:: PDIFF_9 semantic presentation
begin
theorem LM01CPre2: :: PDIFF_9:1
for S, T being RealNormSpace
for f being Point of (R_NormSpace_of_BoundedLinearOperators (S,T))
for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds
||.(f . x).|| <= r * ||.x.|| ) holds
||.f.|| <= r
proof
let S, T be RealNormSpace; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators (S,T))
for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds
||.(f . x).|| <= r * ||.x.|| ) holds
||.f.|| <= r
let f be Point of (R_NormSpace_of_BoundedLinearOperators (S,T)); ::_thesis: for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds
||.(f . x).|| <= r * ||.x.|| ) holds
||.f.|| <= r
let r be Real; ::_thesis: ( 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds
||.(f . x).|| <= r * ||.x.|| ) implies ||.f.|| <= r )
assume AS: ( 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds
||.(f . x).|| <= r * ||.x.|| ) ) ; ::_thesis: ||.f.|| <= r
P1: now__::_thesis:_for_x_being_Point_of_S_st_||.x.||_<=_1_holds_
||.(f_._x).||_<=_r
let x be Point of S; ::_thesis: ( ||.x.|| <= 1 implies ||.(f . x).|| <= r )
assume ||.x.|| <= 1 ; ::_thesis: ||.(f . x).|| <= r
then ( ||.(f . x).|| <= r * ||.x.|| & r * ||.x.|| <= r * 1 ) by AS, XREAL_1:64;
hence ||.(f . x).|| <= r by XXREAL_0:2; ::_thesis: verum
end;
reconsider g = f as Lipschitzian LinearOperator of S,T by LOPBAN_1:def_9;
set PreNormS = PreNorms (modetrans (f,S,T));
Q1: for y being ext-real set st y in PreNorms (modetrans (f,S,T)) holds
y <= r
proof
let y be ext-real set ; ::_thesis: ( y in PreNorms (modetrans (f,S,T)) implies y <= r )
assume y in PreNorms (modetrans (f,S,T)) ; ::_thesis: y <= r
then consider x being VECTOR of S such that
Q2: ( y = ||.((modetrans (f,S,T)) . x).|| & ||.x.|| <= 1 ) ;
y = ||.(g . x).|| by Q2, LOPBAN_1:29;
hence y <= r by P1, Q2; ::_thesis: verum
end;
set UBPreNormS = upper_bound (PreNorms (modetrans (f,S,T)));
set dif = (upper_bound (PreNorms (modetrans (f,S,T)))) - r;
now__::_thesis:_not_upper_bound_(PreNorms_(modetrans_(f,S,T)))_>_r
assume upper_bound (PreNorms (modetrans (f,S,T))) > r ; ::_thesis: contradiction
then D2: (upper_bound (PreNorms (modetrans (f,S,T)))) - r > 0 by XREAL_1:50;
r is UpperBound of PreNorms (modetrans (f,S,T)) by Q1, XXREAL_2:def_1;
then PreNorms (modetrans (f,S,T)) is bounded_above by XXREAL_2:def_10;
then ex w being real set st
( w in PreNorms (modetrans (f,S,T)) & (upper_bound (PreNorms (modetrans (f,S,T)))) - ((upper_bound (PreNorms (modetrans (f,S,T)))) - r) < w ) by D2, SEQ_4:def_1;
hence contradiction by Q1; ::_thesis: verum
end;
then upper_bound (PreNorms g) <= r by LOPBAN_1:def_11;
hence ||.f.|| <= r by LOPBAN_1:30; ::_thesis: verum
end;
theorem NFCONT125: :: PDIFF_9:2
for Z being set
for S being RealNormSpace
for f being PartFunc of S,REAL holds
( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
proof
let Z be set ; ::_thesis: for S being RealNormSpace
for f being PartFunc of S,REAL holds
( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let S be RealNormSpace; ::_thesis: for f being PartFunc of S,REAL holds
( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let f be PartFunc of S,REAL; ::_thesis: ( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
thus ( f is_continuous_on Z implies ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on Z )
proof
assume A1: f is_continuous_on Z ; ::_thesis: ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) )
then A2: Z c= dom f by NFCONT_1:def_8;
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_Z_&_s1_is_convergent_&_lim_s1_in_Z_holds_
(_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= Z & s1 is convergent & lim s1 in Z implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )
assume A3: ( rng s1 c= Z & s1 is convergent & lim s1 in Z ) ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) )
then A6: f | Z is_continuous_in lim s1 by A1, NFCONT_1:def_8;
dom (f | Z) = (dom f) /\ Z by PARTFUN2:15;
then A7: dom (f | Z) = Z by A2, XBOOLE_1:28;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_Z)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | Z) /* s1) . n = (f /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A8: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | Z) /* s1) . n = (f | Z) /. (s1 . n) by A3, A7, FUNCT_2:109
.= f /. (s1 . n) by A3, A7, A8, PARTFUN2:15
.= (f /* s1) . n by A2, A3, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
then A9: (f | Z) /* s1 = f /* s1 by FUNCT_2:63;
f /. (lim s1) = (f | Z) /. (lim s1) by A3, A7, PARTFUN2:15;
hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A7, A6, A9, NFCONT_1:def_6; ::_thesis: verum
end;
hence ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, NFCONT_1:def_8; ::_thesis: verum
end;
assume that
A10: Z c= dom f and
A11: for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on Z
dom (f | Z) = (dom f) /\ Z by PARTFUN2:15;
then A12: dom (f | Z) = Z by A10, XBOOLE_1:28;
now__::_thesis:_for_x1_being_Point_of_S_st_x1_in_Z_holds_
f_|_Z_is_continuous_in_x1
let x1 be Point of S; ::_thesis: ( x1 in Z implies f | Z is_continuous_in x1 )
assume A13: x1 in Z ; ::_thesis: f | Z is_continuous_in x1
now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_(f_|_Z)_&_s1_is_convergent_&_lim_s1_=_x1_holds_
(_(f_|_Z)_/*_s1_is_convergent_&_(f_|_Z)_/._x1_=_lim_((f_|_Z)_/*_s1)_)
let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f | Z) & s1 is convergent & lim s1 = x1 implies ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) ) )
assume A14: ( rng s1 c= dom (f | Z) & s1 is convergent & lim s1 = x1 ) ; ::_thesis: ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_Z)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | Z) /* s1) . n = (f /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A17: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | Z) /* s1) . n = (f | Z) /. (s1 . n) by A14, FUNCT_2:109
.= f /. (s1 . n) by A14, A17, PARTFUN2:15
.= (f /* s1) . n by A10, A12, A14, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
then A18: (f | Z) /* s1 = f /* s1 by FUNCT_2:63;
(f | Z) /. (lim s1) = f /. (lim s1) by A13, A12, A14, PARTFUN2:15;
hence ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) ) by A11, A13, A12, A14, A18; ::_thesis: verum
end;
hence f | Z is_continuous_in x1 by A13, A12, NFCONT_1:def_6; ::_thesis: verum
end;
hence f is_continuous_on Z by A10, NFCONT_1:def_8; ::_thesis: verum
end;
theorem LMXTh0: :: PDIFF_9:3
for i being Element of NAT
for f being PartFunc of (REAL i),REAL holds dom (<>* f) = dom f
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL holds dom (<>* f) = dom f
let f be PartFunc of (REAL i),REAL; ::_thesis: dom (<>* f) = dom f
rng f c= dom ((proj (1,1)) ") by PDIFF_1:2;
hence dom (<>* f) = dom f by RELAT_1:27; ::_thesis: verum
end;
theorem :: PDIFF_9:4
for i being Element of NAT
for Z being set
for f being PartFunc of (REAL i),REAL st Z c= dom f holds
dom ((<>* f) | Z) = Z
proof
let i be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL i),REAL st Z c= dom f holds
dom ((<>* f) | Z) = Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL i),REAL st Z c= dom f holds
dom ((<>* f) | Z) = Z
let f be PartFunc of (REAL i),REAL; ::_thesis: ( Z c= dom f implies dom ((<>* f) | Z) = Z )
assume Z c= dom f ; ::_thesis: dom ((<>* f) | Z) = Z
then Z c= dom (<>* f) by LMXTh0;
hence dom ((<>* f) | Z) = Z by RELAT_1:62; ::_thesis: verum
end;
theorem LMXTh1: :: PDIFF_9:5
for i being Element of NAT
for Z being set
for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z
proof
let i be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z
let f be PartFunc of (REAL i),REAL; ::_thesis: <>* (f | Z) = (<>* f) | Z
set W = (proj (1,1)) " ;
rng (f | Z) c= dom ((proj (1,1)) ") by PDIFF_1:2;
then dom (((proj (1,1)) ") * (f | Z)) = dom (f | Z) by RELAT_1:27
.= (dom f) /\ Z by RELAT_1:61 ;
then P3: dom (((proj (1,1)) ") * (f | Z)) = (dom (<>* f)) /\ Z by LMXTh0;
now__::_thesis:_for_x_being_set_st_x_in_dom_((<>*_f)_|_Z)_holds_
(<>*_(f_|_Z))_._x_=_((<>*_f)_|_Z)_._x
let x be set ; ::_thesis: ( x in dom ((<>* f) | Z) implies (<>* (f | Z)) . x = ((<>* f) | Z) . x )
assume A1: x in dom ((<>* f) | Z) ; ::_thesis: (<>* (f | Z)) . x = ((<>* f) | Z) . x
then x in (dom (<>* f)) /\ Z by RELAT_1:61;
then x in (dom f) /\ Z by LMXTh0;
then A2: ( x in Z & x in dom f ) by XBOOLE_0:def_4;
dom (((proj (1,1)) ") * (f | Z)) = dom ((<>* f) | Z) by P3, RELAT_1:61;
then (<>* (f | Z)) . x = ((proj (1,1)) ") . ((f | Z) . x) by A1, FUNCT_1:12
.= ((proj (1,1)) ") . (f . x) by A2, FUNCT_1:49
.= (((proj (1,1)) ") * f) . x by A2, FUNCT_1:13 ;
hence (<>* (f | Z)) . x = ((<>* f) | Z) . x by A1, FUNCT_1:47; ::_thesis: verum
end;
hence <>* (f | Z) = (<>* f) | Z by P3, FUNCT_1:2, RELAT_1:61; ::_thesis: verum
end;
theorem XTh30: :: PDIFF_9:6
for i being Element of NAT
for f being PartFunc of (REAL i),REAL
for x being Element of REAL i st x in dom f holds
( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> )
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL
for x being Element of REAL i st x in dom f holds
( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> )
let f be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i st x in dom f holds
( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> )
let x be Element of REAL i; ::_thesis: ( x in dom f implies ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) )
set I = (proj (1,1)) " ;
assume A1: x in dom f ; ::_thesis: ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> )
then (<>* f) . x = ((proj (1,1)) ") . (f . x) by FUNCT_1:13;
hence A2: (<>* f) . x = <*(f . x)*> by PDIFF_1:1; ::_thesis: (<>* f) /. x = <*(f /. x)*>
x in dom (<>* f) by A1, LMXTh0;
then (<>* f) /. x = (<>* f) . x by PARTFUN1:def_6;
hence (<>* f) /. x = <*(f /. x)*> by A1, A2, PARTFUN1:def_6; ::_thesis: verum
end;
theorem LMXTh10: :: PDIFF_9:7
for i being Element of NAT
for f, g being PartFunc of (REAL i),REAL holds
( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) )
proof
let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL holds
( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) )
let f, g be PartFunc of (REAL i),REAL; ::_thesis: ( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) )
P1: ( dom (<>* (f + g)) = dom (f + g) & dom (<>* (f - g)) = dom (f - g) & dom (<>* f) = dom f & dom (<>* g) = dom g ) by LMXTh0;
then ( dom (<>* (f + g)) = (dom (<>* f)) /\ (dom (<>* g)) & dom (<>* (f - g)) = (dom (<>* f)) /\ (dom (<>* g)) ) by VALUED_1:12, VALUED_1:def_1;
then P4: ( dom (<>* (f + g)) = dom ((<>* f) + (<>* g)) & dom (<>* (f - g)) = dom ((<>* f) - (<>* g)) ) by VALUED_2:def_45, VALUED_2:def_46;
now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(f_+_g))_holds_
(<>*_(f_+_g))_._x_=_((<>*_f)_+_(<>*_g))_._x
let x be set ; ::_thesis: ( x in dom (<>* (f + g)) implies (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x )
assume A0: x in dom (<>* (f + g)) ; ::_thesis: (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x
then x in (dom f) /\ (dom g) by P1, VALUED_1:def_1;
then ( x in dom f & x in dom g ) by XBOOLE_0:def_4;
then A4: ( <*(f . x)*> = (<>* f) . x & <*(g . x)*> = (<>* g) . x ) by XTh30;
(<>* (f + g)) . x = <*((f + g) . x)*> by XTh30, A0, P1
.= <*((f . x) + (g . x))*> by A0, P1, VALUED_1:def_1
.= ((<>* f) . x) + ((<>* g) . x) by A4, RVSUM_1:13 ;
hence (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x by P4, A0, VALUED_2:def_45; ::_thesis: verum
end;
hence <>* (f + g) = (<>* f) + (<>* g) by P4, FUNCT_1:2; ::_thesis: <>* (f - g) = (<>* f) - (<>* g)
now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(f_-_g))_holds_
(<>*_(f_-_g))_._x_=_((<>*_f)_-_(<>*_g))_._x
let x be set ; ::_thesis: ( x in dom (<>* (f - g)) implies (<>* (f - g)) . x = ((<>* f) - (<>* g)) . x )
assume A0: x in dom (<>* (f - g)) ; ::_thesis: (<>* (f - g)) . x = ((<>* f) - (<>* g)) . x
then x in (dom f) /\ (dom g) by P1, VALUED_1:12;
then ( x in dom f & x in dom g ) by XBOOLE_0:def_4;
then A4: ( <*(f . x)*> = (<>* f) . x & <*(g . x)*> = (<>* g) . x ) by XTh30;
thus (<>* (f - g)) . x = <*((f - g) . x)*> by XTh30, A0, P1
.= <*((f . x) - (g . x))*> by A0, P1, VALUED_1:13
.= ((<>* f) . x) - ((<>* g) . x) by A4, RVSUM_1:29
.= ((<>* f) - (<>* g)) . x by P4, A0, VALUED_2:def_46 ; ::_thesis: verum
end;
hence <>* (f - g) = (<>* f) - (<>* g) by P4, FUNCT_1:2; ::_thesis: verum
end;
theorem LMXTh11: :: PDIFF_9:8
for i being Element of NAT
for f being PartFunc of (REAL i),REAL
for r being real number holds <>* (r (#) f) = r (#) (<>* f)
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL
for r being real number holds <>* (r (#) f) = r (#) (<>* f)
let f be PartFunc of (REAL i),REAL; ::_thesis: for r being real number holds <>* (r (#) f) = r (#) (<>* f)
let r be real number ; ::_thesis: <>* (r (#) f) = r (#) (<>* f)
P1: dom (<>* (r (#) f)) = dom (r (#) f) by LMXTh0;
then P2: dom (<>* (r (#) f)) = dom f by VALUED_1:def_5;
then P4: dom (<>* (r (#) f)) = dom (<>* f) by LMXTh0
.= dom (r (#) (<>* f)) by VALUED_2:def_39 ;
now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(r_(#)_f))_holds_
(<>*_(r_(#)_f))_._x_=_(r_(#)_(<>*_f))_._x
let x be set ; ::_thesis: ( x in dom (<>* (r (#) f)) implies (<>* (r (#) f)) . x = (r (#) (<>* f)) . x )
assume A0: x in dom (<>* (r (#) f)) ; ::_thesis: (<>* (r (#) f)) . x = (r (#) (<>* f)) . x
then (<>* (r (#) f)) . x = <*((r (#) f) . x)*> by P1, XTh30
.= <*(r * (f . x))*> by A0, P1, VALUED_1:def_5
.= r (#) <*(f . x)*> by RVSUM_1:47
.= r (#) ((<>* f) . x) by A0, P2, XTh30 ;
hence (<>* (r (#) f)) . x = (r (#) (<>* f)) . x by A0, P4, VALUED_2:def_39; ::_thesis: verum
end;
hence <>* (r (#) f) = r (#) (<>* f) by P4, FUNCT_1:2; ::_thesis: verum
end;
XTh30D: for x being Real
for y being Element of REAL 1 st <*x*> = y holds
|.x.| = |.y.|
proof
let x be Real; ::_thesis: for y being Element of REAL 1 st <*x*> = y holds
|.x.| = |.y.|
let y be Element of REAL 1; ::_thesis: ( <*x*> = y implies |.x.| = |.y.| )
assume A1: <*x*> = y ; ::_thesis: |.x.| = |.y.|
reconsider I = (proj (1,1)) " as Function of REAL,(REAL 1) by PDIFF_1:2;
reconsider y0 = y as Point of (REAL-NS 1) by REAL_NS1:def_4;
I . x = y by A1, PDIFF_1:1;
then |.x.| = ||.y0.|| by PDIFF_1:3;
hence |.x.| = |.y.| by REAL_NS1:1; ::_thesis: verum
end;
theorem LMXTh13: :: PDIFF_9:9
for i being Element of NAT
for f being PartFunc of (REAL i),REAL
for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds
|.f.| = |.g.|
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL
for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds
|.f.| = |.g.|
let f be PartFunc of (REAL i),REAL; ::_thesis: for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds
|.f.| = |.g.|
let g be PartFunc of (REAL i),(REAL 1); ::_thesis: ( <>* f = g implies |.f.| = |.g.| )
assume AS: <>* f = g ; ::_thesis: |.f.| = |.g.|
A1: dom |.g.| = dom g by NFCONT_4:def_2
.= dom f by AS, LMXTh0 ;
then A2: dom |.g.| = dom |.f.| by VALUED_1:def_11;
now__::_thesis:_for_x_being_Element_of_REAL_i_st_x_in_dom_|.g.|_holds_
|.g.|_._x_=_|.f.|_._x
let x be Element of REAL i; ::_thesis: ( x in dom |.g.| implies |.g.| . x = |.f.| . x )
assume A3: x in dom |.g.| ; ::_thesis: |.g.| . x = |.f.| . x
then A6: g /. x = <*(f /. x)*> by AS, A1, XTh30;
thus |.g.| . x = |.g.| /. x by A3, PARTFUN1:def_6
.= |.(g /. x).| by A3, NFCONT_4:def_2
.= |.(f /. x).| by A6, XTh30D
.= |.(f . x).| by A1, A3, PARTFUN1:def_6
.= |.f.| . x by VALUED_1:18 ; ::_thesis: verum
end;
hence |.f.| = |.g.| by A2, PARTFUN1:5; ::_thesis: verum
end;
theorem OPEN: :: PDIFF_9:10
for m being non empty Element of NAT
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y holds
( X is open iff Y is open )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y holds
( X is open iff Y is open )
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y holds
( X is open iff Y is open )
let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y implies ( X is open iff Y is open ) )
assume A1: X = Y ; ::_thesis: ( X is open iff Y is open )
hereby ::_thesis: ( Y is open implies X is open )
assume X is open ; ::_thesis: Y is open
then ex X0 being Subset of (REAL-NS m) st
( X0 = X & X0 is open ) by PDIFF_7:def_3;
hence Y is open by A1; ::_thesis: verum
end;
thus ( Y is open implies X is open ) by A1, PDIFF_7:def_3; ::_thesis: verum
end;
theorem PDIFF75: :: PDIFF_9:11
for i, j being Element of NAT
for q being Element of REAL st 1 <= i & i <= j holds
|.((reproj (i,(0* j))) . q).| = |.q.|
proof
let i, j be Element of NAT ; ::_thesis: for q being Element of REAL st 1 <= i & i <= j holds
|.((reproj (i,(0* j))) . q).| = |.q.|
let q be Element of REAL ; ::_thesis: ( 1 <= i & i <= j implies |.((reproj (i,(0* j))) . q).| = |.q.| )
assume A1: ( 1 <= i & i <= j ) ; ::_thesis: |.((reproj (i,(0* j))) . q).| = |.q.|
set y = 0* j;
A3: (reproj (i,(0* j))) . q = Replace ((0* j),i,q) by PDIFF_1:def_5;
0* j in j -tuples_on REAL ;
then ex s being Element of REAL * st
( s = 0* j & len s = j ) ;
then A4: (reproj (i,(0* j))) . q = (((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i) by A1, A3, FINSEQ_7:def_1;
sqrt (Sum (sqr ((0* j) | (i -' 1)))) = |.(0* (i -' 1)).| by A1, PDIFF_7:2;
then sqrt (Sum (sqr ((0* j) | (i -' 1)))) = 0 by EUCLID:7;
then A5: Sum (sqr ((0* j) | (i -' 1))) = 0 by RVSUM_1:86, SQUARE_1:24;
sqrt (Sum (sqr ((0* j) /^ i))) = |.(0* (j -' i)).| by PDIFF_7:3;
then A6: sqrt (Sum (sqr ((0* j) /^ i))) = 0 by EUCLID:7;
sqr ((((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i)) = (sqr (((0* j) | (i -' 1)) ^ <*q*>)) ^ (sqr ((0* j) /^ i)) by TOPREAL7:10
.= ((sqr ((0* j) | (i -' 1))) ^ (sqr <*q*>)) ^ (sqr ((0* j) /^ i)) by TOPREAL7:10
.= ((sqr ((0* j) | (i -' 1))) ^ <*(q ^2)*>) ^ (sqr ((0* j) /^ i)) by RVSUM_1:55 ;
then Sum (sqr ((((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i))) = (Sum ((sqr ((0* j) | (i -' 1))) ^ <*(q ^2)*>)) + (Sum (sqr ((0* j) /^ i))) by RVSUM_1:75
.= ((Sum (sqr ((0* j) | (i -' 1)))) + (q ^2)) + (Sum (sqr ((0* j) /^ i))) by RVSUM_1:74
.= q ^2 by A5, A6, RVSUM_1:86, SQUARE_1:24 ;
hence |.((reproj (i,(0* j))) . q).| = |.q.| by A4, COMPLEX1:72; ::_thesis: verum
end;
Lm5: for m being non empty Element of NAT
for i being Element of NAT
for x being Element of REAL m
for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for x being Element of REAL m
for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
let i be Element of NAT ; ::_thesis: for x being Element of REAL m
for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
let x be Element of REAL m; ::_thesis: for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds
ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
let Z be Subset of (REAL m); ::_thesis: ( Z is open & x in Z & 1 <= i & i <= m implies ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z )
assume that
A1: ( Z is open & x in Z ) and
A3: ( 1 <= i & i <= m ) ; ::_thesis: ex N being Neighbourhood of (proj (i,m)) . x st
for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
consider r being Real such that
A4: ( 0 < r & { y where y is Element of REAL m : |.(y - x).| < r } c= Z ) by A1, PDIFF_7:31;
set N = ].(((proj (i,m)) . x) - r),(((proj (i,m)) . x) + r).[;
reconsider N = ].(((proj (i,m)) . x) - r),(((proj (i,m)) . x) + r).[ as Neighbourhood of (proj (i,m)) . x by A4, RCOMP_1:def_6;
take N ; ::_thesis: for z being Element of REAL st z in N holds
(reproj (i,x)) . z in Z
let z be Element of REAL ; ::_thesis: ( z in N implies (reproj (i,x)) . z in Z )
assume z in N ; ::_thesis: (reproj (i,x)) . z in Z
then A5: |.(z - ((proj (i,m)) . x)).| < r by RCOMP_1:1;
|.(((reproj (i,x)) . z) - x).| = |.((reproj (i,(0* m))) . (z - ((proj (i,m)) . x))).| by PDIFF_7:6
.= |.(z - ((proj (i,m)) . x)).| by A3, PDIFF75 ;
then (reproj (i,x)) . z in { y where y is Element of REAL m : |.(y - x).| < r } by A5;
hence (reproj (i,x)) . z in Z by A4; ::_thesis: verum
end;
theorem LMMMTh6: :: PDIFF_9:12
for j, i being Element of NAT
for x being Element of REAL j holds x = (reproj (i,x)) . ((proj (i,j)) . x)
proof
let j, i be Element of NAT ; ::_thesis: for x being Element of REAL j holds x = (reproj (i,x)) . ((proj (i,j)) . x)
let x be Element of REAL j; ::_thesis: x = (reproj (i,x)) . ((proj (i,j)) . x)
set q = (reproj (i,x)) . ((proj (i,j)) . x);
A1: ( dom ((reproj (i,x)) . ((proj (i,j)) . x)) = Seg j & dom x = Seg j ) by FINSEQ_1:89;
A3: len x = j by A1, FINSEQ_1:def_3;
for k being Nat st k in dom x holds
x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k
proof
let k be Nat; ::_thesis: ( k in dom x implies x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k )
assume A5: k in dom x ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k
then A6: ( 1 <= k & k <= j ) by A1, FINSEQ_1:1;
(reproj (i,x)) . ((proj (i,j)) . x) = Replace (x,i,((proj (i,j)) . x)) by PDIFF_1:def_5;
then A9: ((reproj (i,x)) . ((proj (i,j)) . x)) . k = (Replace (x,i,((proj (i,j)) . x))) /. k by A5, A1, PARTFUN1:def_6;
percases ( k = i or k <> i ) ;
supposeA10: k = i ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k
then ((reproj (i,x)) . ((proj (i,j)) . x)) . k = (proj (i,j)) . x by A3, A6, A9, FINSEQ_7:8;
hence x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k by A10, PDIFF_1:def_1; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k
then ((reproj (i,x)) . ((proj (i,j)) . x)) . k = x /. k by A3, A6, A9, FINSEQ_7:10;
hence x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k by A5, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
hence x = (reproj (i,x)) . ((proj (i,j)) . x) by A1, FINSEQ_1:13; ::_thesis: verum
end;
begin
theorem MPDIFF633: :: PDIFF_9:13
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds
X is open
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds
X is open
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds
X is open
let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X implies X is open )
assume f is_differentiable_on X ; ::_thesis: X is open
then ex X0 being Subset of (REAL-NS m) st
( X = X0 & X0 is open ) by PDIFF_6:33;
hence X is open by PDIFF_7:def_3; ::_thesis: verum
end;
theorem MPDIFF632: :: PDIFF_9:14
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st X is open holds
( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st X is open holds
( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st X is open holds
( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( X is open implies ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) ) )
assume X is open ; ::_thesis: ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) )
then ex X0 being Subset of (REAL-NS m) st
( X = X0 & X0 is open ) by PDIFF_7:def_3;
hence ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) ) ) by PDIFF_6:32; ::_thesis: verum
end;
definition
let m, n be non empty Element of NAT ;
let Z be set ;
let f be PartFunc of (REAL m),(REAL n);
assume A1: Z c= dom f ;
funcf `| Z -> PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) means :Def1: :: PDIFF_9:def 1
( dom it = Z & ( for x being Element of REAL m st x in Z holds
it /. x = diff (f,x) ) );
existence
ex b1 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) st
( dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = diff (f,x) ) )
