:: NCFCONT1 semantic presentation
begin
theorem Th1: :: NCFCONT1:1
for CNS being ComplexNormSpace
for seq being sequence of CNS holds - seq = (- 1r) * seq
proof
let CNS be ComplexNormSpace; ::_thesis: for seq being sequence of CNS holds - seq = (- 1r) * seq
let seq be sequence of CNS; ::_thesis: - seq = (- 1r) * seq
now__::_thesis:_for_n_being_Element_of_NAT_holds_((-_1r)_*_seq)_._n_=_(-_seq)_._n
let n be Element of NAT ; ::_thesis: ((- 1r) * seq) . n = (- seq) . n
thus ((- 1r) * seq) . n = (- 1r) * (seq . n) by CLVECT_1:def_14
.= - (seq . n) by CLVECT_1:3
.= (- seq) . n by BHSP_1:44 ; ::_thesis: verum
end;
hence - seq = (- 1r) * seq by FUNCT_2:63; ::_thesis: verum
end;
definition
let CNS be ComplexNormSpace;
let x0 be Point of CNS;
mode Neighbourhood of x0 -> Subset of CNS means :Def1: :: NCFCONT1:def 1
ex g being Real st
( 0 < g & { y where y is Point of CNS : ||.(y - x0).|| < g } c= it );
existence
ex b1 being Subset of CNS ex g being Real st
( 0 < g & { y where y is Point of CNS : ||.(y - x0).|| < g } c= b1 )
proof
set N = { y where y is Point of CNS : ||.(y - x0).|| < 1 } ;
take { y where y is Point of CNS : ||.(y - x0).|| < 1 } ; ::_thesis: ( { y where y is Point of CNS : ||.(y - x0).|| < 1 } is Subset of CNS & ex g being Real st
( 0 < g & { y where y is Point of CNS : ||.(y - x0).|| < g } c= { y where y is Point of CNS : ||.(y - x0).|| < 1 } ) )
{ y where y is Point of CNS : ||.(y - x0).|| < 1 } c= the carrier of CNS
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { y where y is Point of CNS : ||.(y - x0).|| < 1 } or x in the carrier of CNS )
assume x in { y where y is Point of CNS : ||.(y - x0).|| < 1 } ; ::_thesis: x in the carrier of CNS
then ex y being Point of CNS st
( x = y & ||.(y - x0).|| < 1 ) ;
hence x in the carrier of CNS ; ::_thesis: verum
end;
hence ( { y where y is Point of CNS : ||.(y - x0).|| < 1 } is Subset of CNS & ex g being Real st
( 0 < g & { y where y is Point of CNS : ||.(y - x0).|| < g } c= { y where y is Point of CNS : ||.(y - x0).|| < 1 } ) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Neighbourhood NCFCONT1:def_1_:_
for CNS being ComplexNormSpace
for x0 being Point of CNS
for b3 being Subset of CNS holds
( b3 is Neighbourhood of x0 iff ex g being Real st
( 0 < g & { y where y is Point of CNS : ||.(y - x0).|| < g } c= b3 ) );
theorem Th2: :: NCFCONT1:2
for CNS being ComplexNormSpace
for x0 being Point of CNS
for g being Real st 0 < g holds
{ y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0
proof
let CNS be ComplexNormSpace; ::_thesis: for x0 being Point of CNS
for g being Real st 0 < g holds
{ y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0
let x0 be Point of CNS; ::_thesis: for g being Real st 0 < g holds
{ y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0
let g be Real; ::_thesis: ( 0 < g implies { y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0 )
assume A1: g > 0 ; ::_thesis: { y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0
set N = { y where y is Point of CNS : ||.(y - x0).|| < g } ;
{ y where y is Point of CNS : ||.(y - x0).|| < g } c= the carrier of CNS
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { y where y is Point of CNS : ||.(y - x0).|| < g } or x in the carrier of CNS )
assume x in { y where y is Point of CNS : ||.(y - x0).|| < g } ; ::_thesis: x in the carrier of CNS
then ex y being Point of CNS st
( x = y & ||.(y - x0).|| < g ) ;
hence x in the carrier of CNS ; ::_thesis: verum
end;
hence { y where y is Point of CNS : ||.(y - x0).|| < g } is Neighbourhood of x0 by A1, Def1; ::_thesis: verum
end;
theorem Th3: :: NCFCONT1:3
for CNS being ComplexNormSpace
for x0 being Point of CNS
for N being Neighbourhood of x0 holds x0 in N
proof
let CNS be ComplexNormSpace; ::_thesis: for x0 being Point of CNS
for N being Neighbourhood of x0 holds x0 in N
let x0 be Point of CNS; ::_thesis: for N being Neighbourhood of x0 holds x0 in N
let N be Neighbourhood of x0; ::_thesis: x0 in N
consider g being Real such that
A1: 0 < g and
A2: { z where z is Point of CNS : ||.(z - x0).|| < g } c= N by Def1;
||.(x0 - x0).|| = ||.(0. CNS).|| by RLVECT_1:15
.= 0 by CLVECT_1:102 ;
then x0 in { z where z is Point of CNS : ||.(z - x0).|| < g } by A1;
hence x0 in N by A2; ::_thesis: verum
end;
definition
