let a, b be Real; for A being Matrix of REAL holds (a - b) * A = (a * A) - (b * A)
let A be Matrix of REAL; (a - b) * A = (a * A) - (b * A)
A1:
( len ((a - b) * A) = len A & width ((a - b) * A) = width A )
by MATRIXR1:27;
A2:
( len (a * A) = len A & width (a * A) = width A )
by MATRIXR1:27;
A3:
( len (b * A) = len A & width (b * A) = width A )
by MATRIXR1:27;
A4:
for i, j being Nat st [i,j] in Indices ((a - b) * A) holds
((a - b) * A) * (i,j) = ((a * A) - (b * A)) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices ((a - b) * A) implies ((a - b) * A) * (i,j) = ((a * A) - (b * A)) * (i,j) )
assume A5:
[i,j] in Indices ((a - b) * A)
;
((a - b) * A) * (i,j) = ((a * A) - (b * A)) * (i,j)
reconsider i0 =
i,
j0 =
j as
Nat ;
A6:
Indices ((a - b) * A) = Indices A
by A1, MATRIX_4:55;
Indices (a * A) = Indices A
by A2, MATRIX_4:55;
then ((a * A) - (b * A)) * (
i,
j) =
((a * A) * (i0,j0)) - ((b * A) * (i0,j0))
by A2, A3, A5, A6, Th6
.=
(a * (A * (i0,j0))) - ((b * A) * (i0,j0))
by A5, A6, MATRIXR1:29
.=
(a * (A * (i0,j0))) - (b * (A * (i0,j0)))
by A5, A6, MATRIXR1:29
.=
(a - b) * (A * (i,j))
;
hence
((a - b) * A) * (
i,
j)
= ((a * A) - (b * A)) * (
i,
j)
by A5, A6, MATRIXR1:29;
verum
end;
A7: width ((a * A) - (b * A)) =
width (MXF2MXR ((MXR2MXF (a * A)) + (- (MXR2MXF (b * A)))))
by MATRIX_4:def 1
.=
width (a * A)
by MATRIX_3:def 3
;
len ((a * A) - (b * A)) =
len (MXF2MXR ((MXR2MXF (a * A)) + (- (MXR2MXF (b * A)))))
by MATRIX_4:def 1
.=
len (a * A)
by MATRIX_3:def 3
;
hence
(a - b) * A = (a * A) - (b * A)
by A1, A7, A2, A4, MATRIX_0:21; verum