proof
defpred S1[ Element of REAL m, set ] means ( $1 in Z & $2 = diff (f,$1) );
consider F being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) such that
A2: ( ( for x being Element of REAL m holds
( x in dom F iff ex z being Element of Funcs ((REAL m),(REAL n)) st S1[x,z] ) ) & ( for x being Element of REAL m st x in dom F holds
S1[x,F . x] ) ) from SEQ_1:sch_2();
take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) )
A3: Z is Subset of (REAL m) by A1, XBOOLE_1:1;
now__::_thesis:_for_x_being_set_st_x_in_Z_holds_
x_in_dom_F
let x be set ; ::_thesis: ( x in Z implies x in dom F )
assume AS1: x in Z ; ::_thesis: x in dom F
then reconsider z = x as Element of REAL m by A3;
reconsider y = diff (f,z) as Element of Funcs ((REAL m),(REAL n)) by FUNCT_2:8;
S1[z,y] by AS1;
hence x in dom F by A2; ::_thesis: verum
end;
then A4: Z c= dom F by TARSKI:def_3;
for y being set st y in dom F holds
y in Z by A2;
then dom F c= Z by TARSKI:def_3;
hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds
F /. x = diff (f,x)
let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = diff (f,x) )
assume A5: x in Z ; ::_thesis: F /. x = diff (f,x)
then F . x = diff (f,x) by A2, A4;
hence F /. x = diff (f,x) by A5, A4, PARTFUN1:def_6; ::_thesis: verum
end;
uniqueness
for b1, b2 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) st dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = diff (f,x) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds
b2 /. x = diff (f,x) ) holds
b1 = b2
proof
let F, G be PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))); ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ) implies F = G )
assume that
A6: dom F = Z and
A7: for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) and
A8: dom G = Z and
A9: for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ; ::_thesis: F = G
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_
F_/._x_=_G_/._x
let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x )
assume A10: x in dom F ; ::_thesis: F /. x = G /. x
then F /. x = diff (f,x) by A6, A7;
hence F /. x = G /. x by A6, A9, A10; ::_thesis: verum
end;
hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines `| PDIFF_9:def_1_:_
for m, n being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),(REAL n) st Z c= dom f holds
for b5 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) holds
( b5 = f `| Z iff ( dom b5 = Z & ( for x being Element of REAL m st x in Z holds
b5 /. x = diff (f,x) ) ) );
theorem :: PDIFF_9:15
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) )
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X & g is_differentiable_on X implies ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) )
assume AS: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) )
then A1: X is open by MPDIFF633;
then A2: ( X c= dom f & X c= dom g ) by AS, MPDIFF632;
dom (f + g) = (dom f) /\ (dom g) by VALUED_2:def_45;
then A3: X c= dom (f + g) by A2, XBOOLE_1:19;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
f_+_g_is_differentiable_in_x
let x be Element of REAL m; ::_thesis: ( x in X implies f + g is_differentiable_in x )
assume x in X ; ::_thesis: f + g is_differentiable_in x
then ( f is_differentiable_in x & g is_differentiable_in x ) by AS, A1, MPDIFF632;
hence f + g is_differentiable_in x by PDIFF_6:20; ::_thesis: verum
end;
hence f + g is_differentiable_on X by A3, A1, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x))
let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) )
assume A5: x in X ; ::_thesis: ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x))
then ( f is_differentiable_in x & g is_differentiable_in x ) by AS, A1, MPDIFF632;
then diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) by PDIFF_6:20;
hence ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) by A3, A5, Def1; ::_thesis: verum
end;
theorem :: PDIFF_9:16
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) )
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X & g is_differentiable_on X implies ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) )
assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) )
then AS11: X is open by MPDIFF633;
then P1: ( X c= dom f & X c= dom g ) by AS1, MPDIFF632;
dom (f - g) = (dom f) /\ (dom g) by VALUED_2:def_46;
then P3: X c= dom (f - g) by P1, XBOOLE_1:19;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
f_-_g_is_differentiable_in_x
let x be Element of REAL m; ::_thesis: ( x in X implies f - g is_differentiable_in x )
assume x in X ; ::_thesis: f - g is_differentiable_in x
then ( f is_differentiable_in x & g is_differentiable_in x ) by AS1, AS11, MPDIFF632;
hence f - g is_differentiable_in x by PDIFF_6:21; ::_thesis: verum
end;
hence f - g is_differentiable_on X by P3, AS11, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x))
let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) )
assume P7: x in X ; ::_thesis: ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x))
then ( f is_differentiable_in x & g is_differentiable_in x ) by AS1, AS11, MPDIFF632;
then diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) by PDIFF_6:21;
hence ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) by P3, P7, Def1; ::_thesis: verum
end;
theorem :: PDIFF_9:17
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n)
for r being Real st f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n)
for r being Real st f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for r being Real st f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for r being Real st f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) )
let r be Real; ::_thesis: ( f is_differentiable_on X implies ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) )
assume AS1: f is_differentiable_on X ; ::_thesis: ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) )
then AS11: X is open by MPDIFF633;
then X c= dom f by AS1, MPDIFF632;
then P3: X c= dom (r (#) f) by VALUED_2:def_39;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
r_(#)_f_is_differentiable_in_x
let x be Element of REAL m; ::_thesis: ( x in X implies r (#) f is_differentiable_in x )
assume x in X ; ::_thesis: r (#) f is_differentiable_in x
then f is_differentiable_in x by AS1, AS11, MPDIFF632;
hence r (#) f is_differentiable_in x by PDIFF_6:22; ::_thesis: verum
end;
hence r (#) f is_differentiable_on X by P3, AS11, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) (diff (f,x))
let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `| X) /. x = r (#) (diff (f,x)) )
assume P7: x in X ; ::_thesis: ((r (#) f) `| X) /. x = r (#) (diff (f,x))
then f is_differentiable_in x by AS1, AS11, MPDIFF632;
then diff ((r (#) f),x) = r (#) (diff (f,x)) by PDIFF_6:22;
hence ((r (#) f) `| X) /. x = r (#) (diff (f,x)) by P3, P7, Def1; ::_thesis: verum
end;
theorem LM01A: :: PDIFF_9:18
for j being Element of NAT
for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) )
proof
let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) )
let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) )
reconsider One = <*1*> as Element of REAL 1 by FINSEQ_2:98;
reconsider L = f as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS j) by LOPBAN_1:def_9;
the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def_4;
then dom L = REAL 1 by FUNCT_2:def_1;
then reconsider p = f . <*1*> as Point of (REAL-NS j) by FINSEQ_2:98, PARTFUN1:4;
reconsider OneNS = One as VECTOR of (REAL-NS 1) by REAL_NS1:def_4;
A1: now__::_thesis:_for_r_being_Real
for_x_being_Point_of_(REAL-NS_1)_st_x_=_<*r*>_holds_
f_._x_=_r_*_p
let r be Real; ::_thesis: for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p
let x be Point of (REAL-NS 1); ::_thesis: ( x = <*r*> implies f . x = r * p )
assume x = <*r*> ; ::_thesis: f . x = r * p
then P0: f . x = L . <*r*> ;
<*r*> = <*(r * 1)*>
.= r * <*1*> by RVSUM_1:47
.= r * OneNS by REAL_NS1:3 ;
hence f . x = r * p by P0, LOPBAN_1:def_5; ::_thesis: verum
end;
now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_holds_||.(f_._x).||_=_||.p.||_*_||.x.||
let x be Point of (REAL-NS 1); ::_thesis: ||.(f . x).|| = ||.p.|| * ||.x.||
B0: the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def_4;
then reconsider x0 = x as FinSequence of REAL by FINSEQ_2:def_3;
consider r being Element of REAL such that
B2: x0 = <*r*> by B0, FINSEQ_2:97;
thus ||.(f . x).|| = ||.(r * p).|| by A1, B2
.= (abs r) * ||.p.|| by NORMSP_1:def_1
.= ||.p.|| * ||.x.|| by B2, PDIFF_8:2 ; ::_thesis: verum
end;
hence ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by A1; ::_thesis: verum
end;
theorem LM01C: :: PDIFF_9:19
for j being Element of NAT
for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ||.p.|| = ||.f.|| )
proof
let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ||.p.|| = ||.f.|| )
let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ||.p.|| = ||.f.|| )
reconsider g = f as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS j) by LOPBAN_1:def_9;
consider p being Point of (REAL-NS j) such that
P1: ( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by LM01A;
<*1*> in REAL 1 by FINSEQ_2:98;
then reconsider One = <*1*> as Point of (REAL-NS 1) by REAL_NS1:def_4;
||.(g . One).|| <= ||.f.|| * ||.One.|| by LOPBAN_1:32;
then ||.(g . One).|| <= ||.f.|| * (abs 1) by PDIFF_8:2;
then P2: ||.(f . One).|| <= ||.f.|| * 1 by ABSVALUE:def_1;
for x being Point of (REAL-NS 1) st ||.x.|| <= 1 holds
||.(f . x).|| <= ||.p.|| * ||.x.|| by P1;
then ||.f.|| <= ||.p.|| by LM01CPre2;
hence ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ||.p.|| = ||.f.|| ) by P1, P2, XXREAL_0:1; ::_thesis: verum
end;
theorem LM01: :: PDIFF_9:20
for j being Element of NAT
for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j)))
for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.||
proof
let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j)))
for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.||
let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.||
let x be Point of (REAL-NS 1); ::_thesis: ||.(f . x).|| = ||.f.|| * ||.x.||
P1: ex p being Point of (REAL-NS j) st
( p = f . <*1*> & ( for r being Real
for x being Point of (REAL-NS 1) st x = <*r*> holds
f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by LM01A;
ex q being Point of (REAL-NS j) st
( q = f . <*1*> & ||.f.|| = ||.q.|| ) by LM01C;
hence ||.(f . x).|| = ||.f.|| * ||.x.|| by P1; ::_thesis: verum
end;
theorem LM02: :: PDIFF_9:21
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i implies for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> )
assume AS0: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i ) ; ::_thesis: for x being Element of REAL m
for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let x be Element of REAL m; ::_thesis: for y being Point of (REAL-NS m) st x in X & x = y holds
partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
let y be Point of (REAL-NS m); ::_thesis: ( x in X & x = y implies partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> )
assume AS: ( x in X & x = y ) ; ::_thesis: partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*>
then f is_partial_differentiable_in x,i by AS0, PDIFF_7:34;
then ex g0 being PartFunc of (REAL-NS m),(REAL-NS n) ex y0 being Point of (REAL-NS m) st
( f = g0 & x = y0 & partdiff (f,x,i) = (partdiff (g0,y0,i)) . <*1*> ) by PDIFF_1:def_14;
hence partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> by AS0, AS; ::_thesis: verum
end;
theorem LM03: :: PDIFF_9:22
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds
for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i implies for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| )
assume AS0: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i ) ; ::_thesis: for x0, x1 being Element of REAL m
for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let x0, x1 be Element of REAL m; ::_thesis: for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
let y0, y1 be Point of (REAL-NS m); ::_thesis: ( x0 = y0 & x1 = y1 & x0 in X & x1 in X implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| )
assume AS1: ( x0 = y0 & x1 = y1 & x0 in X & x1 in X ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).||
<*1*> is Element of REAL 1 by FINSEQ_2:98;
then reconsider Pt1 = <*1*> as Point of (REAL-NS 1) by REAL_NS1:def_4;
( (f `partial| (X,i)) /. x1 = partdiff (f,x1,i) & (f `partial| (X,i)) /. x0 = partdiff (f,x0,i) ) by AS1, AS0, PDIFF_7:def_5;
then ( (f `partial| (X,i)) /. x1 = (partdiff (g,y1,i)) . Pt1 & (f `partial| (X,i)) /. x0 = (partdiff (g,y0,i)) . Pt1 ) by LM02, AS0, AS1;
then ((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0) = ((partdiff (g,y1,i)) . Pt1) - ((partdiff (g,y0,i)) . Pt1) by REAL_NS1:5;
then A3: ((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0) = ((partdiff (g,y1,i)) - (partdiff (g,y0,i))) . Pt1 by LOPBAN_1:40;
||.Pt1.|| = abs 1 by PDIFF_8:2;
then ||.Pt1.|| = 1 by ABSVALUE:def_1;
then A4: ||.(((partdiff (g,y1,i)) - (partdiff (g,y0,i))) . Pt1).|| = ||.((partdiff (g,y1,i)) - (partdiff (g,y0,i))).|| * 1 by LM01;
g is_partial_differentiable_on Y,i by AS0, PDIFF_7:33;
then ( (g `partial| (Y,i)) /. y1 = partdiff (g,y1,i) & (g `partial| (Y,i)) /. y0 = partdiff (g,y0,i) ) by AS0, AS1, PDIFF_1:def_20;
hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| by A3, A4, REAL_NS1:1; ::_thesis: verum
end;
theorem LM1Direct: :: PDIFF_9:23
for m, n being non empty Element of NAT
for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) )
assume AS: ( 1 <= i & i <= m & X is open & g = f & X = Y ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
hereby ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume A2: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y )
hence g is_partial_differentiable_on Y,i by AS, PDIFF_7:33; ::_thesis: g `partial| (Y,i) is_continuous_on Y
then A3: dom (g `partial| (Y,i)) = Y by PDIFF_1:def_20;
for y0 being Point of (REAL-NS m)
for r being Real st y0 in Y & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) )
proof
let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in Y & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) )
let r be Real; ::_thesis: ( y0 in Y & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) )
reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4;
assume A4: ( y0 in Y & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) )
then consider s being Real such that
A5: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) by AS, A2, PDIFF_7:38;
take s ; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) )
thus 0 < s by A5; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in Y & ||.(y1 - y0).|| < s implies ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r )
reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4;
assume A6: ( y1 in Y & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r
then A7: |.(x1 - x0).| < s by REAL_NS1:1, REAL_NS1:5;
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| by A4, A6, AS, A2, LM03;
hence ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r by A7, A5, A6, AS; ::_thesis: verum
end;
hence g `partial| (Y,i) is_continuous_on Y by A3, NFCONT_1:19; ::_thesis: verum
end;
assume B1: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then B2: f is_partial_differentiable_on X,i by AS, PDIFF_7:33;
then B3: dom (f `partial| (X,i)) = X by PDIFF_7:def_5;
for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) )
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
assume B4: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
then consider s being Real such that
B5: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds
||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) by AS, B1, NFCONT_1:19;
take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
thus 0 < s by B5; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r )
reconsider y1 = x1 as Element of (REAL-NS m) by REAL_NS1:def_4;
assume B6: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r
|.(x1 - x0).| = ||.(y1 - y0).|| by REAL_NS1:1, REAL_NS1:5;
then ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r by B5, B6, AS;
hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by B4, B6, AS, B2, LM03; ::_thesis: verum
end;
hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by B1, B3, AS, PDIFF_7:33, PDIFF_7:38; ::_thesis: verum
end;
theorem ThGdiff: :: PDIFF_9:24
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y & X is open & f = g implies ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) )
assume AS0: ( X = Y & X is open & f = g ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
then A1: Y is open by OPEN;
hereby ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume AS1: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_
(_g_is_partial_differentiable_on_Y,i_&_g_`partial|_(Y,i)_is_continuous_on_Y_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
assume AS2: ( 1 <= i & i <= m ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y )
then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS1;
hence ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by AS0, AS2, LM1Direct; ::_thesis: verum
end;
hence ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by A1, PDIFF_8:22; ::_thesis: verum
end;
assume AS3: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume AS4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by A1, AS3, PDIFF_8:22;
hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, AS4, LM1Direct; ::_thesis: verum
end;
theorem LM2: :: PDIFF_9:25
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds
( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds
( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds
( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds
( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds
( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let Y be Subset of (REAL-NS m); ::_thesis: ( X is open & X c= dom f & g = f & X = Y implies ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) )
assume AS: ( X is open & X c= dom f & g = f & X = Y ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
then O1: Y is open by OPEN;
hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) )
assume AS1: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) )
hence Z1: f is_differentiable_on X by AS, PDIFF_6:30; ::_thesis: for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) )
assume P2: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
reconsider xx0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
consider s being Real such that
P3: ( 0 < s & ( for xx1 being Point of (REAL-NS m) st xx1 in Y & ||.(xx1 - xx0).|| < s holds
||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r ) ) by AS, AS1, P2, NFCONT_1:19;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
thus 0 < s by P3; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| )
assume P4: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
reconsider xx1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4;
||.(xx1 - xx0).|| < s by P4, REAL_NS1:1, REAL_NS1:5;
then P5: ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r by P3, P4, AS;
let v be Element of REAL m; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
reconsider vv = v as Point of (REAL-NS m) by REAL_NS1:def_4;
f is_differentiable_in x0 by P2, AS, Z1, O1, PDIFF_6:32;
then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f = g & x0 = y & diff (f,x0) = diff (g,y) ) by PDIFF_1:def_8;
then P8: ((g `| Y) /. xx0) . vv = (diff (f,x0)) . v by AS, P2, AS1, NDIFF_1:def_9;
f is_differentiable_in x1 by P4, AS, Z1, O1, PDIFF_6:32;
then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f = g & x1 = y & diff (f,x1) = diff (g,y) ) by PDIFF_1:def_8;
then P7: ((g `| Y) /. xx1) . vv = (diff (f,x1)) . v by AS, P4, AS1, NDIFF_1:def_9;
reconsider g10 = ((g `| Y) /. xx1) - ((g `| Y) /. xx0) as Lipschitzian LinearOperator of (REAL-NS m),(REAL-NS n) by LOPBAN_1:def_9;
(((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv) = g10 . vv by LOPBAN_1:40;
then D2: ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| * ||.vv.|| by LOPBAN_1:32;
||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| * ||.vv.|| <= r * ||.vv.|| by P5, XREAL_1:64;
then ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= r * ||.vv.|| by D2, XXREAL_0:2;
then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * ||.vv.|| by P7, P8, REAL_NS1:1, REAL_NS1:5;
hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by REAL_NS1:1; ::_thesis: verum
end;
assume P1: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y )
hence Z1: g is_differentiable_on Y by AS, PDIFF_6:30; ::_thesis: g `| Y is_continuous_on Y
then Z2: dom (g `| Y) = Y by NDIFF_1:def_9;
for x0 being Point of (REAL-NS m)
for r being Real st x0 in Y & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - x0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. x0)).|| < r ) )
proof
let xx0 be Point of (REAL-NS m); ::_thesis: for r being Real st xx0 in Y & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r ) )
let r0 be Real; ::_thesis: ( xx0 in Y & 0 < r0 implies ex s being Real st
( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) ) )
assume P2: ( xx0 in Y & 0 < r0 ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) )
set r = r0 / 2;
P21: ( 0 < r0 / 2 & r0 / 2 < r0 ) by P2, XREAL_1:216;
reconsider x0 = xx0 as Element of REAL m by REAL_NS1:def_4;
consider s being Real such that
P3: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * |.v.| ) ) by P1, AS, P2;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) )
thus 0 < s by P3; ::_thesis: for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds
||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0
let xx1 be Point of (REAL-NS m); ::_thesis: ( xx1 in Y & ||.(xx1 - xx0).|| < s implies ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0 )
assume P4: ( xx1 in Y & ||.(xx1 - xx0).|| < s ) ; ::_thesis: ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0
reconsider x1 = xx1 as Element of REAL m by REAL_NS1:def_4;
P5: |.(x1 - x0).| < s by P4, REAL_NS1:1, REAL_NS1:5;
now__::_thesis:_for_vv_being_Point_of_(REAL-NS_m)_st_||.vv.||_<=_1_holds_
||.((((g_`|_Y)_/._xx1)_-_((g_`|_Y)_/._xx0))_._vv).||_<=_(r0_/_2)_*_||.vv.||
let vv be Point of (REAL-NS m); ::_thesis: ( ||.vv.|| <= 1 implies ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.|| )
assume ||.vv.|| <= 1 ; ::_thesis: ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.||
reconsider v = vv as Element of REAL m by REAL_NS1:def_4;
f is_differentiable_in x0 by P2, AS, P1, O1, PDIFF_6:32;
then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f = g & x0 = y & diff (f,x0) = diff (g,y) ) by PDIFF_1:def_8;
then P8: ((g `| Y) /. xx0) . vv = (diff (f,x0)) . v by AS, Z1, P2, NDIFF_1:def_9;
f is_differentiable_in x1 by P4, AS, P1, O1, PDIFF_6:32;
then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st
( f = g & x1 = y & diff (f,x1) = diff (g,y) ) by PDIFF_1:def_8;
then P7: ((g `| Y) /. xx1) . vv = (diff (f,x1)) . v by AS, Z1, P4, NDIFF_1:def_9;
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * |.v.| by P5, P3, P4, AS;