let CNS be ComplexNormSpace;
let X be Subset of CNS;
attrX is compact means :Def2: :: NCFCONT1:def 2
for s1 being sequence of CNS st rng s1 c= X holds
ex s2 being sequence of CNS st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X );
end;
:: deftheorem Def2 defines compact NCFCONT1:def_2_:_
for CNS being ComplexNormSpace
for X being Subset of CNS holds
( X is compact iff for s1 being sequence of CNS st rng s1 c= X holds
ex s2 being sequence of CNS st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X ) );
definition
let CNS be ComplexNormSpace;
let X be Subset of CNS;
attrX is closed means :: NCFCONT1:def 3
for s1 being sequence of CNS st rng s1 c= X & s1 is convergent holds
lim s1 in X;
end;
:: deftheorem defines closed NCFCONT1:def_3_:_
for CNS being ComplexNormSpace
for X being Subset of CNS holds
( X is closed iff for s1 being sequence of CNS st rng s1 c= X & s1 is convergent holds
lim s1 in X );
definition
let CNS be ComplexNormSpace;
let X be Subset of CNS;
attrX is open means :: NCFCONT1:def 4
X ` is closed ;
end;
:: deftheorem defines open NCFCONT1:def_4_:_
for CNS being ComplexNormSpace
for X being Subset of CNS holds
( X is open iff X ` is closed );
definition
let CNS1, CNS2 be ComplexNormSpace;
let f be PartFunc of CNS1,CNS2;
let x0 be Point of CNS1;
predf is_continuous_in x0 means :Def5: :: NCFCONT1:def 5
( x0 in dom f & ( for seq being sequence of CNS1 st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def5 defines is_continuous_in NCFCONT1:def_5_:_
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of CNS1 st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
definition
let CNS be ComplexNormSpace;
let RNS be RealNormSpace;
let f be PartFunc of CNS,RNS;
let x0 be Point of CNS;
predf is_continuous_in x0 means :Def6: :: NCFCONT1:def 6
( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def6 defines is_continuous_in NCFCONT1:def_6_:_
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
definition
let RNS be RealNormSpace;
let CNS be ComplexNormSpace;
let f be PartFunc of RNS,CNS;
let x0 be Point of RNS;
predf is_continuous_in x0 means :Def7: :: NCFCONT1:def 7
( x0 in dom f & ( for seq being sequence of RNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def7 defines is_continuous_in NCFCONT1:def_7_:_
for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of RNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
definition
let CNS be ComplexNormSpace;
let f be PartFunc of the carrier of CNS,COMPLEX;
let x0 be Point of CNS;
predf is_continuous_in x0 means :Def8: :: NCFCONT1:def 8
( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def8 defines is_continuous_in NCFCONT1:def_8_:_
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,COMPLEX
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
definition
let CNS be ComplexNormSpace;
let f be PartFunc of the carrier of CNS,REAL;
let x0 be Point of CNS;
predf is_continuous_in x0 means :Def9: :: NCFCONT1:def 9
( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def9 defines is_continuous_in NCFCONT1:def_9_:_
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of CNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
definition
let RNS be RealNormSpace;
let f be PartFunc of the carrier of RNS,COMPLEX;
let x0 be Point of RNS;
predf is_continuous_in x0 means :Def10: :: NCFCONT1:def 10
( x0 in dom f & ( for seq being sequence of RNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) );
end;
:: deftheorem Def10 defines is_continuous_in NCFCONT1:def_10_:_
for RNS being RealNormSpace
for f being PartFunc of the carrier of RNS,COMPLEX
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for seq being sequence of RNS st rng seq c= dom f & seq is convergent & lim seq = x0 holds
( f /* seq is convergent & f /. x0 = lim (f /* seq) ) ) ) );
theorem Th4: :: NCFCONT1:4
for n being Element of NAT
for CNS1, CNS2 being ComplexNormSpace
for seq being sequence of CNS1
for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
proof
let n be Element of NAT ; ::_thesis: for CNS1, CNS2 being ComplexNormSpace
for seq being sequence of CNS1
for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for seq being sequence of CNS1
for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let seq be sequence of CNS1; ::_thesis: for h being PartFunc of CNS1,CNS2 st rng seq c= dom h holds
