF of V . (a,x)
let a, x be VECTOR of W; :: thesis: a * x = the multF of V . (a,x)
a * x = the multF of V . [a,x] by ;
hence a * x = the multF of V . (a,x) ; :: thesis: verum
end;
A10: ( W is Abelian & W is add-associative & W is right_zeroed & W is right_complementable & W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
proof
set Mv = the multF of V;
set MV = the Mult of V;
set Av = the addF of V;
hereby :: according to RLVECT_1:def 2 :: thesis: ( W is add-associative & W is right_zeroed & W is right_complementable & W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
let x, y be VECTOR of W; :: thesis: x + y = y + x
reconsider x1 = x, y1 = y as VECTOR of V by ;
( x + y = x1 + y1 & y + x = y1 + x1 ) by A8;
hence x + y = y + x ; :: thesis: verum
end;
hereby :: according to RLVECT_1:def 3 :: thesis: ( W is right_zeroed & W is right_complementable & W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
let x, y, z be VECTOR of W; :: thesis: (x + y) + z = x + (y + z)
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by ;
x + (y + z) = the addF of V . (x1,(y + z)) by A8;
then A11: x + (y + z) = x1 + (y1 + z1) by A8;
(x + y) + z = the addF of V . ((x + y),z1) by A8;
then (x + y) + z = (x1 + y1) + z1 by A8;
hence (x + y) + z = x + (y + z) by ; :: thesis: verum
end;
hereby :: according to RLVECT_1:def 4 :: thesis: ( W is right_complementable & W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
let x be VECTOR of W; :: thesis: x + (0. W) = x
reconsider y = x, z = 0. W as VECTOR of V by ;
thus x + (0. W) = y + (0. V) by A2, A8
.= x by RLVECT_1:4 ; :: thesis: verum
end;
thus W is right_complementable :: thesis: ( W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
proof
let x be Element of W; :: according to ALGSTR_0:def 16 :: thesis:
reconsider x1 = x as Element of V by ;
consider v being Element of V such that
A12: x1 + v = 0. V by ALGSTR_0:def 11;
v = - x1 by ;
then reconsider y = v as Element of W by A1, A7;
take y ; :: according to ALGSTR_0:def 11 :: thesis: x + y = 0. W
thus x + y = 0. W by A2, A8, A12; :: thesis: verum
end;
hereby :: according to GROUP_1:def 12 :: thesis: ( W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )
let v, w be Element of W; :: thesis: v * w = w * v
reconsider v1 = v, w1 = w as Element of V by ;
( v * w = v1 * w1 & w * v = w1 * v1 ) by A9;
hence v * w = w * v ; :: thesis: verum
end;
hereby :: according to GROUP_1:def 3 :: thesis:
let a, b, x be Element of W; :: thesis: (a * b) * x = a * (b * x)
reconsider y = x, a1 = a, b1 = b as Element of V by ;
a * (b * x) = the multF of V . (a,(b * x)) by A9;
then A13: a * (b * x) = a1 * (b1 * y) by A9;
a * b = a1 * b1 by A9;
then (a * b) * x = (a1 * b1) * y by A9;
hence (a * b) * x = a * (b * x) by ; :: thesis: verum
end;
hereby :: according to VECTSP_1:def 4 :: thesis:
let v be Element of W; :: thesis: v * (1. W) = v
reconsider v1 = v as Element of V by ;
v * (1. W) = v1 * (1. V) by A3, A9;
hence v * (1. W) = v ; :: thesis: verum
end;
hereby :: according to VECTSP_1:def 2 :: thesis:
let x, y, z be Element of W; :: thesis: x * (y + z) = (x * y) + (x * z)
reconsider x1 = x, y1 = y, z1 = z as Element of V by ;
y + z = y1 + z1 by A8;
then x * (y + z) = x1 * (y1 + z1) by A9;
then A14: x * (y + z) = (x1 * y1) + (x1 * z1) by VECTSP_1:def 2;
( x * y = x1 * y1 & x * z = x1 * z1 ) by A9;
hence x * (y + z) = (x * y) + (x * z) by ; :: thesis: verum
end;
thus for a being Complex
for x, y being VECTOR of W holds a * (x + y) = (a * x) + (a * y) :: according to CLVECT_1:def 2 :: thesis:
proof
let a be Complex; :: thesis: for x, y being VECTOR of W holds a * (x + y) = (a * x) + (a * y)
let x, y be VECTOR of W; :: thesis: a * (x + y) = (a * x) + (a * y)
reconsider x1 = x, y1 = y as Element of V by ;
A15: a * x = a * x1 by A1, A6, Lm1;
A16: a * y = a * y1 by A1, A6, Lm1;
x + y = x1 + y1 by A8;
then a * (x + y) = a * (x1 + y1) by A1, A6, Lm1;
then a * (x + y) = (a * x1) + (a * y1) by CLVECT_1:def 2;
hence a * (x + y) = (a * x) + (a * y) by A8, A15, A16; :: thesis: verum
end;
thus for a, b being Complex
for v being VECTOR of W holds (a + b) * v = (a * v) + (b * v) :: according to CLVECT_1:def 3 :: thesis: ( W is scalar-associative & W is vector-associative )
proof
let a, b be Complex; :: thesis: for v being VECTOR of W holds (a + b) * v = (a * v) + (b * v)
let x be Element of W; :: thesis: (a + b) * x = (a * x) + (b * x)
reconsider x1 = x as Element of V by ;
A17: a * x = a * x1 by A1, A6, Lm1;
A18: b * x = b * x1 by A1, A6, Lm1;
(a + b) * x = (a + b) * x1 by A1, A6, Lm1;
then (a + b) * x = (a * x1) + (b * x1) by CLVECT_1:def 3;
hence (a + b) * x = (a * x) + (b * x) by A8, A17, A18; :: thesis: verum
end;
thus for a, b being Complex
for v being VECTOR of W holds (a * b) * v = a * (b * v) :: according to CLVECT_1:def 4 :: thesis:
proof
let a, b be Complex; :: thesis: for v being VECTOR of W holds (a * b) * v = a * (b * v)
let x be Element of W; :: thesis: (a * b) * x = a * (b * x)
reconsider x1 = x as Element of V by ;
reconsider y = b * x as Element of W ;
reconsider y1 = b * x1 as Element of V ;
A19: b * x = b * x1 by A1, A6, Lm1;
A20: a * (b * x) = a * (b * x1) by A1, A6, A19, Lm1;
(a * b) * x = (a * b) * x1 by A1, A6, Lm1;
hence (a * b) * x = a * (b * x) by ; :: thesis: verum
end;
thus for x, y being Element of W
for a being Complex holds a * (x * y) = (a * x) * y :: according to CFUNCDOM:def 9 :: thesis: verum
proof
let x, y be Element of W; :: thesis: for a being Complex holds a * (x * y) = (a * x) * y
let a be Complex; :: thesis: a * (x * y) = (a * x) * y
reconsider x1 = x, y1 = y as Element of V by ;
A21: a * x = a * x1 by A1, A6, Lm1;
A22: a * (x * y) = a * (x1 * y1) by A1, A6, Lm1, A9;
a * (x1 * y1) = (a * x1) * y1 by CFUNCDOM:def 9;
hence a * (x * y) = (a * x) * y by A9, A21, A22; :: thesis: verum
end;
thus verum ; :: thesis: verum
end;
( 0. W = 0. V & 1. W = 1. V ) by A2, A3;
hence ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V by A1, A4, A5, A6, A10, Def1; :: thesis: verum