then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * ||.vv.|| by REAL_NS1:1;
then ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= (r0 / 2) * ||.vv.|| by P7, P8, REAL_NS1:1, REAL_NS1:5;
hence ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.|| by LOPBAN_1:40; ::_thesis: verum
end;
then ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| <= r0 / 2 by P2, LM01CPre2;
hence ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0 by P21, XXREAL_0:2; ::_thesis: verum
end;
hence g `| Y is_continuous_on Y by Z2, NFCONT_1:19; ::_thesis: verum
end;
theorem CW01: :: PDIFF_9:26
for m, n being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( X is open & X c= dom f implies ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) )
assume AS0: ( X is open & X c= dom f ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
reconsider Y = X as Subset of (REAL-NS m) by REAL_NS1:def_4;
( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider g = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) )
then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, ThGdiff;
hence ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) by AS0, LM2; ::_thesis: verum
end;
assume ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, LM2;
hence for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, ThGdiff; ::_thesis: verum
end;
theorem :: PDIFF_9:27
for m, n being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds
f `| Z = g `| Z
proof
let m, n be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds
f `| Z = g `| Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds
f `| Z = g `| Z
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds
f `| Z = g `| Z
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: ( f = g & f is_differentiable_on Z implies f `| Z = g `| Z )
assume AS: ( f = g & f is_differentiable_on Z ) ; ::_thesis: f `| Z = g `| Z
then P1: g is_differentiable_on Z by PDIFF_6:30;
then P3: dom (g `| Z) = Z by NDIFF_1:def_9;
AK: Z c= dom f by AS, PDIFF_6:def_4;
then P2: dom (f `| Z) = Z by Def1;
now__::_thesis:_for_x0_being_set_st_x0_in_dom_(f_`|_Z)_holds_
(f_`|_Z)_._x0_=_(g_`|_Z)_._x0
let x0 be set ; ::_thesis: ( x0 in dom (f `| Z) implies (f `| Z) . x0 = (g `| Z) . x0 )
assume P4: x0 in dom (f `| Z) ; ::_thesis: (f `| Z) . x0 = (g `| Z) . x0
then reconsider x = x0 as Element of REAL m ;
reconsider Z1 = Z as Subset of (REAL-NS m) by P1, NDIFF_1:30;
reconsider z = x as Point of (REAL-NS m) by REAL_NS1:def_4;
Z1 is open by P1, NDIFF_1:32;
then f is_differentiable_in x by AS, P2, P4, PDIFF_6:32;
then P6: ex g0 being PartFunc of (REAL-NS m),(REAL-NS n) ex y0 being Point of (REAL-NS m) st
( f = g0 & x = y0 & diff (f,x) = diff (g0,y0) ) by PDIFF_1:def_8;
thus (f `| Z) . x0 = (f `| Z) /. x by P4, PARTFUN1:def_6
.= diff (g,z) by AK, P2, P4, P6, AS, Def1
.= (g `| Z) /. x by P4, P2, P1, NDIFF_1:def_9
.= (g `| Z) . x0 by P4, P2, P3, PARTFUN1:def_6 ; ::_thesis: verum
end;
hence f `| Z = g `| Z by P2, P3, FUNCT_1:2; ::_thesis: verum
end;
theorem :: PDIFF_9:28
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n)
for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m)
for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y & X is open & f = g implies ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) )
assume AS0: ( X = Y & X is open & f = g ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) )
then A1: Y is open by OPEN;
hereby ::_thesis: ( f is_differentiable_on X & g `| Y is_continuous_on Y implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume AS1: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & g `| Y is_continuous_on Y )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_
(_g_is_partial_differentiable_on_Y,i_&_g_`partial|_(Y,i)_is_continuous_on_Y_)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) )
assume AS2: ( 1 <= i & i <= m ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y )
then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS1;
hence ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by AS0, AS2, LM1Direct; ::_thesis: verum
end;
then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by A1, PDIFF_8:22;
hence ( f is_differentiable_on X & g `| Y is_continuous_on Y ) by AS0, PDIFF_6:30; ::_thesis: verum
end;
assume ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then B2: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, PDIFF_6:30;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume AS4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by A1, B2, PDIFF_8:22;
hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, AS4, LM1Direct; ::_thesis: verum
end;
theorem XTh350: :: PDIFF_9:29
for m, n being non empty Element of NAT
for f, g being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds
( f + g is_continuous_in x & f - g is_continuous_in x )
proof
let m, n be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds
( f + g is_continuous_in x & f - g is_continuous_in x )
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds
( f + g is_continuous_in x & f - g is_continuous_in x )
let x be Element of REAL m; ::_thesis: ( f is_continuous_in x & g is_continuous_in x implies ( f + g is_continuous_in x & f - g is_continuous_in x ) )
assume A1: ( f is_continuous_in x & g is_continuous_in x ) ; ::_thesis: ( f + g is_continuous_in x & f - g is_continuous_in x )
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4;
A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),(REAL-NS n) ;
( f1 is_continuous_in y & g1 is_continuous_in y ) by A1, PDIFF_7:35;
then A2: ( f1 + g1 is_continuous_in y & f1 - g1 is_continuous_in y ) by NFCONT_1:15;
( f + g = f1 + g1 & f - g = f1 - g1 ) by NFCONT_4:5, NFCONT_4:10, A20;
hence ( f + g is_continuous_in x & f - g is_continuous_in x ) by A2, PDIFF_7:35; ::_thesis: verum
end;
theorem XTh351: :: PDIFF_9:30
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m
for r being Real st f is_continuous_in x holds
r (#) f is_continuous_in x
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m
for r being Real st f is_continuous_in x holds
r (#) f is_continuous_in x
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m
for r being Real st f is_continuous_in x holds
r (#) f is_continuous_in x
let x be Element of REAL m; ::_thesis: for r being Real st f is_continuous_in x holds
r (#) f is_continuous_in x
let r be Real; ::_thesis: ( f is_continuous_in x implies r (#) f is_continuous_in x )
assume A1: f is_continuous_in x ; ::_thesis: r (#) f is_continuous_in x
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4;
A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
f1 is_continuous_in y by A1, PDIFF_7:35;
then A2: r (#) f1 is_continuous_in y by NFCONT_1:16;
r (#) f = r (#) f1 by NFCONT_4:6, A20;
hence r (#) f is_continuous_in x by A2, PDIFF_7:35; ::_thesis: verum
end;
theorem :: PDIFF_9:31
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x holds
- f is_continuous_in x
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x holds
- f is_continuous_in x
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x holds
- f is_continuous_in x
let x be Element of REAL m; ::_thesis: ( f is_continuous_in x implies - f is_continuous_in x )
A1: - 1 is Real by XREAL_0:def_1;
assume f is_continuous_in x ; ::_thesis: - f is_continuous_in x
then (- 1) (#) f is_continuous_in x by A1, XTh351;
hence - f is_continuous_in x by NFCONT_4:7; ::_thesis: verum
end;
theorem YTh354: :: PDIFF_9:32
for m, n being non empty Element of NAT
for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x holds
|.f.| is_continuous_in x
proof
let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n)
for x being Element of REAL m st f is_continuous_in x holds
|.f.| is_continuous_in x
let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x holds
|.f.| is_continuous_in x
let x be Element of REAL m; ::_thesis: ( f is_continuous_in x implies |.f.| is_continuous_in x )
assume A1: f is_continuous_in x ; ::_thesis: |.f.| is_continuous_in x
reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4;
A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
f1 is_continuous_in y by A1, PDIFF_7:35;
then A2: ||.f1.|| is_continuous_in y by NFCONT_1:17;
|.f.| = ||.f1.|| by NFCONT_4:9, A20;
hence |.f.| is_continuous_in x by A2, NFCONT_4:21; ::_thesis: verum
end;
theorem XTh350X: :: PDIFF_9:33
for m, n being non empty Element of NAT
for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
proof
let m, n be non empty Element of NAT ; ::_thesis: for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z implies ( f + g is_continuous_on Z & f - g is_continuous_on Z ) )
assume A1: ( f is_continuous_on Z & g is_continuous_on Z ) ; ::_thesis: ( f + g is_continuous_on Z & f - g is_continuous_on Z )
A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),(REAL-NS n) ;
( f1 is_continuous_on Z & g1 is_continuous_on Z ) by A1, PDIFF_7:37;
then A3: ( f1 + g1 is_continuous_on Z & f1 - g1 is_continuous_on Z ) by NFCONT_1:25;
( f + g = f1 + g1 & f - g = f1 - g1 ) by NFCONT_4:5, NFCONT_4:10, A2;
hence ( f + g is_continuous_on Z & f - g is_continuous_on Z ) by A3, PDIFF_7:37; ::_thesis: verum
end;
theorem XTh351X: :: PDIFF_9:34
for m, n being non empty Element of NAT
for Z being set
for r being Real
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
r (#) f is_continuous_on Z
proof
let m, n be non empty Element of NAT ; ::_thesis: for Z being set
for r being Real
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
r (#) f is_continuous_on Z
let Z be set ; ::_thesis: for r being Real
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
r (#) f is_continuous_on Z
let r be Real; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
r (#) f is_continuous_on Z
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies r (#) f is_continuous_on Z )
assume A1: f is_continuous_on Z ; ::_thesis: r (#) f is_continuous_on Z
A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
f1 is_continuous_on Z by A1, PDIFF_7:37;
then A3: r (#) f1 is_continuous_on Z by NFCONT_1:27;
r (#) f1 = r (#) f by NFCONT_4:6, A2;
hence r (#) f is_continuous_on Z by A3, PDIFF_7:37; ::_thesis: verum
end;
theorem :: PDIFF_9:35
for m, n being non empty Element of NAT
for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
- f is_continuous_on Z
proof
let m, n be non empty Element of NAT ; ::_thesis: for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
- f is_continuous_on Z
let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
- f is_continuous_on Z
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies - f is_continuous_on Z )
assume A1: f is_continuous_on Z ; ::_thesis: - f is_continuous_on Z
- 1 is Real by XREAL_0:def_1;
then (- 1) (#) f is_continuous_on Z by A1, XTh351X;
hence - f is_continuous_on Z by NFCONT_4:7; ::_thesis: verum
end;
theorem XDef60: :: PDIFF_9:36
for i being Element of NAT
for f being PartFunc of (REAL i),REAL
for x0 being Element of REAL i holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds
|.((f /. x) - (f /. x0)).| < r ) ) ) ) )
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL
for x0 being Element of REAL i holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds
|.((f /. x) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of (REAL i),REAL; ::_thesis: for x0 being Element of REAL i holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds
|.((f /. x) - (f /. x0)).| < r ) ) ) ) )
let x0 be Element of REAL i; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds
|.((f /. x) - (f /. x0)).| < r ) ) ) ) )
hereby ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds
|.((f /. x) - (f /. x0)).| < r ) ) ) implies f is_continuous_in x0 )
assume f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) ) ) )
then consider y0 being Point of (REAL-NS i), g being PartFunc of (REAL-NS i),REAL such that
P1: ( x0 = y0 & f = g & g is_continuous_in y0 ) by NFCONT_4:def_4;
thus x0 in dom f by P1, NFCONT_1:8; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) )
then consider s being Real such that
P3: ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) ) by P1, NFCONT_1:8;
take s = s; ::_thesis: ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) )
thus 0 < s by P3; ::_thesis: for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r
let a be Element of REAL i; ::_thesis: ( a in dom f & |.(a - x0).| < s implies |.((f /. a) - (f /. x0)).| < r )
assume P4: ( a in dom f & |.(a - x0).| < s ) ; ::_thesis: |.((f /. a) - (f /. x0)).| < r
reconsider y1 = a as Point of (REAL-NS i) by REAL_NS1:def_4;
||.(y1 - y0).|| = |.(a - x0).| by P1, REAL_NS1:1, REAL_NS1:5;
hence |.((f /. a) - (f /. x0)).| < r by P1, P3, P4; ::_thesis: verum
end;
assume P1: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) ) ) ) ; ::_thesis: f is_continuous_in x0
reconsider y0 = x0 as Point of (REAL-NS i) by REAL_NS1:def_4;
reconsider g = f as PartFunc of (REAL-NS i),REAL by REAL_NS1:def_4;
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_y1_being_Point_of_(REAL-NS_i)_st_y1_in_dom_g_&_||.(y1_-_y0).||_<_s_holds_
|.((g_/._y1)_-_(g_/._y0)).|_<_r_)_)
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) )
then consider s being Real such that
P3: ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds
|.((f /. a) - (f /. x0)).| < r ) ) by P1;
take s = s; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) )
thus 0 < s by P3; ::_thesis: for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r
hereby ::_thesis: verum
let y1 be Point of (REAL-NS i); ::_thesis: ( y1 in dom g & ||.(y1 - y0).|| < s implies |.((g /. y1) - (g /. y0)).| < r )
assume P4: ( y1 in dom g & ||.(y1 - y0).|| < s ) ; ::_thesis: |.((g /. y1) - (g /. y0)).| < r
reconsider a = y1 as Element of REAL i by REAL_NS1:def_4;
||.(y1 - y0).|| = |.(a - x0).| by REAL_NS1:1, REAL_NS1:5;
hence |.((g /. y1) - (g /. y0)).| < r by P3, P4; ::_thesis: verum
end;
end;
then g is_continuous_in y0 by P1, NFCONT_1:8;
hence f is_continuous_in x0 by NFCONT_4:def_4; ::_thesis: verum
end;
theorem XTh35: :: PDIFF_9:37
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff <>* f is_continuous_in x0 )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m holds
( f is_continuous_in x0 iff <>* f is_continuous_in x0 )
let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m holds
( f is_continuous_in x0 iff <>* f is_continuous_in x0 )
let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 iff <>* f is_continuous_in x0 )
set g = <>* f;
hereby ::_thesis: ( <>* f is_continuous_in x0 implies f is_continuous_in x0 )
assume P1: f is_continuous_in x0 ; ::_thesis: <>* f is_continuous_in x0
then P2: x0 in dom f by XDef60;
then P3: x0 in dom (<>* f) by LMXTh0;
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_dom_(<>*_f)_&_|.(x1_-_x0).|_<_s_holds_
|.(((<>*_f)_/._x1)_-_((<>*_f)_/._x0)).|_<_r_)_)
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) )
then consider s being Real such that
P5: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) by P1, XDef60;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) )
thus 0 < s by P5; ::_thesis: for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r
hereby ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in dom (<>* f) & |.(x1 - x0).| < s implies |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r )
assume P6: ( x1 in dom (<>* f) & |.(x1 - x0).| < s ) ; ::_thesis: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r
then P8: x1 in dom f by LMXTh0;
then P7: |.((f /. x1) - (f /. x0)).| < r by P5, P6;
( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by P2, P8, XTh30;
then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29;
hence |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by P7, XTh30D; ::_thesis: verum
end;
end;
hence <>* f is_continuous_in x0 by P3, PDIFF_7:36; ::_thesis: verum
end;
assume A1: <>* f is_continuous_in x0 ; ::_thesis: f is_continuous_in x0
then x0 in dom (<>* f) by PDIFF_7:36;
then P2: x0 in dom f by LMXTh0;
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_dom_f_&_|.(x1_-_x0).|_<_s_holds_
|.((f_/._x1)_-_(f_/._x0)).|_<_r_)_)
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
P4: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) by A1, PDIFF_7:36;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by P4; ::_thesis: for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
hereby ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in dom f & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume P5: ( x1 in dom f & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
then x1 in dom (<>* f) by LMXTh0;
then P6: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by P4, P5;
( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by P2, P5, XTh30;
then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29;
hence |.((f /. x1) - (f /. x0)).| < r by P6, XTh30D; ::_thesis: verum
end;
end;
hence f is_continuous_in x0 by P2, XDef60; ::_thesis: verum
end;
theorem :: PDIFF_9:38
for m being non empty Element of NAT
for f, g being PartFunc of (REAL m),REAL
for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds
( f + g is_continuous_in x0 & f - g is_continuous_in x0 )
proof
let m be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL
for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds
( f + g is_continuous_in x0 & f - g is_continuous_in x0 )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds
( f + g is_continuous_in x0 & f - g is_continuous_in x0 )
let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 & g is_continuous_in x0 implies ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) )
assume ( f is_continuous_in x0 & g is_continuous_in x0 ) ; ::_thesis: ( f + g is_continuous_in x0 & f - g is_continuous_in x0 )
then ( <>* f is_continuous_in x0 & <>* g is_continuous_in x0 ) by XTh35;
then A2: ( (<>* f) + (<>* g) is_continuous_in x0 & (<>* f) - (<>* g) is_continuous_in x0 ) by XTh350;
( (<>* f) + (<>* g) = <>* (f + g) & (<>* f) - (<>* g) = <>* (f - g) ) by LMXTh10;
hence ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) by A2, XTh35; ::_thesis: verum
end;
theorem :: PDIFF_9:39
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let x0 be Element of REAL m; ::_thesis: for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let r be Real; ::_thesis: ( f is_continuous_in x0 implies r (#) f is_continuous_in x0 )
assume f is_continuous_in x0 ; ::_thesis: r (#) f is_continuous_in x0
then <>* f is_continuous_in x0 by XTh35;
then A2: r (#) (<>* f) is_continuous_in x0 by XTh351;
r (#) (<>* f) = <>* (r (#) f) by LMXTh11;
hence r (#) f is_continuous_in x0 by A2, XTh35; ::_thesis: verum
end;
theorem :: PDIFF_9:40
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m st f is_continuous_in x0 holds
|.f.| is_continuous_in x0
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for x0 being Element of REAL m st f is_continuous_in x0 holds
|.f.| is_continuous_in x0
let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m st f is_continuous_in x0 holds
|.f.| is_continuous_in x0
let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 implies |.f.| is_continuous_in x0 )
assume f is_continuous_in x0 ; ::_thesis: |.f.| is_continuous_in x0
then <>* f is_continuous_in x0 by XTh35;
then |.(<>* f).| is_continuous_in x0 by YTh354;
hence |.f.| is_continuous_in x0 by LMXTh13; ::_thesis: verum
end;
XTh360: for i being Element of NAT
for f being PartFunc of (REAL i),REAL
for g being PartFunc of (REAL-NS i),REAL
for x being Element of REAL i
for y being Point of (REAL-NS i) st f = g & x = y holds
( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
proof
let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL
for g being PartFunc of (REAL-NS i),REAL
for x being Element of REAL i
for y being Point of (REAL-NS i) st f = g & x = y holds
( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
let f be PartFunc of (REAL i),REAL; ::_thesis: for g being PartFunc of (REAL-NS i),REAL
for x being Element of REAL i
for y being Point of (REAL-NS i) st f = g & x = y holds
( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
let g be PartFunc of (REAL-NS i),REAL; ::_thesis: for x being Element of REAL i
for y being Point of (REAL-NS i) st f = g & x = y holds
( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
let x be Element of REAL i; ::_thesis: for y being Point of (REAL-NS i) st f = g & x = y holds
( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
let y be Point of (REAL-NS i); ::_thesis: ( f = g & x = y implies ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) )
assume AS: ( f = g & x = y ) ; ::_thesis: ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) )
hereby ::_thesis: ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) implies f is_continuous_in x )
assume f is_continuous_in x ; ::_thesis: ( y in dom g & ( for s1 being sequence of (REAL-NS i) st rng s1 c= dom g & s1 is convergent & lim s1 = y holds
( g /* s1 is convergent & g /. y = lim (g /* s1) ) ) )
then g is_continuous_in y by AS, NFCONT_4:21;
hence ( y in dom g & ( for s1 being sequence of (REAL-NS i) st rng s1 c= dom g & s1 is convergent & lim s1 = y holds
( g /* s1 is convergent & g /. y = lim (g /* s1) ) ) ) by NFCONT_1:def_6; ::_thesis: verum
end;
hereby ::_thesis: verum
assume ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds
( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ; ::_thesis: f is_continuous_in x
then g is_continuous_in y by NFCONT_1:def_6;
hence f is_continuous_in x by AS, NFCONT_4:21; ::_thesis: verum
end;
end;
theorem :: PDIFF_9:41
for i being Element of NAT
for f, g being PartFunc of (REAL i),REAL
for x being Element of REAL i st f is_continuous_in x & g is_continuous_in x holds
f (#) g is_continuous_in x
proof
let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL
for x being Element of REAL i st f is_continuous_in x & g is_continuous_in x holds
f (#) g is_continuous_in x
let g1, g2 be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i st g1 is_continuous_in x & g2 is_continuous_in x holds
g1 (#) g2 is_continuous_in x
let x be Element of REAL i; ::_thesis: ( g1 is_continuous_in x & g2 is_continuous_in x implies g1 (#) g2 is_continuous_in x )
assume A2: ( g1 is_continuous_in x & g2 is_continuous_in x ) ; ::_thesis: g1 (#) g2 is_continuous_in x
reconsider y = x as Point of (REAL-NS i) by REAL_NS1:def_4;
reconsider f1 = g1, f2 = g2 as PartFunc of (REAL-NS i),REAL by REAL_NS1:def_4;
A3: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4;
( f1 is_continuous_in y & f2 is_continuous_in y ) by A2, NFCONT_4:21;
then X1: ( y in dom f1 & y in dom f2 ) by NFCONT_1:def_6;
then X2: y in dom (f1 (#) f2) by A3, XBOOLE_0:def_4;
now__::_thesis:_for_s1_being_sequence_of_(REAL-NS_i)_st_rng_s1_c=_dom_(f1_(#)_f2)_&_s1_is_convergent_&_lim_s1_=_y_holds_
(_(f1_(#)_f2)_/*_s1_is_convergent_&_(f1_(#)_f2)_/._y_=_lim_((f1_(#)_f2)_/*_s1)_)
let s1 be sequence of (REAL-NS i); ::_thesis: ( rng s1 c= dom (f1 (#) f2) & s1 is convergent & lim s1 = y implies ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1) ) )
assume that
A22: rng s1 c= dom (f1 (#) f2) and
A23: ( s1 is convergent & lim s1 = y ) ; ::_thesis: ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1) )