seq . n in dom h
let h be PartFunc of CNS1,CNS2; ::_thesis: ( rng seq c= dom h implies seq . n in dom h )
n in NAT ;
then A1: n in dom seq by FUNCT_2:def_1;
assume rng seq c= dom h ; ::_thesis: seq . n in dom h
then n in dom (h * seq) by A1, RELAT_1:27;
hence seq . n in dom h by FUNCT_1:11; ::_thesis: verum
end;
theorem Th5: :: NCFCONT1:5
for n being Element of NAT
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for seq being sequence of CNS
for h being PartFunc of CNS,RNS st rng seq c= dom h holds
seq . n in dom h
proof
let n be Element of NAT ; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for seq being sequence of CNS
for h being PartFunc of CNS,RNS st rng seq c= dom h holds
seq . n in dom h
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for seq being sequence of CNS
for h being PartFunc of CNS,RNS st rng seq c= dom h holds
seq . n in dom h
let RNS be RealNormSpace; ::_thesis: for seq being sequence of CNS
for h being PartFunc of CNS,RNS st rng seq c= dom h holds
seq . n in dom h
let seq be sequence of CNS; ::_thesis: for h being PartFunc of CNS,RNS st rng seq c= dom h holds
seq . n in dom h
let h be PartFunc of CNS,RNS; ::_thesis: ( rng seq c= dom h implies seq . n in dom h )
n in NAT ;
then A1: n in dom seq by FUNCT_2:def_1;
assume rng seq c= dom h ; ::_thesis: seq . n in dom h
then n in dom (h * seq) by A1, RELAT_1:27;
hence seq . n in dom h by FUNCT_1:11; ::_thesis: verum
end;
theorem Th6: :: NCFCONT1:6
for n being Element of NAT
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for seq being sequence of RNS
for h being PartFunc of RNS,CNS st rng seq c= dom h holds
seq . n in dom h
proof
let n be Element of NAT ; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for seq being sequence of RNS
for h being PartFunc of RNS,CNS st rng seq c= dom h holds
seq . n in dom h
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for seq being sequence of RNS
for h being PartFunc of RNS,CNS st rng seq c= dom h holds
seq . n in dom h
let RNS be RealNormSpace; ::_thesis: for seq being sequence of RNS
for h being PartFunc of RNS,CNS st rng seq c= dom h holds
seq . n in dom h
let seq be sequence of RNS; ::_thesis: for h being PartFunc of RNS,CNS st rng seq c= dom h holds
seq . n in dom h
let h be PartFunc of RNS,CNS; ::_thesis: ( rng seq c= dom h implies seq . n in dom h )
n in NAT ;
then A1: n in dom seq by FUNCT_2:def_1;
assume rng seq c= dom h ; ::_thesis: seq . n in dom h
then n in dom (h * seq) by A1, RELAT_1:27;
hence seq . n in dom h by FUNCT_1:11; ::_thesis: verum
end;
theorem Th7: :: NCFCONT1:7
for CNS being ComplexNormSpace
for seq being sequence of CNS
for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
proof
let CNS be ComplexNormSpace; ::_thesis: for seq being sequence of CNS
for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
let seq be sequence of CNS; ::_thesis: for x being set holds
( x in rng seq iff ex n being Element of NAT st x = seq . n )
let x be set ; ::_thesis: ( x in rng seq iff ex n being Element of NAT st x = seq . n )
thus ( x in rng seq implies ex n being Element of NAT st x = seq . n ) ::_thesis: ( ex n being Element of NAT st x = seq . n implies x in rng seq )
proof
assume x in rng seq ; ::_thesis: ex n being Element of NAT st x = seq . n
then consider y being set such that
A1: y in dom seq and
A2: x = seq . y by FUNCT_1:def_3;
reconsider m = y as Element of NAT by A1;
take m ; ::_thesis: x = seq . m
thus x = seq . m by A2; ::_thesis: verum
end;
given n being Element of NAT such that A3: x = seq . n ; ::_thesis: x in rng seq
n in NAT ;
then n in dom seq by FUNCT_2:def_1;
hence x in rng seq by A3, FUNCT_1:def_3; ::_thesis: verum
end;
theorem Th8: :: NCFCONT1:8
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 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 x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 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 x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 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 x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let x0 be Point of CNS1; ::_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 x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence x0 in dom f by Def5; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of CNS1 st
