let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) ) )
assume that
A1:
Z c= dom (cos * f)
and
A2:
for x being Real st x in Z holds
f . x = (a * x) + b
; ( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )
for y being object st y in Z holds
y in dom f
by A1, FUNCT_1:11;
then A3:
Z c= dom f
by TARSKI:def 3;
then A4:
f is_differentiable_on Z
by A2, FDIFF_1:23;
A5:
for x being Real st x in Z holds
cos * f is_differentiable_in x
then A7:
cos * f is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))
proof
let x be
Real;
( x in Z implies ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) )
A8:
diff (
cos,
(f . x))
= - (sin . (f . x))
by SIN_COS:63;
A9:
cos is_differentiable_in f . x
by SIN_COS:63;
assume A10:
x in Z
;
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))
then
f is_differentiable_in x
by A4, FDIFF_1:9;
then diff (
(cos * f),
x) =
(diff (cos,(f . x))) * (diff (f,x))
by A9, FDIFF_2:13
.=
- ((sin . (f . x)) * (diff (f,x)))
by A8
.=
- ((sin . ((a * x) + b)) * (diff (f,x)))
by A2, A10
.=
- ((sin . ((a * x) + b)) * ((f `| Z) . x))
by A4, A10, FDIFF_1:def 7
.=
- (a * (sin . ((a * x) + b)))
by A2, A3, A10, FDIFF_1:23
;
hence
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))
by A7, A10, FDIFF_1:def 7;
verum
end;
hence
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )
by A1, A5, FDIFF_1:9; verum