( dom (f1 (#) f2) c= dom f1 & dom (f1 (#) f2) c= dom f2 ) by A3, XBOOLE_1:17;
then A24: ( rng s1 c= dom f1 & rng s1 c= dom f2 ) by A22, XBOOLE_1:1;
then A25: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A2, A23, XTh360;
then (f1 /* s1) (#) (f2 /* s1) is convergent ;
hence (f1 (#) f2) /* s1 is convergent by A3, A22, RFUNCT_2:8; ::_thesis: (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1)
( f1 . y = f1 /. y & f2 . y = f2 /. y ) by X1, PARTFUN1:def_6;
then A29: ( f1 . y = lim (f1 /* s1) & f2 . y = lim (f2 /* s1) ) by A2, A23, A24, XTh360;
thus (f1 (#) f2) /. y = (f1 (#) f2) . y by X2, PARTFUN1:def_6
.= (f1 . y) * (f2 . y) by VALUED_1:5
.= lim ((f1 /* s1) (#) (f2 /* s1)) by A29, A25, SEQ_2:15
.= lim ((f1 (#) f2) /* s1) by A3, A22, RFUNCT_2:8 ; ::_thesis: verum
end;
hence g1 (#) g2 is_continuous_in x by XTh360, X2; ::_thesis: verum
end;
definition
let m be non empty Element of NAT ;
let Z be set ;
let f be PartFunc of (REAL m),REAL;
predf is_continuous_on Z means :XDef7: :: PDIFF_9:def 2
for x0 being Element of REAL m st x0 in Z holds
f | Z is_continuous_in x0;
end;
:: deftheorem XDef7 defines is_continuous_on PDIFF_9:def_2_:_
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL holds
( f is_continuous_on Z iff for x0 being Element of REAL m st x0 in Z holds
f | Z is_continuous_in x0 );
theorem XTh360B: :: PDIFF_9:42
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) )
assume AS: f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) )
assume P2: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z )
assume P0: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z
now__::_thesis:_for_x0_being_Point_of_(REAL-NS_m)_st_x0_in_Z_holds_
g_|_Z_is_continuous_in_x0
let x0 be Point of (REAL-NS m); ::_thesis: ( x0 in Z implies g | Z is_continuous_in x0 )
assume P3: x0 in Z ; ::_thesis: g | Z is_continuous_in x0
reconsider y0 = x0 as Element of REAL m by REAL_NS1:def_4;
f | Z is_continuous_in y0 by P3, P0, XDef7;
hence g | Z is_continuous_in x0 by AS, NFCONT_4:21; ::_thesis: verum
end;
hence g is_continuous_on Z by AS, P2, NFCONT_1:def_8; ::_thesis: verum
end;
assume P1: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z )
hence Z c= dom f by AS, NFCONT_1:def_8; ::_thesis: f is_continuous_on Z
let x0 be Element of REAL m; :: according to PDIFF_9:def_2 ::_thesis: ( x0 in Z implies f | Z is_continuous_in x0 )
assume P3: x0 in Z ; ::_thesis: f | Z is_continuous_in x0
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
g | Z is_continuous_in y0 by P3, P1, NFCONT_1:def_8;
hence f | Z is_continuous_in x0 by AS, NFCONT_4:21; ::_thesis: verum
end;
theorem XTh360C: :: PDIFF_9:43
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds
( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds
( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds
( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds
( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) )
let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g & Z c= dom f implies ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) )
assume AS: f = g ; ::_thesis: ( not Z c= dom f or ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) )
assume A0: Z c= dom f ; ::_thesis: ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) )
hereby ::_thesis: ( ( for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) implies f is_continuous_on Z )
assume f is_continuous_on Z ; ::_thesis: for s1 being sequence of (REAL-NS m) st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( g /* s1 is convergent & g /. (lim s1) = lim (g /* s1) )
then g is_continuous_on Z by A0, XTh360B, AS;
hence for s1 being sequence of (REAL-NS m) st rng s1 c= Z & s1 is convergent & lim s1 in Z holds
( g /* s1 is convergent & g /. (lim s1) = lim (g /* s1) ) by NFCONT125; ::_thesis: verum
end;
assume for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds
( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ; ::_thesis: f is_continuous_on Z
then g is_continuous_on Z by AS, A0, NFCONT125;
hence f is_continuous_on Z by XTh360B, AS; ::_thesis: verum
end;
theorem XTh37: :: PDIFF_9:44
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) )
assume A1: <>* f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
then A0: <>* (f | Z) = g | Z by LMXTh1;
hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) )
assume AK: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z )
assume A2: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z
A3: Z c= dom g by LMXTh0, A1, AK;
now__::_thesis:_for_x0_being_Element_of_REAL_m_st_x0_in_Z_holds_
g_|_Z_is_continuous_in_x0
let x0 be Element of REAL m; ::_thesis: ( x0 in Z implies g | Z is_continuous_in x0 )
assume x0 in Z ; ::_thesis: g | Z is_continuous_in x0
then f | Z is_continuous_in x0 by A2, XDef7;
hence g | Z is_continuous_in x0 by A0, XTh35; ::_thesis: verum
end;
hence g is_continuous_on Z by A3, PDIFF_7:def_7; ::_thesis: verum
end;
assume A5: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z )
then Z c= dom g by PDIFF_7:def_7;
hence Z c= dom f by LMXTh0, A1; ::_thesis: f is_continuous_on Z
let x0 be Element of REAL m; :: according to PDIFF_9:def_2 ::_thesis: ( x0 in Z implies f | Z is_continuous_in x0 )
assume x0 in Z ; ::_thesis: f | Z is_continuous_in x0
then g | Z is_continuous_in x0 by A5, PDIFF_7:def_7;
hence f | Z is_continuous_in x0 by A0, XTh35; ::_thesis: verum
end;
theorem XTh38: :: PDIFF_9:45
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL st Z c= dom f holds
( f is_continuous_on Z iff for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL st Z c= dom f holds
( f is_continuous_on Z iff for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z c= dom f holds
( f is_continuous_on Z iff for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z c= dom f implies ( f is_continuous_on Z iff for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
set g = <>* f;
assume A2: Z c= dom f ; ::_thesis: ( f is_continuous_on Z iff for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
hereby ::_thesis: ( ( for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_on Z )
assume f is_continuous_on Z ; ::_thesis: for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then A1: <>* f is_continuous_on Z by A2, XTh37;
thus for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ::_thesis: verum
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume A3: ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
A4: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) by A1, PDIFF_7:38;
take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A4; ::_thesis: for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
hereby ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume A5: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
then A6: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by A4;
( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by A5, A2, A3, XTh30;
then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29;
hence |.((f /. x1) - (f /. x0)).| < r by A6, XTh30D; ::_thesis: verum
end;
end;
end;
assume A7: for x0 being Element of REAL m
for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_on Z
A70: Z c= dom (<>* f) by A2, LMXTh0;
for y0 being Element of REAL m
for r being Real st y0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - y0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. y0)).| < r ) )
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) ) )
assume A8: ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) )
then consider s being Real such that
A9: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) by A7;
take s ; ::_thesis: ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) )
thus 0 < s by A9; ::_thesis: for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds
|.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r
hereby ::_thesis: verum
let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r )
assume A10: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r
then A11: |.((f /. x1) - (f /. x0)).| < r by A9;
( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by A2, A10, A8, XTh30;
then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29;
hence |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by A11, XTh30D; ::_thesis: verum
end;
end;
then <>* f is_continuous_on Z by A70, PDIFF_7:38;
hence f is_continuous_on Z by XTh37; ::_thesis: verum
end;
theorem :: PDIFF_9:46
for m being non empty Element of NAT
for Z being set
for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
( f + g is_continuous_on Z & f - g is_continuous_on Z )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g implies ( f + g is_continuous_on Z & f - g is_continuous_on Z ) )
assume ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g ) ; ::_thesis: ( f + g is_continuous_on Z & f - g is_continuous_on Z )
then ( <>* f is_continuous_on Z & <>* g is_continuous_on Z ) by XTh37;
then P2: ( (<>* f) + (<>* g) is_continuous_on Z & (<>* f) - (<>* g) is_continuous_on Z ) by XTh350X;
( (<>* f) + (<>* g) = <>* (f + g) & (<>* f) - (<>* g) = <>* (f - g) ) by LMXTh10;
hence ( f + g is_continuous_on Z & f - g is_continuous_on Z ) by P2, XTh37; ::_thesis: verum
end;
theorem :: PDIFF_9:47
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for r being Real st Z c= dom f & f is_continuous_on Z holds
r (#) f is_continuous_on Z
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for r being Real st Z c= dom f & f is_continuous_on Z holds
r (#) f is_continuous_on Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for r being Real st Z c= dom f & f is_continuous_on Z holds
r (#) f is_continuous_on Z
let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st Z c= dom f & f is_continuous_on Z holds
r (#) f is_continuous_on Z
let r be Real; ::_thesis: ( Z c= dom f & f is_continuous_on Z implies r (#) f is_continuous_on Z )
assume ( Z c= dom f & f is_continuous_on Z ) ; ::_thesis: r (#) f is_continuous_on Z
then <>* f is_continuous_on Z by XTh37;
then P2: r (#) (<>* f) is_continuous_on Z by XTh351X;
r (#) (<>* f) = <>* (r (#) f) by LMXTh11;
hence r (#) f is_continuous_on Z by P2, XTh37; ::_thesis: verum
end;
theorem :: PDIFF_9:48
for m being non empty Element of NAT
for Z being set
for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
f (#) g is_continuous_on Z
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
f (#) g is_continuous_on Z
let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds
f (#) g is_continuous_on Z
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g implies f (#) g is_continuous_on Z )
assume A1: ( f is_continuous_on Z & g is_continuous_on Z ) ; ::_thesis: ( not Z c= dom f or not Z c= dom g or f (#) g is_continuous_on Z )
assume AK: ( Z c= dom f & Z c= dom g ) ; ::_thesis: f (#) g is_continuous_on Z
reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),REAL by REAL_NS1:def_4;
P2: Z c= (dom f1) /\ (dom g1) by AK, XBOOLE_1:19;
AA: dom (f1 (#) g1) = (dom f1) /\ (dom g1) by VALUED_1:def_4;
now__::_thesis:_for_s1_being_sequence_of_(REAL-NS_m)_st_rng_s1_c=_Z_&_s1_is_convergent_&_lim_s1_in_Z_holds_
(_(f1_(#)_g1)_/*_s1_is_convergent_&_(f1_(#)_g1)_/._(lim_s1)_=_lim_((f1_(#)_g1)_/*_s1)_)
let s1 be sequence of (REAL-NS m); ::_thesis: ( rng s1 c= Z & s1 is convergent & lim s1 in Z implies ( (f1 (#) g1) /* s1 is convergent & (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1) ) )
assume A23: ( rng s1 c= Z & s1 is convergent & lim s1 in Z ) ; ::_thesis: ( (f1 (#) g1) /* s1 is convergent & (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1) )
then A25: ( f1 /* s1 is convergent & g1 /* s1 is convergent ) by AK, XTh360C, A1;
then A28: (f1 /* s1) (#) (g1 /* s1) is convergent ;
A26: rng s1 c= (dom f1) /\ (dom g1) by P2, A23, XBOOLE_1:1;
hence (f1 (#) g1) /* s1 is convergent by A28, RFUNCT_2:8; ::_thesis: (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1)
set y = lim s1;
( f1 . (lim s1) = f1 /. (lim s1) & g1 . (lim s1) = g1 /. (lim s1) ) by A23, AK, PARTFUN1:def_6;
then A29: ( f1 . (lim s1) = lim (f1 /* s1) & g1 . (lim s1) = lim (g1 /* s1) ) by A23, AK, XTh360C, A1;
thus (f1 (#) g1) /. (lim s1) = (f1 (#) g1) . (lim s1) by A23, P2, AA, PARTFUN1:def_6
.= (f1 . (lim s1)) * (g1 . (lim s1)) by VALUED_1:5
.= lim ((f1 /* s1) (#) (g1 /* s1)) by A29, A25, SEQ_2:15
.= lim ((f1 (#) g1) /* s1) by A26, RFUNCT_2:8 ; ::_thesis: verum
end;
hence f (#) g is_continuous_on Z by P2, AA, XTh360C; ::_thesis: verum
end;
theorem PDIFF736X: :: PDIFF_9:49
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g holds
( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) )
assume AS0: f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z )
hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) )
assume P0: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z )
assume Q0: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z
now__::_thesis:_for_y0_being_Point_of_(REAL-NS_m)
for_r_being_Real_st_y0_in_Z_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_y1_being_Point_of_(REAL-NS_m)_st_y1_in_Z_&_||.(y1_-_y0).||_<_s_holds_
|.((g_/._y1)_-_(g_/._y0)).|_<_r_)_)
let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) )
let r be Real; ::_thesis: ( y0 in Z & 0 < r implies ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) ) )
reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4;
assume ( y0 in Z & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) )
then consider s being Real such that
A7: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) by P0, Q0, XTh38;
take s = s; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) )
thus 0 < s by A7; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r
let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in Z & ||.(y1 - y0).|| < s implies |.((g /. y1) - (g /. y0)).| < r )
assume A8: ( y1 in Z & ||.(y1 - y0).|| < s ) ; ::_thesis: |.((g /. y1) - (g /. y0)).| < r
reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4;
||.(y1 - y0).|| = |.(x1 - x0).| by REAL_NS1:1, REAL_NS1:5;
hence |.((g /. y1) - (g /. y0)).| < r by AS0, A8, A7; ::_thesis: verum
end;
hence g is_continuous_on Z by AS0, P0, NFCONT_1:20; ::_thesis: verum
end;
assume Q1: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z )
then A60: Z c= dom f by AS0, NFCONT_1:20;
now__::_thesis:_for_x0_being_Element_of_REAL_m
for_r_being_Real_st_x0_in_Z_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_Z_&_|.(x1_-_x0).|_<_s_holds_
|.((f_/._x1)_-_(f_/._x0)).|_<_r_)_)
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4;
assume ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
A7: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds
|.((g /. y1) - (g /. y0)).| < r ) ) by Q1, NFCONT_1:20;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A7; ::_thesis: for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds
|.((f /. x1) - (f /. x0)).| < r
let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume A8: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
reconsider y1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4;
||.(y1 - y0).|| = |.(x1 - x0).| by REAL_NS1:1, REAL_NS1:5;
hence |.((f /. x1) - (f /. x0)).| < r by AS0, A8, A7; ::_thesis: verum
end;
hence ( Z c= dom f & f is_continuous_on Z ) by A60, XTh38; ::_thesis: verum
end;
theorem :: PDIFF_9:50
for m, n being non empty Element of NAT
for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
|.f.| is_continuous_on Z
proof
let m, n be non empty Element of NAT ; ::_thesis: for Z being set
for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
|.f.| is_continuous_on Z
let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds
|.f.| is_continuous_on Z
let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies |.f.| is_continuous_on Z )
assume A1: f is_continuous_on Z ; ::_thesis: |.f.| is_continuous_on Z
A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4;
then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ;
f1 is_continuous_on Z by A1, PDIFF_7:37;
then A3: ||.f1.|| is_continuous_on Z by NFCONT_1:28;
||.f1.|| = |.f.| by NFCONT_4:9, A2;
hence |.f.| is_continuous_on Z by A3, PDIFF736X; ::_thesis: verum
end;
theorem PDIFF620X: :: PDIFF_9:51
for m being non empty Element of NAT
for f, g being PartFunc of (REAL m),REAL
for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds
( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
proof
let m be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL
for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds
( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds
( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
let x be Element of REAL m; ::_thesis: ( f is_differentiable_in x & g is_differentiable_in x implies ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) )
assume ( f is_differentiable_in x & g is_differentiable_in x ) ; ::_thesis: ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
then P2: ( <>* f is_differentiable_in x & <>* g is_differentiable_in x ) by PDIFF_7:def_1;
then P4: ( (<>* f) + (<>* g) is_differentiable_in x & (<>* f) - (<>* g) is_differentiable_in x ) by PDIFF_6:20, PDIFF_6:21;
(<>* f) + (<>* g) = <>* (f + g) by LMXTh10;
hence f + g is_differentiable_in x by P4, PDIFF_7:def_1; ::_thesis: ( diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
thus diff ((f + g),x) = (proj (1,1)) * (diff (((<>* f) + (<>* g)),x)) by LMXTh10
.= (proj (1,1)) * ((diff ((<>* f),x)) + (diff ((<>* g),x))) by P2, PDIFF_6:20
.= (diff (f,x)) + (diff (g,x)) by INTEGR15:15 ; ::_thesis: ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
(<>* f) - (<>* g) = <>* (f - g) by LMXTh10;
hence f - g is_differentiable_in x by P4, PDIFF_7:def_1; ::_thesis: diff ((f - g),x) = (diff (f,x)) - (diff (g,x))
thus diff ((f - g),x) = (proj (1,1)) * (diff (((<>* f) - (<>* g)),x)) by LMXTh10
.= (proj (1,1)) * ((diff ((<>* f),x)) - (diff ((<>* g),x))) by P2, PDIFF_6:21
.= (diff (f,x)) - (diff (g,x)) by INTEGR15:15 ; ::_thesis: verum
end;
theorem PDIFF622X: :: PDIFF_9:52
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for r being Real
for x being Element of REAL m st f is_differentiable_in x holds
( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for r being Real
for x being Element of REAL m st f is_differentiable_in x holds
( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real
for x being Element of REAL m st f is_differentiable_in x holds
( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
let r be Real; ::_thesis: for x being Element of REAL m st f is_differentiable_in x holds
( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
let x be Element of REAL m; ::_thesis: ( f is_differentiable_in x implies ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) )
assume f is_differentiable_in x ; ::_thesis: ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
then P1: <>* f is_differentiable_in x by PDIFF_7:def_1;
then P2: r (#) (<>* f) is_differentiable_in x by PDIFF_6:22;
r (#) (<>* f) = <>* (r (#) f) by LMXTh11;
hence r (#) f is_differentiable_in x by P2, PDIFF_7:def_1; ::_thesis: diff ((r (#) f),x) = r (#) (diff (f,x))
thus diff ((r (#) f),x) = (proj (1,1)) * (diff ((r (#) (<>* f)),x)) by LMXTh11
.= (proj (1,1)) * (r (#) (diff ((<>* f),x))) by P1, PDIFF_6:22
.= r (#) (diff (f,x)) by INTEGR15:16 ; ::_thesis: verum
end;
definition
let Z be set ;
let m be non empty Element of NAT ;
let f be PartFunc of (REAL m),REAL;
predf is_differentiable_on Z means :XDef4: :: PDIFF_9:def 3
for x being Element of REAL m st x in Z holds
f | Z is_differentiable_in x;
end;
:: deftheorem XDef4 defines is_differentiable_on PDIFF_9:def_3_:_
for Z being set
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL holds
( f is_differentiable_on Z iff for x being Element of REAL m st x in Z holds
f | Z is_differentiable_in x );
theorem YTh30: :: PDIFF_9:53
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds
( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z )
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g implies ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) )
assume A1: <>* f = g ; ::_thesis: ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z )
AN: dom (<>* f) = dom f by LMXTh0;
hereby ::_thesis: ( g is_differentiable_on Z implies ( Z c= dom f & f is_differentiable_on Z ) )
assume AK: Z c= dom f ; ::_thesis: ( f is_differentiable_on Z implies g is_differentiable_on Z )
assume A3: f is_differentiable_on Z ; ::_thesis: g is_differentiable_on Z
A40: Z c= dom g by AK, LMXTh0, A1;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_Z_holds_
g_|_Z_is_differentiable_in_x
let x be Element of REAL m; ::_thesis: ( x in Z implies g | Z is_differentiable_in x )
assume x in Z ; ::_thesis: g | Z is_differentiable_in x
then f | Z is_differentiable_in x by A3, XDef4;
then <>* (f | Z) is_differentiable_in x by PDIFF_7:def_1;
hence g | Z is_differentiable_in x by A1, LMXTh1; ::_thesis: verum
end;
hence g is_differentiable_on Z by A40, PDIFF_6:def_4; ::_thesis: verum
end;
assume A5: g is_differentiable_on Z ; ::_thesis: ( Z c= dom f & f is_differentiable_on Z )
hence Z c= dom f by AN, A1, PDIFF_6:def_4; ::_thesis: f is_differentiable_on Z
hereby :: according to PDIFF_9:def_3 ::_thesis: verum
let x be Element of REAL m; ::_thesis: ( x in Z implies f | Z is_differentiable_in x )
assume x in Z ; ::_thesis: f | Z is_differentiable_in x
then A60: g | Z is_differentiable_in x by A5, PDIFF_6:def_4;
g | Z = <>* (f | Z) by A1, LMXTh1;
hence f | Z is_differentiable_in x by A60, PDIFF_7:def_1; ::_thesis: verum
end;
end;
theorem YTh32: :: PDIFF_9:54
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds
( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds
( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds
( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X is open implies ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x ) )
set g = <>* f;
assume AK: X c= dom f ; ::_thesis: ( not X is open or ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x ) )
assume X is open ; ::_thesis: ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds
f is_differentiable_in x )
then ex Z0 being Subset of (REAL-NS m) st
( Z0 = X & Z0 is open ) by PDIFF_7:def_3;
then A2: ( <>* f is_differentiable_on X iff ( X c= dom (<>* f) & ( for x being Element of REAL m st x in X holds
<>* f is_differentiable_in x ) ) ) by PDIFF_6:32;
hereby ::_thesis: ( ( for x being Element of REAL m st x in X holds
f is_differentiable_in x ) implies f is_differentiable_on X )
assume A3: f is_differentiable_on X ; ::_thesis: for x being Element of REAL m st x in X holds
f is_differentiable_in x
let x be Element of REAL m; ::_thesis: ( x in X implies f is_differentiable_in x )
assume x in X ; ::_thesis: f is_differentiable_in x
then <>* f is_differentiable_in x by AK, A2, A3, YTh30;
hence f is_differentiable_in x by PDIFF_7:def_1; ::_thesis: verum
end;
assume A4: for x being Element of REAL m st x in X holds
f is_differentiable_in x ; ::_thesis: f is_differentiable_on X
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
<>*_f_is_differentiable_in_x
let x be Element of REAL m; ::_thesis: ( x in X implies <>* f is_differentiable_in x )