( x1 in dom f & ||.(x1 - x0).|| < s & not ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of CNS1] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not ||.((f /. $2) - (f /. x0)).|| < r );
A4: for n being Element of NAT ex p being Point of CNS1 st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS1 st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of CNS1 such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not ||.((f /. p) - (f /. x0)).|| < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of CNS1 such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of CNS1 ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_||.(((f_/*_s1)_._n)_-_(f_/._x0)).||_<_r
let n be Element of NAT ; ::_thesis: not ||.(((f /* s1) . n) - (f /. x0)).|| < r
not ||.((f /. (s1 . n)) - (f /. x0)).|| < r by A6;
hence not ||.(((f /* s1) . n) - (f /. x0)).|| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by CLVECT_1:def_15;
then lim s1 = x0 by A10, CLVECT_1:def_16;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def5;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < r by A2, CLVECT_1:def_16;
||.(((f /* s1) . n) - (f /. x0)).|| < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.(((f_/*_s1)_._m)_-_(f_/._x0)).||_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < p by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, CLVECT_1:def_16;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume k <= m ; ::_thesis: ||.(((f /* s1) . m) - (f /. x0)).|| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then ||.((f /. (s1 . m)) - (f /. x0)).|| < p by A18, A22, A24;
hence ||.(((f /* s1) . m) - (f /. x0)).|| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by CLVECT_1:def_15;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, CLVECT_1:def_16; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def5; ::_thesis: verum
end;
theorem Th9: :: NCFCONT1:9
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let x0 be Point of CNS; ::_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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence x0 in dom f by Def6; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of CNS st
( x1 in dom f & ||.(x1 - x0).|| < s & not ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of CNS] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not ||.((f /. $2) - (f /. x0)).|| < r );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of CNS such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not ||.((f /. p) - (f /. x0)).|| < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of CNS such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of CNS ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_||.(((f_/*_s1)_._n)_-_(f_/._x0)).||_<_r
let n be Element of NAT ; ::_thesis: not ||.(((f /* s1) . n) - (f /. x0)).|| < r
not ||.((f /. (s1 . n)) - (f /. x0)).|| < r by A6;
hence not ||.(((f /* s1) . n) - (f /. x0)).|| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by CLVECT_1:def_15;
then lim s1 = x0 by A10, CLVECT_1:def_16;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def6;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < r by A2, NORMSP_1:def_7;
||.(((f /* s1) . n) - (f /. x0)).|| < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.(((f_/*_s1)_._m)_-_(f_/._x0)).||_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < p by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, CLVECT_1:def_16;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume k <= m ; ::_thesis: ||.(((f /* s1) . m) - (f /. x0)).|| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then ||.((f /. (s1 . m)) - (f /. x0)).|| < p by A18, A22, A24;
hence ||.(((f /* s1) . m) - (f /. x0)).|| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by NORMSP_1:def_6;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, NORMSP_1:def_7; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def6; ::_thesis: verum
end;
theorem Th10: :: NCFCONT1:10
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let x0 be Point of RNS; ::_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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence x0 in dom f by Def7; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of RNS st
