let A be symmetrical Subset of COMPLEX; :: thesis: for F being PartFunc of REAL,REAL st A c= dom F & ( for x being Real st x in A holds
F . x = F . |.x.| ) holds
F is_even_on A
let F be PartFunc of REAL,REAL; :: thesis: ( A c= dom F & ( for x being Real st x in A holds
F . x = F . |.x.| ) implies F is_even_on A )
assume that
A1: A c= dom F and
A2: for x being Real st x in A holds
F . x = F . |.x.| ; :: thesis: F is_even_on A
A3: dom (F | A) = A by A1, RELAT_1:62;
A4: for x being Real st x in A holds
- x in A by Def1;
A5: for x being Real st x in A holds
F . (- x) = F . x
proof
let x be Real; :: thesis: ( x in A implies F . (- x) = F . x )
assume A6: x in A ; :: thesis: F . (- x) = F . x
per cases ( x < 0 or 0 < x or x = 0 ) ;
suppose x < 0 ; :: thesis: F . (- x) = F . x
then F . (- x) = F . |.x.| by ABSVALUE:def 1
.= F . x by A2, A6 ;
hence F . (- x) = F . x ; :: thesis: verum
end;
suppose 0 < x ; :: thesis: F . (- x) = F . x
then |.(- x).| = - (- x) by ABSVALUE:def 1
.= x ;
hence F . (- x) = F . x by A2, A4, A6; :: thesis: verum
end;
suppose x = 0 ; :: thesis: F . (- x) = F . x
hence F . (- x) = F . x ; :: thesis: verum
end;
end;
end;
for x being Real st x in dom (F | A) & - x in dom (F | A) holds
(F | A) . (- x) = (F | A) . x
proof
let x be Real; :: thesis: ( x in dom (F | A) & - x in dom (F | A) implies (F | A) . (- x) = (F | A) . x )
assume that
A7: x in dom (F | A) and
A8: - x in dom (F | A) ; :: thesis: (F | A) . (- x) = (F | A) . x
reconsider x = x as Element of REAL by XREAL_0:def 1;
(F | A) . (- x) = (F | A) /. (- x) by A8, PARTFUN1:def 6
.= F /. (- x) by A1, A3, A8, PARTFUN2:17
.= F . (- x) by A1, A8, PARTFUN1:def 6
.= F . x by A5, A7
.= F /. x by A1, A7, PARTFUN1:def 6
.= (F | A) /. x by A1, A3, A7, PARTFUN2:17
.= (F | A) . x by A7, PARTFUN1:def 6 ;
hence (F | A) . (- x) = (F | A) . x ; :: thesis: verum
end;
then ( F | A is with_symmetrical_domain & F | A is quasi_even ) by A3;
hence F is_even_on A by A1; :: thesis: verum