assume x in X ; ::_thesis: <>* f is_differentiable_in x
then f is_differentiable_in x by A4;
hence <>* f is_differentiable_in x by PDIFF_7:def_1; ::_thesis: verum
end;
hence f is_differentiable_on X by AK, A2, LMXTh0, YTh30; ::_thesis: verum
end;
theorem YTh33: :: PDIFF_9:55
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds
X is open
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds
X is open
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds
X is open
let f be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & f is_differentiable_on X implies X is open )
reconsider g = <>* f as PartFunc of (REAL m),(REAL 1) ;
assume ( X c= dom f & f is_differentiable_on X ) ; ::_thesis: X is open
then g is_differentiable_on X by YTh30;
then ex Z0 being Subset of (REAL-NS m) st
( X = Z0 & Z0 is open ) by PDIFF_6:33;
hence X is open by PDIFF_7:def_3; ::_thesis: verum
end;
definition
let m be non empty Element of NAT ;
let Z be set ;
let f be PartFunc of (REAL m),REAL;
assume AK: Z c= dom f ;
funcf `| Z -> PartFunc of (REAL m),(Funcs ((REAL m),REAL)) means :XDef1: :: PDIFF_9:def 4
( dom it = Z & ( for x being Element of REAL m st x in Z holds
it /. x = diff (f,x) ) );
existence
ex b1 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) st
( dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = diff (f,x) ) )
proof
defpred S1[ Element of REAL m, set ] means ( $1 in Z & $2 = diff (f,$1) );
consider F being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) such that
A2: ( ( for x being Element of REAL m holds
( x in dom F iff ex z being Element of Funcs ((REAL m),REAL) st S1[x,z] ) ) & ( for x being Element of REAL m st x in dom F holds
S1[x,F . x] ) ) from SEQ_1:sch_2();
take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) )
A3: Z is Subset of (REAL m) by AK, XBOOLE_1:1;
now__::_thesis:_for_x_being_set_st_x_in_Z_holds_
x_in_dom_F
let x be set ; ::_thesis: ( x in Z implies x in dom F )
assume AS1: x in Z ; ::_thesis: x in dom F
then reconsider z = x as Element of REAL m by A3;
reconsider y = diff (f,z) as Element of Funcs ((REAL m),REAL) by FUNCT_2:8;
S1[z,y] by AS1;
hence x in dom F by A2; ::_thesis: verum
end;
then A4: Z c= dom F by TARSKI:def_3;
for y being set st y in dom F holds
y in Z by A2;
then dom F c= Z by TARSKI:def_3;
hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds
F /. x = diff (f,x)
hereby ::_thesis: verum
let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = diff (f,x) )
assume A5: x in Z ; ::_thesis: F /. x = diff (f,x)
then F . x = diff (f,x) by A2, A4;
hence F /. x = diff (f,x) by A5, A4, PARTFUN1:def_6; ::_thesis: verum
end;
end;
uniqueness
for b1, b2 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) st dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = diff (f,x) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds
b2 /. x = diff (f,x) ) holds
b1 = b2
proof
let F, G be PartFunc of (REAL m),(Funcs ((REAL m),REAL)); ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ) implies F = G )
assume that
A6: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = diff (f,x) ) ) and
A8: ( dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = diff (f,x) ) ) ; ::_thesis: F = G
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_
F_/._x_=_G_/._x
let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x )
assume A10: x in dom F ; ::_thesis: F /. x = G /. x
then F /. x = diff (f,x) by A6;
hence F /. x = G /. x by A6, A8, A10; ::_thesis: verum
end;
hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum
end;
end;
:: deftheorem XDef1 defines `| PDIFF_9:def_4_:_
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL st Z c= dom f holds
for b4 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) holds
( b4 = f `| Z iff ( dom b4 = Z & ( for x being Element of REAL m st x in Z holds
b4 /. x = diff (f,x) ) ) );
theorem :: PDIFF_9:56
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds
( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds
( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds
( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds
( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) )
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X c= dom f & f is_differentiable_on X implies ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) )
assume AS: ( <>* f = g & X c= dom f & f is_differentiable_on X ) ; ::_thesis: ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) )
hence g is_differentiable_on X by YTh30; ::_thesis: for x being Element of REAL m st x in X holds
(f `| X) /. x = (proj (1,1)) * ((g `| X) /. x)
AA: dom f = dom (<>* f) by LMXTh0;
let x be Element of REAL m; ::_thesis: ( x in X implies (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) )
assume P4: x in X ; ::_thesis: (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x)
then (f `| X) /. x = diff (f,x) by AS, XDef1;
hence (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) by AA, AS, P4, Def1; ::_thesis: verum
end;
theorem :: PDIFF_9:57
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X implies ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) )
assume AK: ( X c= dom f & X c= dom g ) ; ::_thesis: ( not f is_differentiable_on X or not g is_differentiable_on X or ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) )
assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) )
then P0: X is open by AK, YTh33;
dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1;
then P3: X c= dom (f + g) by AK, XBOOLE_1:19;
P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_f_+_g_is_differentiable_in_x_&_diff_((f_+_g),x)_=_(diff_(f,x))_+_(diff_(g,x))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) ) )
assume x in X ; ::_thesis: ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) )
then ( f is_differentiable_in x & g is_differentiable_in x ) by AK, AS1, P0, YTh32;
hence ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) ) by PDIFF620X; ::_thesis: verum
end;
then for x being Element of REAL m st x in X holds
f + g is_differentiable_in x ;
hence f + g is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds
((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x)
let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) )
assume P7: x in X ; ::_thesis: ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x)
then ((f + g) `| X) /. x = diff ((f + g),x) by P3, XDef1;
then ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) by P7, P5;
then ((f + g) `| X) /. x = ((f `| X) /. x) + (diff (g,x)) by AK, P7, XDef1;
hence ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) by AK, P7, XDef1; ::_thesis: verum
end;
theorem :: PDIFF_9:58
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X implies ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) )
assume AK: ( X c= dom f & X c= dom g ) ; ::_thesis: ( not f is_differentiable_on X or not g is_differentiable_on X or ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) )
assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
then P0: X is open by AK, YTh33;
dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12;
then P3: X c= dom (f - g) by AK, XBOOLE_1:19;
P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_f_-_g_is_differentiable_in_x_&_diff_((f_-_g),x)_=_(diff_(f,x))_-_(diff_(g,x))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) )
assume x in X ; ::_thesis: ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
then ( f is_differentiable_in x & g is_differentiable_in x ) by AK, AS1, P0, YTh32;
hence ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) by PDIFF620X; ::_thesis: verum
end;
then for x being Element of REAL m st x in X holds
f - g is_differentiable_in x ;
hence f - g is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x)
let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) )
assume P7: x in X ; ::_thesis: ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x)
then ((f - g) `| X) /. x = diff ((f - g),x) by P3, XDef1;
then ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) by P7, P5;
then ((f - g) `| X) /. x = ((f `| X) /. x) - (diff (g,x)) by AK, P7, XDef1;
hence ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) by AK, P7, XDef1; ::_thesis: verum
end;
theorem :: PDIFF_9:59
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for r being Real st X c= dom f & f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for r being Real st X c= dom f & f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL
for r being Real st X c= dom f & f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st X c= dom f & f is_differentiable_on X holds
( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) )
let r be Real; ::_thesis: ( X c= dom f & f is_differentiable_on X implies ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) )
assume AK: X c= dom f ; ::_thesis: ( not f is_differentiable_on X or ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) )
assume AS1: f is_differentiable_on X ; ::_thesis: ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) )
then P0: X is open by AK, YTh33;
P3: X c= dom (r (#) f) by AK, VALUED_1:def_5;
P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_r_(#)_f_is_differentiable_in_x_&_diff_((r_(#)_f),x)_=_r_(#)_(diff_(f,x))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) )
assume x in X ; ::_thesis: ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) )
then f is_differentiable_in x by AS1, P0, AK, YTh32;
hence ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) by PDIFF622X; ::_thesis: verum
end;
then for x being Element of REAL m st x in X holds
r (#) f is_differentiable_in x ;
hence r (#) f is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds
((r (#) f) `| X) /. x = r (#) ((f `| X) /. x)
let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) )
assume P7: x in X ; ::_thesis: ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x)
then ((r (#) f) `| X) /. x = diff ((r (#) f),x) by P3, XDef1;
hence ((r (#) f) `| X) /. x = r (#) (diff (f,x)) by P7, P5
.= r (#) ((f `| X) /. x) by AK, P7, XDef1 ;
::_thesis: verum
end;
definition
let m be non empty Element of NAT ;
let Z be set ;
let i be Element of NAT ;
let f be PartFunc of (REAL m),REAL;
predf is_partial_differentiable_on Z,i means :CWDef19: :: PDIFF_9:def 5
( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f | Z is_partial_differentiable_in x,i ) );
end;
:: deftheorem CWDef19 defines is_partial_differentiable_on PDIFF_9:def_5_:_
for m being non empty Element of NAT
for Z being set
for i being Element of NAT
for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds
f | Z is_partial_differentiable_in x,i ) ) );
definition
let m be non empty Element of NAT ;
let Z be set ;
let i be Element of NAT ;
let f be PartFunc of (REAL m),REAL;
assume A1: f is_partial_differentiable_on Z,i ;
funcf `partial| (Z,i) -> PartFunc of (REAL m),REAL means :DefPDX: :: PDIFF_9:def 6
( dom it = Z & ( for x being Element of REAL m st x in Z holds
it /. x = partdiff (f,x,i) ) );
existence
ex b1 being PartFunc of (REAL m),REAL st
( dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = partdiff (f,x,i) ) )
proof
deffunc H1( Element of REAL m) -> Element of REAL = partdiff (f,$1,i);
defpred S1[ Element of REAL m] means $1 in Z;
consider F being PartFunc of (REAL m),REAL such that
A2: ( ( for x being Element of REAL m holds
( x in dom F iff S1[x] ) ) & ( for x being Element of REAL m st x in dom F holds
F . x = H1(x) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = partdiff (f,x,i) ) )
now__::_thesis:_for_y_being_set_st_y_in_Z_holds_
y_in_dom_F
Z c= dom f by A1, CWDef19;
then A3: Z is Subset of (REAL m) by XBOOLE_1:1;
let y be set ; ::_thesis: ( y in Z implies y in dom F )
assume y in Z ; ::_thesis: y in dom F
hence y in dom F by A2, A3; ::_thesis: verum
end;
then A4: Z c= dom F by TARSKI:def_3;
for y being set st y in dom F holds
y in Z by A2;
then dom F c= Z by TARSKI:def_3;
hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds
F /. x = partdiff (f,x,i)
hereby ::_thesis: verum
let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = partdiff (f,x,i) )
assume x in Z ; ::_thesis: F /. x = partdiff (f,x,i)
then A5: x in dom F by A2;
then F . x = partdiff (f,x,i) by A2;
hence F /. x = partdiff (f,x,i) by A5, PARTFUN1:def_6; ::_thesis: verum
end;
end;
uniqueness
for b1, b2 being PartFunc of (REAL m),REAL st dom b1 = Z & ( for x being Element of REAL m st x in Z holds
b1 /. x = partdiff (f,x,i) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds
b2 /. x = partdiff (f,x,i) ) holds
b1 = b2
proof
let F, G be PartFunc of (REAL m),REAL; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = partdiff (f,x,i) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = partdiff (f,x,i) ) implies F = G )
assume that
A6: ( dom F = Z & ( for x being Element of REAL m st x in Z holds
F /. x = partdiff (f,x,i) ) ) and
A8: ( dom G = Z & ( for x being Element of REAL m st x in Z holds
G /. x = partdiff (f,x,i) ) ) ; ::_thesis: F = G
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_
F_/._x_=_G_/._x
let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x )
assume A10: x in dom F ; ::_thesis: F /. x = G /. x
then F /. x = partdiff (f,x,i) by A6;
hence F /. x = G /. x by A6, A8, A10; ::_thesis: verum
end;
hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum
end;
end;
:: deftheorem DefPDX defines `partial| PDIFF_9:def_6_:_
for m being non empty Element of NAT
for Z being set
for i being Element of NAT
for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_on Z,i holds
for b5 being PartFunc of (REAL m),REAL holds
( b5 = f `partial| (Z,i) iff ( dom b5 = Z & ( for x being Element of REAL m st x in Z holds
b5 /. x = partdiff (f,x,i) ) ) );
theorem PDIFF734: :: PDIFF_9:60
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) ) )
assume that
A1: X is open and
A2: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) )
thus ( f is_partial_differentiable_on X,i implies ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) ) ) ::_thesis: ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) implies f is_partial_differentiable_on X,i )
proof
assume A3: f is_partial_differentiable_on X,i ; ::_thesis: ( X c= dom f & ( for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i ) )
hence A4: X c= dom f by CWDef19; ::_thesis: for x being Element of REAL m st x in X holds
f is_partial_differentiable_in x,i
let nx0 be Element of REAL m; ::_thesis: ( nx0 in X implies f is_partial_differentiable_in nx0,i )
reconsider x0 = (proj (i,m)) . nx0 as Element of REAL ;
assume A5: nx0 in X ; ::_thesis: f is_partial_differentiable_in nx0,i
then f | X is_partial_differentiable_in nx0,i by A3, CWDef19;
then (f | X) * (reproj (i,nx0)) is_differentiable_in x0 by PDIFF_1:def_11;
then consider N0 being Neighbourhood of x0 such that
A6: N0 c= dom ((f | X) * (reproj (i,nx0))) and
A7: ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N0 holds
(((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by FDIFF_1:def_4;
consider L being LinearFunc, R being RestFunc such that
A8: for x being Element of REAL st x in N0 holds
(((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A7;
consider N1 being Neighbourhood of x0 such that
A9: for x being Element of REAL st x in N1 holds
(reproj (i,nx0)) . x in X by A1, A2, A5, Lm5;
A10: now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N1_holds_
(reproj_(i,nx0))_._x_in_dom_(f_|_X)
let x be Element of REAL ; ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | X) )
assume x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | X)
then (reproj (i,nx0)) . x in X by A9;
then (reproj (i,nx0)) . x in (dom f) /\ X by A4, XBOOLE_0:def_4;
hence (reproj (i,nx0)) . x in dom (f | X) by RELAT_1:61; ::_thesis: verum
end;
consider N being Neighbourhood of x0 such that
NXX: ( N c= N0 & N c= N1 ) by RCOMP_1:17;
(f | X) * (reproj (i,nx0)) c= f * (reproj (i,nx0)) by RELAT_1:29, RELAT_1:59;
then A11: dom ((f | X) * (reproj (i,nx0))) c= dom (f * (reproj (i,nx0))) by RELAT_1:11;
N c= dom ((f | X) * (reproj (i,nx0))) by A6, NXX, XBOOLE_1:1;
then A12: N c= dom (f * (reproj (i,nx0))) by A11, XBOOLE_1:1;
now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N_holds_
((f_*_(reproj_(i,nx0)))_._x)_-_((f_*_(reproj_(i,nx0)))_._x0)_=_(L_._(x_-_x0))_+_(R_._(x_-_x0))
let x be Element of REAL ; ::_thesis: ( x in N implies ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume A13: x in N ; ::_thesis: ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0))
then (reproj (i,nx0)) . x in dom (f | X) by A10, NXX;
then A17: ( (reproj (i,nx0)) . x in dom f & (reproj (i,nx0)) . x in X ) by RELAT_1:57;
(reproj (i,nx0)) . x0 in dom (f | X) by A10, RCOMP_1:16;
then A19: ( (reproj (i,nx0)) . x0 in dom f & (reproj (i,nx0)) . x0 in X ) by RELAT_1:57;
A15: dom (reproj (i,nx0)) = REAL by FUNCT_2:def_1;
then A20: ((f | X) * (reproj (i,nx0))) . x = (f | X) . ((reproj (i,nx0)) . x) by FUNCT_1:13
.= f . ((reproj (i,nx0)) . x) by A17, FUNCT_1:49
.= (f * (reproj (i,nx0))) . x by A15, FUNCT_1:13 ;
((f | X) * (reproj (i,nx0))) . x0 = (f | X) . ((reproj (i,nx0)) . x0) by A15, FUNCT_1:13
.= f . ((reproj (i,nx0)) . x0) by A19, FUNCT_1:49
.= (f * (reproj (i,nx0))) . x0 by A15, FUNCT_1:13 ;
hence ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A8, A13, NXX, A20; ::_thesis: verum
end;
then f * (reproj (i,nx0)) is_differentiable_in x0 by A12, FDIFF_1:def_4;
hence f is_partial_differentiable_in nx0,i by PDIFF_1:def_11; ::_thesis: verum
end;
assume that
A21: X c= dom f and
A22: for nx being Element of REAL m st nx in X holds
f is_partial_differentiable_in nx,i ; ::_thesis: f is_partial_differentiable_on X,i
thus X c= dom f by A21; :: according to PDIFF_9:def_5 ::_thesis: for x being Element of REAL m st x in X holds
f | X is_partial_differentiable_in x,i
now__::_thesis:_for_nx0_being_Element_of_REAL_m_st_nx0_in_X_holds_
f_|_X_is_partial_differentiable_in_nx0,i
let nx0 be Element of REAL m; ::_thesis: ( nx0 in X implies f | X is_partial_differentiable_in nx0,i )
assume A23: nx0 in X ; ::_thesis: f | X is_partial_differentiable_in nx0,i
then A24: f is_partial_differentiable_in nx0,i by A22;
reconsider x0 = (proj (i,m)) . nx0 as Element of REAL ;
f * (reproj (i,nx0)) is_differentiable_in x0 by A24, PDIFF_1:def_11;
then consider N0 being Neighbourhood of x0 such that
N0 c= dom (f * (reproj (i,nx0))) and
A25: ex L being LinearFunc ex R being RestFunc st
for x being Element of REAL st x in N0 holds
((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by FDIFF_1:def_4;
consider N1 being Neighbourhood of x0 such that
A26: for x being Element of REAL st x in N1 holds
(reproj (i,nx0)) . x in X by A1, A2, A23, Lm5;
A27: now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N1_holds_
(reproj_(i,nx0))_._x_in_dom_(f_|_X)
let x be Element of REAL ; ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | X) )
assume x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | X)
then (reproj (i,nx0)) . x in X by A26;
then (reproj (i,nx0)) . x in (dom f) /\ X by A21, XBOOLE_0:def_4;
hence (reproj (i,nx0)) . x in dom (f | X) by RELAT_1:61; ::_thesis: verum
end;
A28: N1 c= dom ((f | X) * (reproj (i,nx0)))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in N1 or z in dom ((f | X) * (reproj (i,nx0))) )
assume A29: z in N1 ; ::_thesis: z in dom ((f | X) * (reproj (i,nx0)))
then A30: z in REAL ;
reconsider x = z as Element of REAL by A29;
A31: (reproj (i,nx0)) . x in dom (f | X) by A29, A27;
z in dom (reproj (i,nx0)) by A30, FUNCT_2:def_1;
hence z in dom ((f | X) * (reproj (i,nx0))) by A31, FUNCT_1:11; ::_thesis: verum
end;
consider N being Neighbourhood of x0 such that
NXX: ( N c= N0 & N c= N1 ) by RCOMP_1:17;
A32: N c= dom ((f | X) * (reproj (i,nx0))) by NXX, A28, XBOOLE_1:1;
consider L being LinearFunc, R being RestFunc such that
A33: for x being Element of REAL st x in N0 holds
((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A25;
now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N_holds_
(((f_|_X)_*_(reproj_(i,nx0)))_._x)_-_(((f_|_X)_*_(reproj_(i,nx0)))_._x0)_=_(L_._(x_-_x0))_+_(R_._(x_-_x0))
let x be Element of REAL ; ::_thesis: ( x in N implies (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) )
assume A34: x in N ; ::_thesis: (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0))
A36: dom (reproj (i,nx0)) = REAL by FUNCT_2:def_1;
(reproj (i,nx0)) . x in dom (f | X) by A27, A34, NXX;
then A38: (reproj (i,nx0)) . x in (dom f) /\ X by RELAT_1:61;
(reproj (i,nx0)) . x0 in dom (f | X) by A27, RCOMP_1:16;
then A41: (reproj (i,nx0)) . x0 in (dom f) /\ X by RELAT_1:61;
A43: ((f | X) * (reproj (i,nx0))) . x = (f | X) . ((reproj (i,nx0)) /. x) by A36, FUNCT_1:13
.= f . ((reproj (i,nx0)) . x) by A38, FUNCT_1:48
.= (f * (reproj (i,nx0))) . x by A36, FUNCT_1:13 ;
((f | X) * (reproj (i,nx0))) . x0 = (f | X) . ((reproj (i,nx0)) . x0) by A36, FUNCT_1:13
.= f . ((reproj (i,nx0)) . x0) by A41, FUNCT_1:48
.= (f * (reproj (i,nx0))) . x0 by A36, FUNCT_1:13 ;
hence (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A43, A34, A33, NXX; ::_thesis: verum
end;
then (f | X) * (reproj (i,nx0)) is_differentiable_in x0 by A32, FDIFF_1:def_4;
hence f | X is_partial_differentiable_in nx0,i by PDIFF_1:def_11; ::_thesis: verum
end;
hence for x being Element of REAL m st x in X holds
f | X is_partial_differentiable_in x,i ; ::_thesis: verum
end;
theorem CW020: :: PDIFF_9:61
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) )
assume AS: ( <>* f = g & X is open & 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i )
hereby ::_thesis: ( g is_partial_differentiable_on X,i implies f is_partial_differentiable_on X,i )
assume P2: f is_partial_differentiable_on X,i ; ::_thesis: g is_partial_differentiable_on X,i
then X c= dom f by AS, PDIFF734;
then P3: X c= dom g by LMXTh0, AS;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
g_is_partial_differentiable_in_x,i
let x be Element of REAL m; ::_thesis: ( x in X implies g is_partial_differentiable_in x,i )
assume x in X ; ::_thesis: g is_partial_differentiable_in x,i
then f is_partial_differentiable_in x,i by P2, AS, PDIFF734;
hence g is_partial_differentiable_in x,i by AS, PDIFF_1:18; ::_thesis: verum
end;
hence g is_partial_differentiable_on X,i by P3, AS, PDIFF_7:34; ::_thesis: verum
end;
hereby ::_thesis: verum
assume P3: g is_partial_differentiable_on X,i ; ::_thesis: f is_partial_differentiable_on X,i
then X c= dom g by AS, PDIFF_7:34;
then P4: X c= dom f by LMXTh0, AS;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
f_is_partial_differentiable_in_x,i
let x be Element of REAL m; ::_thesis: ( x in X implies f is_partial_differentiable_in x,i )
assume x in X ; ::_thesis: f is_partial_differentiable_in x,i
then g is_partial_differentiable_in x,i by P3, AS, PDIFF_7:34;
hence f is_partial_differentiable_in x,i by AS, PDIFF_1:18; ::_thesis: verum
end;
hence f is_partial_differentiable_on X,i by AS, PDIFF734, P4; ::_thesis: verum
end;
end;
theorem CW021: :: PDIFF_9:62
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i implies ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) )