( x1 in dom f & ||.(x1 - x0).|| < s & not ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of RNS] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not ||.((f /. $2) - (f /. x0)).|| < r );
A4: for n being Element of NAT ex p being Point of RNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of RNS st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of RNS such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not ||.((f /. p) - (f /. x0)).|| < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of RNS such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of RNS ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_||.(((f_/*_s1)_._n)_-_(f_/._x0)).||_<_r
let n be Element of NAT ; ::_thesis: not ||.(((f /* s1) . n) - (f /. x0)).|| < r
not ||.((f /. (s1 . n)) - (f /. x0)).|| < r by A6;
hence not ||.(((f /* s1) . n) - (f /. x0)).|| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by NORMSP_1:def_6;
then lim s1 = x0 by A10, NORMSP_1:def_7;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def7;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < r by A2, CLVECT_1:def_16;
||.(((f /* s1) . n) - (f /. x0)).|| < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.(((f_/*_s1)_._m)_-_(f_/._x0)).||_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < p by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, NORMSP_1:def_7;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.(((f /* s1) . m) - (f /. x0)).|| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s1) . m) - (f /. x0)).|| < p )
assume k <= m ; ::_thesis: ||.(((f /* s1) . m) - (f /. x0)).|| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then ||.((f /. (s1 . m)) - (f /. x0)).|| < p by A18, A22, A24;
hence ||.(((f /* s1) . m) - (f /. x0)).|| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by CLVECT_1:def_15;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, CLVECT_1:def_16; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def7; ::_thesis: verum
end;
theorem Th11: :: NCFCONT1:11
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,REAL
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let x0 be Point of CNS; ::_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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) )
hence x0 in dom f by Def9; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of CNS st
( x1 in dom f & ||.(x1 - x0).|| < s & not abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of CNS] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not abs ((f /. $2) - (f /. x0)) < r );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of CNS such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not abs ((f /. p) - (f /. x0)) < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of CNS such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of CNS ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_abs_(((f_/*_s1)_._n)_-_(f_/._x0))_<_r
let n be Element of NAT ; ::_thesis: not abs (((f /* s1) . n) - (f /. x0)) < r
not abs ((f /. (s1 . n)) - (f /. x0)) < r by A6;
hence not abs (((f /* s1) . n) - (f /. x0)) < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by CLVECT_1:def_15;
then lim s1 = x0 by A10, CLVECT_1:def_16;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def9;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
abs (((f /* s1) . m) - (f /. x0)) < r by A2, SEQ_2:def_7;
abs (((f /* s1) . n) - (f /. x0)) < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
abs_(((f_/*_s1)_._m)_-_(f_/._x0))_<_p
let p be real number ; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p )
reconsider pp = p as Real by XREAL_0:def_1;
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < pp by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, CLVECT_1:def_16;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
abs (((f /* s1) . m) - (f /. x0)) < p
let m be Element of NAT ; ::_thesis: ( k <= m implies abs (((f /* s1) . m) - (f /. x0)) < p )
assume k <= m ; ::_thesis: abs (((f /* s1) . m) - (f /. x0)) < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then abs ((f /. (s1 . m)) - (f /. x0)) < p by A18, A22, A24;
hence abs (((f /* s1) . m) - (f /. x0)) < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by SEQ_2:def_6;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, SEQ_2:def_7; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def9; ::_thesis: verum
