set A = C_Algebra_of_BoundedLinearOperators X;
set MULT = FuncMult X;
set UNIT = FuncUnit X;
set BLOP = BoundedLinearOperators X,X;
set ADD = Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X);
set RRL = CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #);
thus C_Algebra_of_BoundedLinearOperators X is Abelian :: thesis: ( C_Algebra_of_BoundedLinearOperators X is add-associative & C_Algebra_of_BoundedLinearOperators X is right_zeroed & C_Algebra_of_BoundedLinearOperators X is right_complementable & C_Algebra_of_BoundedLinearOperators X is associative & C_Algebra_of_BoundedLinearOperators X is right_unital & C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x, y be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 5 :: thesis: x + y = y + x
reconsider f = x, g = y as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus x + y = f + g
.= y + x by RLVECT_1:5 ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is add-associative :: thesis: ( C_Algebra_of_BoundedLinearOperators X is right_zeroed & C_Algebra_of_BoundedLinearOperators X is right_complementable & C_Algebra_of_BoundedLinearOperators X is associative & C_Algebra_of_BoundedLinearOperators X is right_unital & C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x, y, z be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 6 :: thesis: (x + y) + z = x + (y + z)
reconsider f = x, g = y, h = z as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus (x + y) + z = (f + g) + h
.= f + (g + h) by RLVECT_1:def 6
.= x + (y + z) ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is right_zeroed :: thesis: ( C_Algebra_of_BoundedLinearOperators X is right_complementable & C_Algebra_of_BoundedLinearOperators X is associative & C_Algebra_of_BoundedLinearOperators X is right_unital & C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 7 :: thesis: x + (0. (C_Algebra_of_BoundedLinearOperators X)) = x
reconsider f = x as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus x + (0. (C_Algebra_of_BoundedLinearOperators X)) = f + (0. CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #))
.= x by RLVECT_1:def 7 ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is right_complementable :: thesis: ( C_Algebra_of_BoundedLinearOperators X is associative & C_Algebra_of_BoundedLinearOperators X is right_unital & C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to ALGSTR_0:def 16 :: thesis: x is right_complementable
reconsider f = x as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
consider s being Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) such that
A1: f + s = 0. CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) by ALGSTR_0:def 11;
reconsider t = s as Element of (C_Algebra_of_BoundedLinearOperators X) ;
take t ; :: according to ALGSTR_0:def 11 :: thesis: x + t = 0. (C_Algebra_of_BoundedLinearOperators X)
thus x + t = 0. (C_Algebra_of_BoundedLinearOperators X) by A1; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is associative :: thesis: ( C_Algebra_of_BoundedLinearOperators X is right_unital & C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x, y, z be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to GROUP_1:def 4 :: thesis: (x * y) * z = x * (y * z)
reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators X,X ;
thus (x * y) * z = (FuncMult X) . (xx * yy),zz by Def4
.= (xx * yy) * zz by Def4
.= xx * (yy * zz) by Th7
.= (FuncMult X) . xx,(yy * zz) by Def4
.= x * (y * z) by Def4 ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is right_unital :: thesis: ( C_Algebra_of_BoundedLinearOperators X is right-distributive & C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to VECTSP_1:def 13 :: thesis: x * (1. (C_Algebra_of_BoundedLinearOperators X)) = x
reconsider xx = x as Element of BoundedLinearOperators X,X ;
thus x * (1. (C_Algebra_of_BoundedLinearOperators X)) = xx * (FuncUnit X) by Def4
.= x by Th8 ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is right-distributive :: thesis: ( C_Algebra_of_BoundedLinearOperators X is vector-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let x, y, z be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to VECTSP_1:def 11 :: thesis: x * (y + z) = (x * y) + (x * z)
reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators X,X ;
thus x * (y + z) = xx * (yy + zz) by Def4
.= (xx * yy) + (xx * zz) by Th9
.= (Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) . (xx * yy),((FuncMult X) . xx,zz) by Def4
.= (x * y) + (x * z) by Def4 ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is vector-distributive :: thesis: ( C_Algebra_of_BoundedLinearOperators X is scalar-distributive & C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let a be Complex; :: according to CLVECT_1:def 2 :: thesis: for b₁, b₂ being Element of the carrier of (C_Algebra_of_BoundedLinearOperators X) holds a * (b₁ + b₂) = (a * b₁) + (a * b₂)
let x, y be Element of (C_Algebra_of_BoundedLinearOperators X); :: thesis: a * (x + y) = (a * x) + (a * y)
reconsider f = x, g = y as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus a * (x + y) = a * (f + g)
.= (a * f) + (a * g) by CLVECT_1:def 2
.= (a * x) + (a * y) ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is scalar-distributive :: thesis: ( C_Algebra_of_BoundedLinearOperators X is scalar-associative & C_Algebra_of_BoundedLinearOperators X is vector-associative )

proof

let a, b be Complex; :: according to CLVECT_1:def 3 :: thesis: for b₁ being Element of the carrier of (C_Algebra_of_BoundedLinearOperators X) holds (a + b) * b₁ = (a * b₁) + (b * b₁)
let x be Element of (C_Algebra_of_BoundedLinearOperators X); :: thesis: (a + b) * x = (a * x) + (b * x)
reconsider f = x as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus (a + b) * x = (a + b) * f
.= (a * f) + (b * f) by CLVECT_1:def 3
.= (a * x) + (b * x) ; :: thesis: verum

end;

thus C_Algebra_of_BoundedLinearOperators X is scalar-associative :: thesis: C_Algebra_of_BoundedLinearOperators X is vector-associative

proof

let a, b be Complex; :: according to CLVECT_1:def 4 :: thesis: for b₁ being Element of the carrier of (C_Algebra_of_BoundedLinearOperators X) holds (a * b) * b₁ = a * (b * b₁)
let x be Element of (C_Algebra_of_BoundedLinearOperators X); :: thesis: (a * b) * x = a * (b * x)
reconsider f = x as Element of CLSStruct(# (BoundedLinearOperators X,X),(Zero_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Add_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)),(Mult_ (BoundedLinearOperators X,X),(C_VectorSpace_of_LinearOperators X,X)) #) ;
thus (a * b) * x = (a * b) * f
.= a * (b * f) by CLVECT_1:def 4
.= a * (b * x) ; :: thesis: verum

end;

let x, y be Element of (C_Algebra_of_BoundedLinearOperators X); :: according to CFUNCDOM:def 9 :: thesis: for b₁ being Element of COMPLEX holds b₁ * (x * y) = (b₁ * x) * y
let a be Complex; :: thesis: a * (x * y) = (a * x) * y
reconsider xx = x, yy = y as Element of BoundedLinearOperators X,X ;
thus a * (x * y) = a * (xx * yy) by Def4
.= (a * xx) * yy by Th12
.= (a * x) * y by Def4 ; :: thesis: verum