assume AS: ( <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i ) ; ::_thesis: ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X )
then P1: g is_partial_differentiable_on X,i by CW020;
set ff = f `partial| (X,i);
set gg = g `partial| (X,i);
EQ1: for x, y being Element of REAL m st x in X & y in X holds
|.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).|
proof
let x, y be Element of REAL m; ::_thesis: ( x in X & y in X implies |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| )
assume EQ2: ( x in X & y in X ) ; ::_thesis: |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).|
then EQ3: ( (f `partial| (X,i)) /. x = partdiff (f,x,i) & (f `partial| (X,i)) /. y = partdiff (f,y,i) ) by AS, DefPDX;
EQ5: ( (g `partial| (X,i)) /. x = partdiff (g,x,i) & (g `partial| (X,i)) /. y = partdiff (g,y,i) ) by P1, EQ2, PDIFF_7:def_5;
( g is_partial_differentiable_in x,i & g is_partial_differentiable_in y,i ) by P1, EQ2, AS, PDIFF_7:34;
then ( partdiff (g,x,i) = <*(partdiff (f,x,i))*> & partdiff (g,y,i) = <*(partdiff (f,y,i))*> ) by AS, PDIFF_1:19;
then ((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y) = <*(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y))*> by EQ3, EQ5, RVSUM_1:29;
hence |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| by XTh30D; ::_thesis: verum
end;
D1: dom (g `partial| (X,i)) = X by P1, PDIFF_7:def_5;
D2: dom (f `partial| (X,i)) = X by DefPDX, AS;
hereby ::_thesis: ( g `partial| (X,i) is_continuous_on X implies f `partial| (X,i) is_continuous_on X )
assume Q2: f `partial| (X,i) is_continuous_on X ; ::_thesis: g `partial| (X,i) is_continuous_on X
now__::_thesis:_for_x0_being_Element_of_REAL_m
for_r_being_Real_st_x0_in_X_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_
|.(((g_`partial|_(X,i))_/._x1)_-_((g_`partial|_(X,i))_/._x0)).|_<_r_)_)
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) )
assume Q40: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) )
then consider s being Real such that
Q41: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) by D2, Q2, XTh38;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) )
thus 0 < s by Q41; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r )
assume Q42: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r
then |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by Q41;
hence |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r by Q40, Q42, EQ1; ::_thesis: verum
end;
hence g `partial| (X,i) is_continuous_on X by D1, PDIFF_7:38; ::_thesis: verum
end;
hereby ::_thesis: verum
assume Q2: g `partial| (X,i) is_continuous_on X ; ::_thesis: f `partial| (X,i) is_continuous_on X
now__::_thesis:_for_x0_being_Element_of_REAL_m
for_r_being_Real_st_x0_in_X_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_
|.(((f_`partial|_(X,i))_/._x1)_-_((f_`partial|_(X,i))_/._x0)).|_<_r_)_)
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) )
assume Q40: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
then consider s being Real such that
Q41: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) by Q2, PDIFF_7:38;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) )
thus 0 < s by Q41; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
|.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r )
assume Q42: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r
then |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r by Q41;
hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by Q40, Q42, EQ1; ::_thesis: verum
end;
hence f `partial| (X,i) is_continuous_on X by XTh38, D2; ::_thesis: verum
end;
end;
CW022: for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1)
for x1, x0, v being Element of REAL m st <>* f = g holds
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).|
proof
let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL
for g being PartFunc of (REAL m),(REAL 1)
for x1, x0, v being Element of REAL m st <>* f = g holds
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).|
let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1)
for x1, x0, v being Element of REAL m st <>* f = g holds
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).|
let g be PartFunc of (REAL m),(REAL 1); ::_thesis: for x1, x0, v being Element of REAL m st <>* f = g holds
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).|
let x1, x0, v be Element of REAL m; ::_thesis: ( <>* f = g implies |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| )
assume AS: <>* f = g ; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).|
set I = proj (1,1);
reconsider w0 = (diff (g,x0)) . v, w1 = (diff (g,x1)) . v as Point of (REAL-NS 1) by REAL_NS1:def_4;
( dom (diff (g,x1)) = REAL m & dom (diff (g,x0)) = REAL m ) by FUNCT_2:def_1;
then ( (diff (f,x0)) . v = (proj (1,1)) . w0 & (diff (f,x1)) . v = (proj (1,1)) . w1 ) by AS, FUNCT_1:13;
then ((diff (f,x1)) . v) - ((diff (f,x0)) . v) = (proj (1,1)) . (w1 - w0) by PDIFF_1:4;
then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = ||.(w1 - w0).|| by PDIFF_1:4;
hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| by REAL_NS1:1, REAL_NS1:5; ::_thesis: verum
end;
theorem :: PDIFF_9:63
for m being non empty Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds
( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & X c= dom f implies ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) )
set g = <>* f;
assume AS1: ( X is open & X c= dom f ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) )
then AS2: X c= dom (<>* f) by LMXTh0;
hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume P1: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) )
P3: for i being Element of NAT st 1 <= i & i <= m holds
( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) )
assume P20: ( 1 <= i & i <= m ) ; ::_thesis: ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X )
then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by P1;
hence ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) by AS1, CW020, CW021, P20; ::_thesis: verum
end;
then <>* f is_differentiable_on X by CW01, AS1, AS2;
hence f is_differentiable_on X by YTh30; ::_thesis: for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
thus for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ::_thesis: verum
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) )
assume ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
then consider s being Real such that
Q2: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) by P3, CW01, AS1, AS2;
take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) )
thus 0 < s by Q2; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| )
assume Q3: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
let v be Element of REAL m; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.|
|.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| by Q3, Q2;
hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by CW022; ::_thesis: verum
end;
end;
now__::_thesis:_(_f_is_differentiable_on_X_&_(_for_x0_being_Element_of_REAL_m
for_r_being_Real_st_x0_in_X_&_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_
for_v_being_Element_of_REAL_m_holds_|.(((diff_(f,x1))_._v)_-_((diff_(f,x0))_._v)).|_<=_r_*_|.v.|_)_)_)_implies_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_
(_f_is_partial_differentiable_on_X,i_&_f_`partial|_(X,i)_is_continuous_on_X_)_)
assume P1: ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then P2: <>* f is_differentiable_on X by AS1, YTh30;
P3: for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) )
proof
let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) )
assume ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) )
then consider s being Real such that
Q2: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) by P1;
take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) )
thus 0 < s by Q2; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.|
let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| )
assume Q3: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.|
let v be Element of REAL m; ::_thesis: |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.|
|.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by Q3, Q2;
hence |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| by CW022; ::_thesis: verum
end;
thus for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) )
assume P4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X )
then P5: ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) by P3, CW01, AS2, AS1, P2;
hence f is_partial_differentiable_on X,i by P4, CW020, AS1; ::_thesis: f `partial| (X,i) is_continuous_on X
hence f `partial| (X,i) is_continuous_on X by P4, P5, CW021, AS1; ::_thesis: verum
end;
end;
hence ( f is_differentiable_on X & ( for x0 being Element of REAL m
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds
for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds
( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) ; ::_thesis: verum
end;
LM1291: for i, k being Element of NAT
for f, g being PartFunc of (REAL i),REAL
for x being Element of REAL i holds (f * (reproj (k,x))) (#) (g * (reproj (k,x))) = (f (#) g) * (reproj (k,x))
proof
let i, k be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL
for x being Element of REAL i holds (f * (reproj (k,x))) (#) (g * (reproj (k,x))) = (f (#) g) * (reproj (k,x))
let f1, f2 be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i holds (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x))
let x be Element of REAL i; ::_thesis: (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x))
A1: dom (reproj (k,x)) = REAL by FUNCT_2:def_1;
A2: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4;
for s being Element of REAL holds
( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) )
proof
let s be Element of REAL ; ::_thesis: ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) )
( s in dom ((f1 (#) f2) * (reproj (k,x))) iff (reproj (k,x)) . s in (dom f1) /\ (dom f2) ) by A2, A1, FUNCT_1:11;
then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff ( (reproj (k,x)) . s in dom f1 & (reproj (k,x)) . s in dom f2 ) ) by XBOOLE_0:def_4;
then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff ( s in dom (f1 * (reproj (k,x))) & s in dom (f2 * (reproj (k,x))) ) ) by A1, FUNCT_1:11;
then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in (dom (f1 * (reproj (k,x)))) /\ (dom (f2 * (reproj (k,x)))) ) by XBOOLE_0:def_4;
hence ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) by VALUED_1:def_4; ::_thesis: verum
end;
then for s being set holds
( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) ;
then A3: dom ((f1 (#) f2) * (reproj (k,x))) = dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) by TARSKI:1;
for z being Element of REAL st z in dom ((f1 (#) f2) * (reproj (k,x))) holds
((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z
proof
let z be Element of REAL ; ::_thesis: ( z in dom ((f1 (#) f2) * (reproj (k,x))) implies ((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z )
assume A5: z in dom ((f1 (#) f2) * (reproj (k,x))) ; ::_thesis: ((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z
then (reproj (k,x)) . z in (dom f1) /\ (dom f2) by A2, FUNCT_1:11;
then ( (reproj (k,x)) . z in dom f1 & (reproj (k,x)) . z in dom f2 ) by XBOOLE_0:def_4;
then ( z in dom (f1 * (reproj (k,x))) & z in dom (f2 * (reproj (k,x))) ) by A1, FUNCT_1:11;
then A13: ( f1 . ((reproj (k,x)) . z) = (f1 * (reproj (k,x))) . z & f2 . ((reproj (k,x)) . z) = (f2 * (reproj (k,x))) . z ) by FUNCT_1:12;
thus ((f1 (#) f2) * (reproj (k,x))) . z = (f1 (#) f2) . ((reproj (k,x)) . z) by A5, FUNCT_1:12
.= (f1 . ((reproj (k,x)) . z)) * (f2 . ((reproj (k,x)) . z)) by VALUED_1:5
.= ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z by A3, A5, A13, VALUED_1:def_4 ; ::_thesis: verum
end;
hence (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x)) by A3, PARTFUN1:5; ::_thesis: verum
end;
theorem MPDIFF129: :: PDIFF_9:64
for m being non empty Element of NAT
for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL
for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds
( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL
for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds
( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL
for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds
( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds
( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
let x be Element of REAL m; ::_thesis: ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i implies ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
assume ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) ; ::_thesis: ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
then P1: ( f * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x & g * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x ) by PDIFF_1:def_11;
set y = (proj (i,m)) . x;
dom (reproj (i,x)) = REAL by FUNCT_2:def_1;
then P7: ( (f * (reproj (i,x))) . ((proj (i,m)) . x) = f . ((reproj (i,x)) . ((proj (i,m)) . x)) & (g * (reproj (i,x))) . ((proj (i,m)) . x) = g . ((reproj (i,x)) . ((proj (i,m)) . x)) ) by FUNCT_1:13;
then P6: (f * (reproj (i,x))) . ((proj (i,m)) . x) = f . x by LMMMTh6;
(f * (reproj (i,x))) (#) (g * (reproj (i,x))) = (f (#) g) * (reproj (i,x)) by LM1291;
then (f (#) g) * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x by P1, FDIFF_1:16;
hence f (#) g is_partial_differentiable_in x,i by PDIFF_1:def_11; ::_thesis: partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i)))
thus partdiff ((f (#) g),x,i) = diff (((f * (reproj (i,x))) (#) (g * (reproj (i,x)))),((proj (i,m)) . x)) by LM1291
.= (((g * (reproj (i,x))) . ((proj (i,m)) . x)) * (partdiff (f,x,i))) + (((f * (reproj (i,x))) . ((proj (i,m)) . x)) * (diff ((g * (reproj (i,x))),((proj (i,m)) . x)))) by P1, FDIFF_1:16
.= ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by P6, P7, LMMMTh6 ; ::_thesis: verum
end;
theorem XXX1: :: PDIFF_9:65
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) )
assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734;
Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS;
dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1;
then P3: X c= dom (f + g) by P1, XBOOLE_1:19;
XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_f_+_g_is_partial_differentiable_in_x,i_&_partdiff_((f_+_g),x,i)_=_(partdiff_(f,x,i))_+_(partdiff_(g,x,i))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) )
assume x in X ; ::_thesis: ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) )
then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734;
hence ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) by PDIFF_1:29; ::_thesis: verum
end;
then P7: for x being Element of REAL m st x in X holds
f + g is_partial_differentiable_in x,i ;
then P8: f + g is_partial_differentiable_on X,i by P3, PDIFF734, AS;
then P9: dom ((f + g) `partial| (X,i)) = X by DefPDX;
P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_+_g)_`partial|_(X,i))_/._x_=_(partdiff_(f,x,i))_+_(partdiff_(g,x,i))
let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) )
assume P10: x in X ; ::_thesis: ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i))
then ((f + g) `partial| (X,i)) /. x = partdiff ((f + g),x,i) by P8, DefPDX;
hence ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) by XX1, P10; ::_thesis: verum
end;
P11: dom ((f `partial| (X,i)) + (g `partial| (X,i))) = (dom (f `partial| (X,i))) /\ (dom (g `partial| (X,i))) by VALUED_1:def_1;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_+_g)_`partial|_(X,i))_._x_=_((f_`partial|_(X,i))_+_(g_`partial|_(X,i)))_._x
let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x )
assume A1: x in X ; ::_thesis: ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x
thus ((f + g) `partial| (X,i)) . x = ((f + g) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6
.= (partdiff (f,x,i)) + (partdiff (g,x,i)) by P10, A1
.= ((f `partial| (X,i)) /. x) + (partdiff (g,x,i)) by A1, DefPDX, AS
.= ((f `partial| (X,i)) /. x) + ((g `partial| (X,i)) /. x) by A1, DefPDX, AS
.= ((f `partial| (X,i)) . x) + ((g `partial| (X,i)) /. x) by A1, Q1, PARTFUN1:def_6
.= ((f `partial| (X,i)) . x) + ((g `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6
.= ((f `partial| (X,i)) + (g `partial| (X,i))) . x by A1, P11, Q1, VALUED_1:def_1 ; ::_thesis: verum
end;
hence ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) by P7, P3, PDIFF734, AS, P9, P10, P11, Q1, PARTFUN1:5; ::_thesis: verum
end;
theorem XXX2: :: PDIFF_9:66
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) )
assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734;
Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS;
dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12;
then P3: X c= dom (f - g) by P1, XBOOLE_1:19;
XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_f_-_g_is_partial_differentiable_in_x,i_&_partdiff_((f_-_g),x,i)_=_(partdiff_(f,x,i))_-_(partdiff_(g,x,i))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) )
assume x in X ; ::_thesis: ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) )
then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734;
hence ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) by PDIFF_1:31; ::_thesis: verum
end;
then P7: for x being Element of REAL m st x in X holds
f - g is_partial_differentiable_in x,i ;
then P8: f - g is_partial_differentiable_on X,i by P3, PDIFF734, AS;
then B1: dom ((f - g) `partial| (X,i)) = X by DefPDX;
P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_-_g)_`partial|_(X,i))_/._x_=_(partdiff_(f,x,i))_-_(partdiff_(g,x,i))
let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) )
assume P10: x in X ; ::_thesis: ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i))
then ((f - g) `partial| (X,i)) /. x = partdiff ((f - g),x,i) by P8, DefPDX;
hence ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) by XX1, P10; ::_thesis: verum
end;
B2: dom ((f `partial| (X,i)) - (g `partial| (X,i))) = (dom (f `partial| (X,i))) /\ (dom (g `partial| (X,i))) by VALUED_1:12;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_-_g)_`partial|_(X,i))_._x_=_((f_`partial|_(X,i))_-_(g_`partial|_(X,i)))_._x
let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `partial| (X,i)) . x = ((f `partial| (X,i)) - (g `partial| (X,i))) . x )
assume A1: x in X ; ::_thesis: ((f - g) `partial| (X,i)) . x = ((f `partial| (X,i)) - (g `partial| (X,i))) . x
thus ((f - g) `partial| (X,i)) . x = ((f - g) `partial| (X,i)) /. x by A1, B1, PARTFUN1:def_6
.= (partdiff (f,x,i)) - (partdiff (g,x,i)) by P10, A1
.= ((f `partial| (X,i)) /. x) - (partdiff (g,x,i)) by A1, DefPDX, AS
.= ((f `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. x) by A1, DefPDX, AS
.= ((f `partial| (X,i)) . x) - ((g `partial| (X,i)) /. x) by A1, Q1, PARTFUN1:def_6
.= ((f `partial| (X,i)) . x) - ((g `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6
.= ((f `partial| (X,i)) - (g `partial| (X,i))) . x by A1, B2, Q1, VALUED_1:13 ; ::_thesis: verum
end;
hence ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) by B1, B2, Q1, P7, P10, P3, PDIFF734, AS, PARTFUN1:5; ::_thesis: verum
end;
theorem XXX3: :: PDIFF_9:67
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for r being Real
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for r being Real
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for r being Real
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
let X be Subset of (REAL m); ::_thesis: for r being Real
for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
let r be Real; ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds
( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i implies ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) )
assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i ) ; ::_thesis: ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) )
Q1: dom (f `partial| (X,i)) = X by DefPDX, AS;
dom (r (#) f) = dom f by VALUED_1:def_5;
then P3: X c= dom (r (#) f) by AS, PDIFF734;
XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_r_(#)_f_is_partial_differentiable_in_x,i_&_partdiff_((r_(#)_f),x,i)_=_r_*_(partdiff_(f,x,i))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) ) )
assume x in X ; ::_thesis: ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) )
then f is_partial_differentiable_in x,i by AS, PDIFF734;
hence ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) ) by PDIFF_1:33; ::_thesis: verum
end;
then P7: for x being Element of REAL m st x in X holds
r (#) f is_partial_differentiable_in x,i ;
then P8: r (#) f is_partial_differentiable_on X,i by P3, PDIFF734, AS;
then P9: dom ((r (#) f) `partial| (X,i)) = X by DefPDX;
P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((r_(#)_f)_`partial|_(X,i))_/._x_=_r_*_(partdiff_(f,x,i))
let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) )
assume P10: x in X ; ::_thesis: ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i))
then ((r (#) f) `partial| (X,i)) /. x = partdiff ((r (#) f),x,i) by P8, DefPDX;
hence ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) by XX1, P10; ::_thesis: verum
end;
dom (r (#) (f `partial| (X,i))) = dom (f `partial| (X,i)) by VALUED_1:def_5;
then P11: dom (r (#) (f `partial| (X,i))) = X by DefPDX, AS;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((r_(#)_f)_`partial|_(X,i))_._x_=_(r_(#)_(f_`partial|_(X,i)))_._x
let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `partial| (X,i)) . x = (r (#) (f `partial| (X,i))) . x )
assume A1: x in X ; ::_thesis: ((r (#) f) `partial| (X,i)) . x = (r (#) (f `partial| (X,i))) . x
thus ((r (#) f) `partial| (X,i)) . x = ((r (#) f) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6
.= r * (partdiff (f,x,i)) by P10, A1
.= r * ((f `partial| (X,i)) /. x) by A1, DefPDX, AS
.= r * ((f `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6
.= (r (#) (f `partial| (X,i))) . x by A1, P11, VALUED_1:def_5 ; ::_thesis: verum
end;
hence ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds
((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) by P9, P11, P7, P10, P3, PDIFF734, AS, PARTFUN1:5; ::_thesis: verum
end;
theorem XXX4: :: PDIFF_9:68
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds
( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) )
assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734;
Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS;
dom (f (#) g) = (dom f) /\ (dom g) by VALUED_1:def_4;
then P3: X c= dom (f (#) g) by P1, XBOOLE_1:19;
XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
(_f_(#)_g_is_partial_differentiable_in_x,i_&_partdiff_((f_(#)_g),x,i)_=_((partdiff_(f,x,i))_*_(g_._x))_+_((f_._x)_*_(partdiff_(g,x,i)))_)
let x be Element of REAL m; ::_thesis: ( x in X implies ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) )
assume x in X ; ::_thesis: ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734;
hence ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) by MPDIFF129; ::_thesis: verum
end;
then P7: for x being Element of REAL m st x in X holds
f (#) g is_partial_differentiable_in x,i ;
then P8: f (#) g is_partial_differentiable_on X,i by P3, PDIFF734, AS;
then P9: dom ((f (#) g) `partial| (X,i)) = X by DefPDX;
P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_(#)_g)_`partial|_(X,i))_/._x_=_((partdiff_(f,x,i))_*_(g_._x))_+_((f_._x)_*_(partdiff_(g,x,i)))
let x be Element of REAL m; ::_thesis: ( x in X implies ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) )
assume P10: x in X ; ::_thesis: ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i)))
then ((f (#) g) `partial| (X,i)) /. x = partdiff ((f (#) g),x,i) by P8, DefPDX;
hence ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by XX1, P10; ::_thesis: verum
end;
( dom ((f `partial| (X,i)) (#) g) = (dom (f `partial| (X,i))) /\ (dom g) & dom (f (#) (g `partial| (X,i))) = (dom f) /\ (dom (g `partial| (X,i))) ) by VALUED_1:def_4;
then P12: ( dom ((f `partial| (X,i)) (#) g) = X & dom (f (#) (g `partial| (X,i))) = X ) by Q1, P1, XBOOLE_1:28;
P14: dom (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) = (dom ((f `partial| (X,i)) (#) g)) /\ (dom (f (#) (g `partial| (X,i)))) by VALUED_1:def_1;
now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_
((f_(#)_g)_`partial|_(X,i))_._x_=_(((f_`partial|_(X,i))_(#)_g)_+_(f_(#)_(g_`partial|_(X,i))))_._x
let x be Element of REAL m; ::_thesis: ( x in X implies ((f (#) g) `partial| (X,i)) . x = (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x )
assume A1: x in X ; ::_thesis: ((f (#) g) `partial| (X,i)) . x = (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x
thus ((f (#) g) `partial| (X,i)) . x = ((f (#) g) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6
.= ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by P10, A1
.= (((f `partial| (X,i)) /. x) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by A1, DefPDX, AS
.= (((f `partial| (X,i)) /. x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) /. x)) by A1, DefPDX, AS
.= (((f `partial| (X,i)) . x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) /. x)) by A1, Q1, PARTFUN1:def_6
.= (((f `partial| (X,i)) . x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) . x)) by A1, Q1, PARTFUN1:def_6
.= (((f `partial| (X,i)) (#) g) . x) + ((f . x) * ((g `partial| (X,i)) . x)) by VALUED_1:5
.= (((f `partial| (X,i)) (#) g) . x) + ((f (#) (g `partial| (X,i))) . x) by VALUED_1:5
.= (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x by A1, P14, P12, VALUED_1:def_1 ; ::_thesis: verum
end;
hence ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds
((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) by P7, P10, P3, PDIFF734, AS, P9, P12, P14, PARTFUN1:5; ::_thesis: verum
end;
begin
definition
let m be non empty Element of NAT ;
let Z be set ;
let I be FinSequence of NAT ;
let f be PartFunc of (REAL m),REAL;
func PartDiffSeq (f,Z,I) -> Functional_Sequence of (REAL m),REAL means :TDef5: :: PDIFF_9:def 7
( it . 0 = f & ( for i being natural number holds it . (i + 1) = (it . i) `partial| (Z,(I /. (i + 1))) ) );
existence
ex b1 being Functional_Sequence of (REAL m),REAL st
( b1 . 0 = f & ( for i being natural number holds b1 . (i + 1) = (b1 . i) `partial| (Z,(I /. (i + 1))) ) )
proof
reconsider fZ = f as Element of PFuncs ((REAL m),REAL) by PARTFUN1:45;
defpred S1[ set , set , set ] means ex k being Element of NAT ex h being PartFunc of (REAL m),REAL st
( $1 = k & $2 = h & $3 = h `partial| (Z,(I /. (k + 1))) );
A1: for n being Element of NAT
for x being Element of PFuncs ((REAL m),REAL) ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being Element of PFuncs ((REAL m),REAL) ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y]
let x be Element of PFuncs ((REAL m),REAL); ::_thesis: ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y]
reconsider x9 = x as PartFunc of (REAL m),REAL by PARTFUN1:46;
reconsider y = x9 `partial| (Z,(I /. (n + 1))) as Element of PFuncs ((REAL m),REAL) by PARTFUN1:45;
ex h being PartFunc of (REAL m),REAL st
( x = h & y = h `partial| (Z,(I /. (n + 1))) ) ;
hence ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y] ; ::_thesis: verum
end;
consider g being Function of NAT,(PFuncs ((REAL m),REAL)) such that
A2: ( g . 0 = fZ & ( for n being Element of NAT holds S1[n,g . n,g . (n + 1)] ) ) from RECDEF_1:sch_2(A1);
reconsider g = g as Functional_Sequence of (REAL m),REAL ;
take g ; ::_thesis: ( g . 0 = f & ( for i being natural number holds g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) ) )
thus g . 0 = f by A2; ::_thesis: for i being natural number holds g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1)))
let i be natural number ; ::_thesis: g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1)))
i is Element of NAT by ORDINAL1:def_12;
then S1[i,g . i,g . (i + 1)] by A2;
hence g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Functional_Sequence of (REAL m),REAL st b1 . 0 = f & ( for i being natural number holds b1 . (i + 1) = (b1 . i) `partial| (Z,(I /. (i + 1))) ) & b2 . 0 = f & ( for i being natural number holds b2 . (i + 1) = (b2 . i) `partial| (Z,(I /. (i + 1))) ) holds
b1 = b2
proof
let seq1, seq2 be Functional_Sequence of (REAL m),REAL; ::_thesis: ( seq1 . 0 = f & ( for i being natural number holds seq1 . (i + 1) = (seq1 . i) `partial| (Z,(I /. (i + 1))) ) & seq2 . 0 = f & ( for i being natural number holds seq2 . (i + 1) = (seq2 . i) `partial| (Z,(I /. (i + 1))) ) implies seq1 = seq2 )
assume that
A3: seq1 . 0 = f and
A4: for n being natural number holds seq1 . (n + 1) = (seq1 . n) `partial| (Z,(I /. (n + 1))) and
A5: seq2 . 0 = f and
A6: for n being natural number holds seq2 . (n + 1) = (seq2 . n) `partial| (Z,(I /. (n + 1))) ; ::_thesis: seq1 = seq2
defpred S1[ Element of NAT ] means seq1 . $1 = seq2 . $1;
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
seq1 . (k + 1) = (seq1 . k) `partial| (Z,(I /. (k + 1))) by A4;
hence seq1 . (k + 1) = seq2 . (k + 1) by A6, A8; ::_thesis: verum
end;
A9: S1[ 0 ] by A3, A5;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7);
hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem TDef5 defines PartDiffSeq PDIFF_9:def_7_:_
for m being non empty Element of NAT
for Z being set
for I being FinSequence of NAT
for f being PartFunc of (REAL m),REAL
for b5 being Functional_Sequence of (REAL m),REAL holds
( b5 = PartDiffSeq (f,Z,I) iff ( b5 . 0 = f & ( for i being natural number holds b5 . (i + 1) = (b5 . i) `partial| (Z,(I /. (i + 1))) ) ) );
definition
let m be non empty Element of NAT ;
let Z be set ;
let I be FinSequence of NAT ;
let f be PartFunc of (REAL m),REAL;
predf is_partial_differentiable_on Z,I means :TDef6: :: PDIFF_9:def 8
for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq (f,Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1);
end;
:: deftheorem TDef6 defines is_partial_differentiable_on PDIFF_9:def_8_:_
for m being non empty Element of NAT
for Z being set
for I being FinSequence of NAT
for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,I iff for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq (f,Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) );
definition
let m be non empty Element of NAT ;
let Z be set ;
let I be FinSequence of NAT ;
let f be PartFunc of (REAL m),REAL;
funcf `partial| (Z,I) -> PartFunc of (REAL m),REAL equals :: PDIFF_9:def 9
(PartDiffSeq (f,Z,I)) . (len I);
correctness
coherence
(PartDiffSeq (f,Z,I)) . (len I) is PartFunc of (REAL m),REAL;
;
end;
:: deftheorem defines `partial| PDIFF_9:def_9_:_
for m being non empty Element of NAT
for Z being set
for I being FinSequence of NAT
for f being PartFunc of (REAL m),REAL holds f `partial| (Z,I) = (PartDiffSeq (f,Z,I)) . (len I);
XCWLM1: for i being Element of NAT
for I being non empty FinSequence of NAT
for X being set st 1 <= i & i <= len I & rng I c= X holds
I /. i in X
proof
let i be Element of NAT ; ::_thesis: for I being non empty FinSequence of NAT
for X being set st 1 <= i & i <= len I & rng I c= X holds
I /. i in X
let I be non empty FinSequence of NAT ; ::_thesis: for X being set st 1 <= i & i <= len I & rng I c= X holds
I /. i in X
let X be set ; ::_thesis: ( 1 <= i & i <= len I & rng I c= X implies I /. i in X )
assume AS: ( 1 <= i & i <= len I & rng I c= X ) ; ::_thesis: I /. i in X
then i in Seg (len I) ;
then X1: i in dom I by FINSEQ_1:def_3;
then I . i in rng I by FUNCT_1:3;
then I /. i in rng I by X1, PARTFUN1:def_6;
hence I /. i in X by AS; ::_thesis: verum
end;
theorem XCW010: :: PDIFF_9:69
for m being non empty Element of NAT
for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f + g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) + ((PartDiffSeq (g,X,I)) . i) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f + g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) + ((PartDiffSeq (g,X,I)) . i) )
let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) )
let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) )
thus for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) ::_thesis: verum
proof
defpred S1[ Element of NAT ] means ( $1 <= (len I) - 1 implies ( (PartDiffSeq ((f + g),Z,I)) . $1 is_partial_differentiable_on Z,I /. ($1 + 1) & (PartDiffSeq ((f + g),Z,I)) . $1 = ((PartDiffSeq (f,Z,I)) . $1) + ((PartDiffSeq (g,Z,I)) . $1) ) );
reconsider Z0 = 0 as Element of NAT ;
A9: S1[ 0 ]
proof
assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f + g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f + g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) + ((PartDiffSeq (g,Z,I)) . 0) )
then Q2: ( (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) & (PartDiffSeq (g,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) ) by AS, TDef6;
Q0: ( f = (PartDiffSeq (f,Z,I)) . Z0 & (PartDiffSeq ((f + g),Z,I)) . Z0 = f + g ) by TDef5;
1 <= len I by FINSEQ_1:20;
then I /. 1 in Seg m by AS, XCWLM1;
then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1;
then ((PartDiffSeq (f,Z,I)) . Z0) + ((PartDiffSeq (g,Z,I)) . Z0) is_partial_differentiable_on Z,I /. (Z0 + 1) by AS, Q2, XXX1;
hence ( (PartDiffSeq ((f + g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f + g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) + ((PartDiffSeq (g,Z,I)) . 0) ) by Q0, TDef5; ::_thesis: verum
end;
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f + g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) )
A83: k <= k + 1 by NAT_1:11;
then A82: k <= (len I) - 1 by A81, XXREAL_0:2;
A84: ( (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq (g,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) ) by A81, AS, TDef6;
k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6;
then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
A840: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by A82, AS, TDef6;
R1: (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5;
(k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6;
then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1;
A86: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = (((PartDiffSeq (f,Z,I)) . k) + ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by A83, A8, A81, TDef5, XXREAL_0:2
.= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) + (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A840, AS, Q4, XXX1
.= ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) by R1, TDef5 ;
hence (PartDiffSeq ((f + g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX1; ::_thesis: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1))
thus (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) by A86; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7);
hence for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) ; ::_thesis: verum
end;
end;
theorem XCW011: :: PDIFF_9:70
for m being non empty Element of NAT
for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
( f + g is_partial_differentiable_on X,I & (f + g) `partial| (X,I) = (f `partial| (X,I)) + (g `partial| (X,I)) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
( f + g is_partial_differentiable_on X,I & (f + g) `partial| (X,I) = (f `partial| (X,I)) + (g `partial| (X,I)) )
let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) )
let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) )
then for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW010;
hence f + g is_partial_differentiable_on Z,I by TDef6; ::_thesis: (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I))
1 <= len I by FINSEQ_1:20;
then reconsider k = (len I) - 1 as Element of NAT by INT_1:5;
P1: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by AS, TDef6;
1 <= k + 1 by NAT_1:11;
then I /. (k + 1) in Seg m by AS, XCWLM1;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
R1: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq ((f + g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5
.= (((PartDiffSeq (f,Z,I)) . k) + ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW010
.= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) + (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by P1, AS, Q4, XXX1 ;
(PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5;
hence (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) by R1, TDef5; ::_thesis: verum
end;
theorem XCW020: :: PDIFF_9:71
for m being non empty Element of NAT
for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f - g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) - ((PartDiffSeq (g,X,I)) . i) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f - g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) - ((PartDiffSeq (g,X,I)) . i) )
let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) )
let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) )
defpred S1[ Element of NAT ] means ( $1 <= (len I) - 1 implies ( (PartDiffSeq ((f - g),Z,I)) . $1 is_partial_differentiable_on Z,I /. ($1 + 1) & (PartDiffSeq ((f - g),Z,I)) . $1 = ((PartDiffSeq (f,Z,I)) . $1) - ((PartDiffSeq (g,Z,I)) . $1) ) );
reconsider Z0 = 0 as Element of NAT ;
A9: S1[ 0 ]
proof
assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f - g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f - g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) - ((PartDiffSeq (g,Z,I)) . 0) )
then Q2: ( (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) & (PartDiffSeq (g,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) ) by AS, TDef6;
( f = (PartDiffSeq (f,Z,I)) . Z0 & f - g = (PartDiffSeq ((f - g),Z,I)) . Z0 ) by TDef5;
then Q5: (PartDiffSeq ((f - g),Z,I)) . Z0 = ((PartDiffSeq (f,Z,I)) . Z0) - ((PartDiffSeq (g,Z,I)) . Z0) by TDef5;
1 <= len I by FINSEQ_1:20;
then I /. 1 in Seg m by AS, XCWLM1;
then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1;
hence ( (PartDiffSeq ((f - g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f - g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) - ((PartDiffSeq (g,Z,I)) . 0) ) by Q5, AS, Q2, XXX2; ::_thesis: verum
end;
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f - g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) )
B1: k <= k + 1 by NAT_1:11;
then A82: k <= (len I) - 1 by A81, XXREAL_0:2;
A84: ( (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq (g,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) ) by A81, AS, TDef6;
k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6;
then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
A85: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by A82, AS, TDef6;
R1: (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5;
(k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6;
then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1;
A86: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq ((f - g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5
.= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) - (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A85, AS, Q4, XXX2, B1, A8, A81, XXREAL_0:2
.= ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) by R1, TDef5 ;
hence (PartDiffSeq ((f - g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX2; ::_thesis: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1))
thus (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) by A86; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7);
hence for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) ; ::_thesis: verum
end;
theorem XCW021: :: PDIFF_9:72
for m being non empty Element of NAT
for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
( f - g is_partial_differentiable_on X,I & (f - g) `partial| (X,I) = (f `partial| (X,I)) - (g `partial| (X,I)) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds
( f - g is_partial_differentiable_on X,I & (f - g) `partial| (X,I) = (f `partial| (X,I)) - (g `partial| (X,I)) )
let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT
for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) )
let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds
( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) )
then for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW020;
hence f - g is_partial_differentiable_on Z,I by TDef6; ::_thesis: (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I))
1 <= len I by FINSEQ_1:20;
then reconsider k = (len I) - 1 as Element of NAT by INT_1:5;
P1: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by AS, TDef6;
1 <= k + 1 by NAT_1:11;
then I /. (k + 1) in Seg m by AS, XCWLM1;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
(PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq ((f - g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5
.= (((PartDiffSeq (f,Z,I)) . k) - ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW020 ;
then R1: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) - (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by P1, AS, Q4, XXX2;
(PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5;
hence (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) by R1, TDef5; ::_thesis: verum
end;
theorem XCW030: :: PDIFF_9:73
for m being non empty Element of NAT
for X being Subset of (REAL m)
for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((r (#) f),X,I)) . i = r (#) ((PartDiffSeq (f,X,I)) . i) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((r (#) f),X,I)) . i = r (#) ((PartDiffSeq (f,X,I)) . i) )
let Z be Subset of (REAL m); ::_thesis: for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) )
let r be Real; ::_thesis: for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) )
let I be non empty FinSequence of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) )
defpred S1[ Element of NAT ] means ( $1 <= (len I) - 1 implies ( (PartDiffSeq ((r (#) f),Z,I)) . $1 is_partial_differentiable_on Z,I /. ($1 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . $1 = r (#) ((PartDiffSeq (f,Z,I)) . $1) ) );
reconsider Z0 = 0 as Element of NAT ;
A9: S1[ 0 ]
proof
assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((r (#) f),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . 0 = r (#) ((PartDiffSeq (f,Z,I)) . 0) )
then Q2: (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) by AS, TDef6;
(PartDiffSeq ((r (#) f),Z,I)) . Z0 = r (#) f by TDef5;
then Q5: (PartDiffSeq ((r (#) f),Z,I)) . Z0 = r (#) ((PartDiffSeq (f,Z,I)) . Z0) by TDef5;
1 <= len I by FINSEQ_1:20;
then I /. 1 in Seg m by AS, XCWLM1;
then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1;
hence ( (PartDiffSeq ((r (#) f),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . 0 = r (#) ((PartDiffSeq (f,Z,I)) . 0) ) by Q5, AS, Q2, XXX3; ::_thesis: verum
end;
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) )
B1: k <= k + 1 by NAT_1:11;
then A82: k <= (len I) - 1 by A81, XXREAL_0:2;
A84: (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by A81, AS, TDef6;
k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6;
then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
A85: (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by A82, AS, TDef6;
(k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6;
then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11;
then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1;
A86: (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = (r (#) ((PartDiffSeq (f,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by B1, A81, A8, TDef5, XXREAL_0:2
.= r (#) (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A85, AS, Q4, XXX3
.= r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) by TDef5 ;
hence (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX3; ::_thesis: (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1))
thus (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) by A86; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7);
hence for i being Element of NAT st i <= (len I) - 1 holds
( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) ; ::_thesis: verum
end;
theorem XCW031: :: PDIFF_9:74
for m being non empty Element of NAT
for X being Subset of (REAL m)
for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds
( r (#) f is_partial_differentiable_on X,I & (r (#) f) `partial| (X,I) = r (#) (f `partial| (X,I)) )
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds
( r (#) f is_partial_differentiable_on X,I & (r (#) f) `partial| (X,I) = r (#) (f `partial| (X,I)) )
let Z be Subset of (REAL m); ::_thesis: for r being Real
for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) )
let r be Real; ::_thesis: for I being non empty FinSequence of NAT
for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) )
let I be non empty FinSequence of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds
( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I implies ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) )
assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I ) ; ::_thesis: ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) )
then for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW030;
hence r (#) f is_partial_differentiable_on Z,I by TDef6; ::_thesis: (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I))
1 <= len I by FINSEQ_1:20;
then reconsider k = (len I) - 1 as Element of NAT by INT_1:5;
PP1: (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by AS, TDef6;
1 <= k + 1 by NAT_1:11;
then I /. (k + 1) in Seg m by AS, XCWLM1;
then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1;
(PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = ((PartDiffSeq ((r (#) f),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5
.= (r (#) ((PartDiffSeq (f,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW030
.= r (#) (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by PP1, AS, Q4, XXX3 ;
hence (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) by TDef5; ::_thesis: verum
end;
definition
let m be non empty Element of NAT ;
let f be PartFunc of (REAL m),REAL;
let k be Element of NAT ;
let Z be set ;
predf is_partial_differentiable_up_to_order k,Z means :TDef9: :: PDIFF_9:def 10
for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds
f is_partial_differentiable_on Z,I;
end;
:: deftheorem TDef9 defines is_partial_differentiable_up_to_order PDIFF_9:def_10_:_
for m being non empty Element of NAT
for f being PartFunc of (REAL m),REAL
for k being Element of NAT
for Z being set holds
( f is_partial_differentiable_up_to_order k,Z iff for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds
f is_partial_differentiable_on Z,I );
theorem XCW040: :: PDIFF_9:75
for m being non empty Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for I, G being non empty FinSequence of NAT holds
( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) )
proof
let m be non empty Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for I, G being non empty FinSequence of NAT holds
( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for I, G being non empty FinSequence of NAT holds
( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) )
let f be PartFunc of (REAL m),REAL; ::_thesis: for I, G being non empty FinSequence of NAT holds
( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) )
let I, G be non empty FinSequence of NAT ; ::_thesis: ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) )
set g = f `partial| (Z,G);
reconsider Z0 = 0 as Element of NAT ;
S1: dom G c= dom (G ^ I) by FINSEQ_1:26;
Y0: for i being Element of NAT st i <= (len G) - 1 holds
(G ^ I) /. (i + 1) = G /. (i + 1)
proof
let i be Element of NAT ; ::_thesis: ( i <= (len G) - 1 implies (G ^ I) /. (i + 1) = G /. (i + 1) )
assume i <= (len G) - 1 ; ::_thesis: (G ^ I) /. (i + 1) = G /. (i + 1)
then ( 1 <= i + 1 & i + 1 <= len G ) by NAT_1:11, XREAL_1:19;
then D3: i + 1 in dom G by FINSEQ_3:25;
then (G ^ I) /. (i + 1) = (G ^ I) . (i + 1) by S1, PARTFUN1:def_6;
then (G ^ I) /. (i + 1) = G . (i + 1) by D3, FINSEQ_1:def_7;
hence (G ^ I) /. (i + 1) = G /. (i + 1) by D3, PARTFUN1:def_6; ::_thesis: verum
end;
X2: len (G ^ I) = (len G) + (len I) by FINSEQ_1:22;
X3: for i being Element of NAT st i <= (len I) - 1 holds
(G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1)
proof
let i be Element of NAT ; ::_thesis: ( i <= (len I) - 1 implies (G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1) )
assume i <= (len I) - 1 ; ::_thesis: (G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1)
then D2: i + 1 <= len I by XREAL_1:19;
1 <= i + 1 by NAT_1:11;
then D3: i + 1 in dom I by D2, FINSEQ_3:25;
D9: 1 <= (len G) + (i + 1) by NAT_1:11, XREAL_1:38;
(len G) + (i + 1) <= len (G ^ I) by X2, D2, XREAL_1:7;
then (len G) + (i + 1) in dom (G ^ I) by D9, FINSEQ_3:25;
hence (G ^ I) /. ((len G) + (i + 1)) = (G ^ I) . ((len G) + (i + 1)) by PARTFUN1:def_6
.= I . (i + 1) by D3, FINSEQ_1:def_7
.= I /. (i + 1) by D3, PARTFUN1:def_6 ;
::_thesis: verum
end;
defpred S1[ Element of NAT ] means ( $1 <= (len G) - 1 implies (PartDiffSeq (f,Z,(G ^ I))) . $1 = (PartDiffSeq (f,Z,G)) . $1 );
B1: S1[ 0 ]
proof
assume 0 <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . 0 = (PartDiffSeq (f,Z,G)) . 0
(PartDiffSeq (f,Z,(G ^ I))) . 0 = f by TDef5;
hence (PartDiffSeq (f,Z,(G ^ I))) . 0 = (PartDiffSeq (f,Z,G)) . 0 by TDef5; ::_thesis: verum
end;
B2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume D1: S1[k] ; ::_thesis: S1[k + 1]
assume D2: k + 1 <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . (k + 1) = (PartDiffSeq (f,Z,G)) . (k + 1)
D20: k <= k + 1 by NAT_1:11;
thus (PartDiffSeq (f,Z,(G ^ I))) . (k + 1) = ((PartDiffSeq (f,Z,(G ^ I))) . k) `partial| (Z,((G ^ I) /. (k + 1))) by TDef5
.= ((PartDiffSeq (f,Z,G)) . k) `partial| (Z,(G /. (k + 1))) by D20, Y0, D1, D2, XXREAL_0:2
.= (PartDiffSeq (f,Z,G)) . (k + 1) by TDef5 ; ::_thesis: verum
end;
Y1: for n being Element of NAT holds S1[n] from NAT_1:sch_1(B1, B2);
1 <= len G by FINSEQ_1:20;
then reconsider j = (len G) - 1 as Element of NAT by INT_1:5;
Y11: (PartDiffSeq (f,Z,(G ^ I))) . (len G) = ((PartDiffSeq (f,Z,(G ^ I))) . j) `partial| (Z,((G ^ I) /. (j + 1))) by TDef5
.= ((PartDiffSeq (f,Z,G)) . j) `partial| (Z,((G ^ I) /. (j + 1))) by Y1
.= ((PartDiffSeq (f,Z,G)) . j) `partial| (Z,(G /. (j + 1))) by Y0
.= (PartDiffSeq (f,Z,G)) . (len G) by TDef5 ;
defpred S2[ Element of NAT ] means ( $1 <= (len I) - 1 implies (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . $1 = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + $1) );
A1: S2[ 0 ] by Y11, TDef5;
A2: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A3: S2[k] ; ::_thesis: S2[k + 1]
set i = (len G) + k;
assume P0: k + 1 <= (len I) - 1 ; ::_thesis: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . (k + 1) = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1))
P1: k <= k + 1 by NAT_1:11;
(G ^ I) /. (((len G) + k) + 1) = (G ^ I) /. ((len G) + (k + 1)) ;
then P2: (G ^ I) /. (((len G) + k) + 1) = I /. (k + 1) by X3, P1, P0, XXREAL_0:2;
(PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1)) = ((PartDiffSeq (f,Z,(G ^ I))) . ((len G) + k)) `partial| (Z,((G ^ I) /. (((len G) + k) + 1))) by TDef5;
hence (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . (k + 1) = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1)) by P1, P0, A3, P2, TDef5, XXREAL_0:2; ::_thesis: verum
end;
X1: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A1, A2);
hereby ::_thesis: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I implies f is_partial_differentiable_on Z,G ^ I )
assume P1: f is_partial_differentiable_on Z,G ^ I ; ::_thesis: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I )
now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_G)_-_1_holds_
(PartDiffSeq_(f,Z,G))_._i_is_partial_differentiable_on_Z,G_/._(i_+_1)
let i be Element of NAT ; ::_thesis: ( i <= (len G) - 1 implies (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) )
assume D1: i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1)
then i + Z0 <= ((len G) - 1) + (len I) by XREAL_1:7;
then i <= (len (G ^ I)) - 1 by X2;
then P2: (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by TDef6, P1;
(G ^ I) /. (i + 1) = G /. (i + 1) by D1, Y0;
hence (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) by P2, D1, Y1; ::_thesis: verum
end;
hence f is_partial_differentiable_on Z,G by TDef6; ::_thesis: f `partial| (Z,G) is_partial_differentiable_on Z,I
now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_I)_-_1_holds_
(PartDiffSeq_((f_`partial|_(Z,G)),Z,I))_._i_is_partial_differentiable_on_Z,I_/._(i_+_1)
let i be Element of NAT ; ::_thesis: ( i <= (len I) - 1 implies (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) )
assume S1: i <= (len I) - 1 ; ::_thesis: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1)
then (len G) + i <= (len G) + ((len I) - 1) by XREAL_1:6;
then (len G) + i <= (len (G ^ I)) - 1 by X2;
then X4: (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + i) is_partial_differentiable_on Z,(G ^ I) /. (((len G) + i) + 1) by TDef6, P1;
(G ^ I) /. (((len G) + i) + 1) = (G ^ I) /. ((len G) + (i + 1)) ;
then (G ^ I) /. (((len G) + i) + 1) = I /. (i + 1) by S1, X3;
hence (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by X4, S1, X1; ::_thesis: verum
end;
hence f `partial| (Z,G) is_partial_differentiable_on Z,I by TDef6; ::_thesis: verum
end;
now__::_thesis:_(_f_is_partial_differentiable_on_Z,G_&_f_`partial|_(Z,G)_is_partial_differentiable_on_Z,I_implies_f_is_partial_differentiable_on_Z,G_^_I_)
assume P0: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ; ::_thesis: f is_partial_differentiable_on Z,G ^ I
now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_(G_^_I))_-_1_holds_
(PartDiffSeq_(f,Z,(G_^_I)))_._i_is_partial_differentiable_on_Z,(G_^_I)_/._(i_+_1)
let i be Element of NAT ; ::_thesis: ( i <= (len (G ^ I)) - 1 implies (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1) )
assume Q1: i <= (len (G ^ I)) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1)
percases ( i <= (len G) - 1 or not i <= (len G) - 1 ) ;
supposeQ2: i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1)
then Q3: (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) by TDef6, P0;
G /. (i + 1) = (G ^ I) /. (i + 1) by Q2, Y0;
hence (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by Q2, Y1, Q3; ::_thesis: verum
end;
suppose not i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1)
then len G < i + 1 by XREAL_1:19;
then len G <= i by NAT_1:13;
then reconsider k = i - (len G) as Element of NAT by INT_1:5;
R5: i - (len G) <= (((len G) + (len I)) - 1) - (len G) by Q1, X2, XREAL_1:9;
then Q5: ( k <= (len I) - 1 & i = k + (len G) ) ;
then Q6: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by TDef6, P0;
i + 1 = (k + 1) + (len G) ;
then I /. (k + 1) = (G ^ I) /. (i + 1) by R5, X3;
hence (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by Q6, Q5, X1; ::_thesis: verum
end;
end;
end;
hence f is_partial_differentiable_on Z,G ^ I by TDef6; ::_thesis: verum
end;
hence ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I implies f is_partial_differentiable_on Z,G ^ I ) ; ::_thesis: verum
end;
set Z0 = 0 ;
theorem XCW041: :: PDIFF_9:76
for m being non empty Element of NAT
for i being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i )
let i be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i )
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL holds
( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i )
let f be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i )
set I = <*i*>;
P0: len <*i*> = 1 by FINSEQ_1:39;
P1: (PartDiffSeq (f,Z,<*i*>)) . 0 = f by TDef5;
1 in Seg 1 ;
then X2: 1 in dom <*i*> by FINSEQ_1:38;
<*i*> /. (0 + 1) = <*i*> . 1 by X2, PARTFUN1:def_6;
then Q1: <*i*> /. (0 + 1) = i by FINSEQ_1:40;
hereby ::_thesis: ( f is_partial_differentiable_on Z,i implies f is_partial_differentiable_on Z,<*i*> )
assume PX1: f is_partial_differentiable_on Z,<*i*> ; ::_thesis: f is_partial_differentiable_on Z,i
0 <= (len <*i*>) - 1 by P0;
hence f is_partial_differentiable_on Z,i by Q1, P1, PX1, TDef6; ::_thesis: verum
end;
assume P3: f is_partial_differentiable_on Z,i ; ::_thesis: f is_partial_differentiable_on Z,<*i*>
now__::_thesis:_for_k_being_Element_of_NAT_st_k_<=_(len_<*i*>)_-_1_holds_
(PartDiffSeq_(f,Z,<*i*>))_._k_is_partial_differentiable_on_Z,<*i*>_/._(k_+_1)
let k be Element of NAT ; ::_thesis: ( k <= (len <*i*>) - 1 implies (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1) )
assume k <= (len <*i*>) - 1 ; ::_thesis: (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1)
then P5: k = 0 by P0;
then <*i*> /. (k + 1) = <*i*> . 1 by X2, PARTFUN1:def_6;
then <*i*> /. (k + 1) = i by FINSEQ_1:40;
hence (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1) by P3, P5, TDef5; ::_thesis: verum
end;
hence f is_partial_differentiable_on Z,<*i*> by TDef6; ::_thesis: verum
end;
theorem XCW042: :: PDIFF_9:77
for m being non empty Element of NAT
for i being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i)
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i)
let i be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i)
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i)
let f be PartFunc of (REAL m),REAL; ::_thesis: f `partial| (Z,<*i*>) = f `partial| (Z,i)
set I = <*i*>;
1 in Seg 1 ;
then 1 in dom <*i*> by FINSEQ_1:38;
then <*i*> /. (0 + 1) = <*i*> . 1 by PARTFUN1:def_6;
then Q1: <*i*> /. (0 + 1) = i by FINSEQ_1:40;
thus f `partial| (Z,<*i*>) = (PartDiffSeq (f,Z,<*i*>)) . 1 by FINSEQ_1:39
.= ((PartDiffSeq (f,Z,<*i*>)) . 0) `partial| (Z,(<*i*> /. (0 + 1))) by TDef5
.= f `partial| (Z,i) by Q1, TDef5 ; ::_thesis: verum
end;
theorem XCW0400: :: PDIFF_9:78
for m being non empty Element of NAT
for i, j being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds
f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
proof
let m be non empty Element of NAT ; ::_thesis: for i, j being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL
for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds
f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
let i, j be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL
for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds
f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL
for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds
f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
let f be PartFunc of (REAL m),REAL; ::_thesis: for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds
f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
let I be non empty FinSequence of NAT ; ::_thesis: ( f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j implies f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z )
assume AS: ( f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j ) ; ::_thesis: f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z
let J be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len J <= i & rng J c= Seg m implies f `partial| (Z,I) is_partial_differentiable_on Z,J )
assume AS1: ( len J <= i & rng J c= Seg m ) ; ::_thesis: f `partial| (Z,I) is_partial_differentiable_on Z,J
reconsider G = I ^ J as non empty FinSequence of NAT ;
P1: rng G = (rng I) \/ (rng J) by FINSEQ_1:31;
len G = (len I) + (len J) by FINSEQ_1:22;
then ( len G <= i + j & rng G c= Seg m ) by AS1, P1, AS, XBOOLE_1:8, XREAL_1:6;
then f is_partial_differentiable_on Z,G by AS, TDef9;
hence f `partial| (Z,I) is_partial_differentiable_on Z,J by XCW040; ::_thesis: verum
end;
theorem XCW0410: :: PDIFF_9:79
for m being non empty Element of NAT
for i, j being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds
f is_partial_differentiable_up_to_order j,Z
proof
let m be non empty Element of NAT ; ::_thesis: for i, j being Element of NAT
for Z being set
for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds
f is_partial_differentiable_up_to_order j,Z
let i, j be Element of NAT ; ::_thesis: for Z being set
for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds
f is_partial_differentiable_up_to_order j,Z
let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds
f is_partial_differentiable_up_to_order j,Z
let f be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order i,Z & j <= i implies f is_partial_differentiable_up_to_order j,Z )
assume AS: ( f is_partial_differentiable_up_to_order i,Z & j <= i ) ; ::_thesis: f is_partial_differentiable_up_to_order j,Z
let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= j & rng I c= Seg m implies f is_partial_differentiable_on Z,I )
assume AS1: ( len I <= j & rng I c= Seg m ) ; ::_thesis: f is_partial_differentiable_on Z,I
then len I <= i by AS, XXREAL_0:2;
hence f is_partial_differentiable_on Z,I by AS, AS1, TDef9; ::_thesis: verum
end;
theorem :: PDIFF_9:80
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds
( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X )
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds
( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X )
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds
( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X )
let Z be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds
( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z )
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z implies ( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z ) )
assume AS: ( Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z ) ; ::_thesis: ( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z )
hereby :: according to PDIFF_9:def_10 ::_thesis: f - g is_partial_differentiable_up_to_order i,Z
let I be non empty FinSequence of NAT ; ::_thesis: ( len I <= i & rng I c= Seg m implies f + g is_partial_differentiable_on Z,I )
assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: f + g is_partial_differentiable_on Z,I
then ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by AS, TDef9;
hence f + g is_partial_differentiable_on Z,I by AS, P1, XCW011; ::_thesis: verum
end;
let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= i & rng I c= Seg m implies f - g is_partial_differentiable_on Z,I )
assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: f - g is_partial_differentiable_on Z,I
then ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by AS, TDef9;
hence f - g is_partial_differentiable_on Z,I by AS, P1, XCW021; ::_thesis: verum
end;
theorem :: PDIFF_9:81
for m being non empty Element of NAT
for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds
r (#) f is_partial_differentiable_up_to_order i,X
proof
let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT
for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds
r (#) f is_partial_differentiable_up_to_order i,X
let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds
r (#) f is_partial_differentiable_up_to_order i,X
let Z be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL
for r being Real st Z is open & f is_partial_differentiable_up_to_order i,Z holds
r (#) f is_partial_differentiable_up_to_order i,Z
let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st Z is open & f is_partial_differentiable_up_to_order i,Z holds
r (#) f is_partial_differentiable_up_to_order i,Z
let r be Real; ::_thesis: ( Z is open & f is_partial_differentiable_up_to_order i,Z implies r (#) f is_partial_differentiable_up_to_order i,Z )
assume AS: ( Z is open & f is_partial_differentiable_up_to_order i,Z ) ; ::_thesis: r (#) f is_partial_differentiable_up_to_order i,Z
let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= i & rng I c= Seg m implies r (#) f is_partial_differentiable_on Z,I )
assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: r (#) f is_partial_differentiable_on Z,I
then f is_partial_differentiable_on Z,I by AS, TDef9;
hence r (#) f is_partial_differentiable_on Z,I by AS, P1, XCW031; ::_thesis: verum
end;
theorem :: PDIFF_9:82
for m being non empty Element of NAT
for X being Subset of (REAL m) st X is open holds
for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds
f (#) g is_partial_differentiable_up_to_order i,X
proof
let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) st X is open holds
for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds
f (#) g is_partial_differentiable_up_to_order i,X
let Z be Subset of (REAL m); ::_thesis: ( Z is open implies for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds
f (#) g is_partial_differentiable_up_to_order i,Z )
assume AS: Z is open ; ::_thesis: for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds
f (#) g is_partial_differentiable_up_to_order i,Z
defpred S1[ Element of NAT ] means for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order $1,Z & g is_partial_differentiable_up_to_order $1,Z holds
f (#) g is_partial_differentiable_up_to_order $1,Z;
P9: S1[ 0 ]
proof
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order 0 ,Z & g is_partial_differentiable_up_to_order 0 ,Z implies f (#) g is_partial_differentiable_up_to_order 0 ,Z )
assume ( f is_partial_differentiable_up_to_order 0 ,Z & g is_partial_differentiable_up_to_order 0 ,Z ) ; ::_thesis: f (#) g is_partial_differentiable_up_to_order 0 ,Z
for I being non empty FinSequence of NAT st len I <= 0 & rng I c= Seg m holds
f (#) g is_partial_differentiable_on Z,I by FINSEQ_1:20;
hence f (#) g is_partial_differentiable_up_to_order 0 ,Z by TDef9; ::_thesis: verum
end;
P7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume P71: S1[k] ; ::_thesis: S1[k + 1]
let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order k + 1,Z & g is_partial_differentiable_up_to_order k + 1,Z implies f (#) g is_partial_differentiable_up_to_order k + 1,Z )
assume R71: ( f is_partial_differentiable_up_to_order k + 1,Z & g is_partial_differentiable_up_to_order k + 1,Z ) ; ::_thesis: f (#) g is_partial_differentiable_up_to_order k + 1,Z
then R711: ( f is_partial_differentiable_up_to_order k,Z & g is_partial_differentiable_up_to_order k,Z ) by XCW0410, NAT_1:11;
now__::_thesis:_for_I_being_non_empty_FinSequence_of_NAT_st_len_I_<=_k_+_1_&_rng_I_c=_Seg_m_holds_
f_(#)_g_is_partial_differentiable_on_Z,I
let I be non empty FinSequence of NAT ; ::_thesis: ( len I <= k + 1 & rng I c= Seg m implies f (#) g is_partial_differentiable_on Z,b1 )
assume P72: ( len I <= k + 1 & rng I c= Seg m ) ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1
then R721: ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by R71, TDef9;
RRRR: 1 <= len I by FINSEQ_1:20;
then T1: 1 in dom I by FINSEQ_3:25;
then T4: I /. 1 = I . 1 by PARTFUN1:def_6;
T2: I . 1 in rng I by T1, FUNCT_1:3;
then I . 1 in Seg m by P72;
then reconsider i = I . 1 as Element of NAT ;
P75: ( 1 <= i & i <= m ) by T2, P72, FINSEQ_1:1;
percases ( 1 = len I or 1 <> len I ) ;
suppose 1 = len I ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1
then T3: I = <*(I /. 1)*> by FINSEQ_5:14;
then ( f is_partial_differentiable_on Z,i & g is_partial_differentiable_on Z,i ) by XCW041, R721, T4;
then f (#) g is_partial_differentiable_on Z,i by P75, XXX4, AS;
hence f (#) g is_partial_differentiable_on Z,I by XCW041, T3, T4; ::_thesis: verum
end;
suppose 1 <> len I ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1
then 1 < len I by RRRR, XXREAL_0:1;
then 1 + 1 <= len I by NAT_1:13;
then 1 <= (len I) - 1 by XREAL_1:19;
then 1 <= len (I /^ 1) by RRRR, RFINSEQ:def_1;
then reconsider J = I /^ 1 as non empty FinSequence of NAT by FINSEQ_1:20;
set I1 = <*i*>;
(len I) - 1 <= k by P72, XREAL_1:20;
then P74: len J <= k by RRRR, RFINSEQ:def_1;
V1: I = <*(I /. 1)*> ^ (I /^ 1) by FINSEQ_5:29;
then U1: ( rng <*i*> c= rng I & rng J c= rng I ) by T4, FINSEQ_1:29, FINSEQ_1:30;
then P76: rng J c= Seg m by P72, XBOOLE_1:1;
I = <*i*> ^ J by T4, FINSEQ_5:29;
then ( f is_partial_differentiable_on Z,<*i*> & g is_partial_differentiable_on Z,<*i*> ) by XCW040, R721;
then P79: ( f is_partial_differentiable_on Z,i & g is_partial_differentiable_on Z,i ) by XCW041;
then f (#) g is_partial_differentiable_on Z,i by P75, XXX4, AS;
then P86: f (#) g is_partial_differentiable_on Z,<*i*> by XCW041;
P87: (f (#) g) `partial| (Z,<*i*>) = (f (#) g) `partial| (Z,i) by XCW042
.= ((f `partial| (Z,i)) (#) g) + (f (#) (g `partial| (Z,i))) by P79, P75, XXX4, AS
.= ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,i))) by XCW042
.= ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,<*i*>))) by XCW042 ;
( len <*i*> = 1 & rng <*i*> c= Seg m ) by U1, P72, FINSEQ_1:39, XBOOLE_1:1;
then ( f `partial| (Z,<*i*>) is_partial_differentiable_up_to_order k,Z & g `partial| (Z,<*i*>) is_partial_differentiable_up_to_order k,Z ) by XCW0400, R71;
then ( (f `partial| (Z,<*i*>)) (#) g is_partial_differentiable_up_to_order k,Z & f (#) (g `partial| (Z,<*i*>)) is_partial_differentiable_up_to_order k,Z ) by P71, R711;
then ( (f `partial| (Z,<*i*>)) (#) g is_partial_differentiable_on Z,J & f (#) (g `partial| (Z,<*i*>)) is_partial_differentiable_on Z,J ) by P74, P76, TDef9;
then ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,<*i*>))) is_partial_differentiable_on Z,J by AS, P76, XCW011;
hence f (#) g is_partial_differentiable_on Z,I by P86, V1, T4, P87, XCW040; ::_thesis: verum
end;
end;
end;
hence f (#) g is_partial_differentiable_up_to_order k + 1,Z by TDef9; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(P9, P7);
hence for i being Element of NAT
for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds
f (#) g is_partial_differentiable_up_to_order i,Z ; ::_thesis: verum
end;