end;
theorem Th12: :: NCFCONT1:12
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,COMPLEX
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,COMPLEX
for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of the carrier of CNS,COMPLEX; ::_thesis: for x0 being Point of CNS 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let x0 be Point of CNS; ::_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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
hence x0 in dom f by Def8; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of CNS st
( x1 in dom f & ||.(x1 - x0).|| < s & not |.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of CNS] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not |.((f /. $2) - (f /. x0)).| < r );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of CNS such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not |.((f /. p) - (f /. x0)).| < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of CNS such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of CNS ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_|.(((f_/*_s1)_._n)_-_(f_/._x0)).|_<_r
let n be Element of NAT ; ::_thesis: not |.(((f /* s1) . n) - (f /. x0)).| < r
not |.((f /. (s1 . n)) - (f /. x0)).| < r by A6;
hence not |.(((f /* s1) . n) - (f /. x0)).| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by CLVECT_1:def_15;
then lim s1 = x0 by A10, CLVECT_1:def_16;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def8;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < r by A2, COMSEQ_2:def_6;
|.(((f /* s1) . n) - (f /. x0)).| < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
|.(((f_/*_s1)_._m)_-_(f_/._x0)).|_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p )
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < p by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, CLVECT_1:def_16;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies |.(((f /* s1) . m) - (f /. x0)).| < p )
assume k <= m ; ::_thesis: |.(((f /* s1) . m) - (f /. x0)).| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then |.((f /. (s1 . m)) - (f /. x0)).| < p by A18, A22, A24;
hence |.(((f /* s1) . m) - (f /. x0)).| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by COMSEQ_2:def_5;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, COMSEQ_2:def_6; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def8; ::_thesis: verum
end;
theorem Th13: :: NCFCONT1:13
for RNS being RealNormSpace
for f being PartFunc of the carrier of RNS,COMPLEX
for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of RNS,COMPLEX
for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of the carrier of RNS,COMPLEX; ::_thesis: for x0 being Point of RNS 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let x0 be Point of RNS; ::_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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_in x0 )
proof
assume A1: 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 x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
hence x0 in dom f by Def10; ::_thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
given r being Real such that A2: 0 < r and
A3: for s being Real holds
( not 0 < s or ex x1 being Point of RNS st
( x1 in dom f & ||.(x1 - x0).|| < s & not |.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: contradiction
defpred S1[ Element of NAT , Point of RNS] means ( $2 in dom f & ||.($2 - x0).|| < 1 / ($1 + 1) & not |.((f /. $2) - (f /. x0)).| < r );
A4: for n being Element of NAT ex p being Point of RNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of RNS st S1[n,p]
0 < n + 1 by NAT_1:3;
then 0 < (n + 1) " by XREAL_1:122;
then 0 < 1 / (n + 1) by XCMPLX_1:215;
then consider p being Point of RNS such that
A5: ( p in dom f & ||.(p - x0).|| < 1 / (n + 1) & not |.((f /. p) - (f /. x0)).| < r ) by A3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5; ::_thesis: verum
end;
consider s1 being Function of NAT, the carrier of RNS such that
A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4);
reconsider s1 = s1 as sequence of RNS ;
A7: rng s1 c= dom f
proof
A8: dom s1 = NAT by FUNCT_2:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng s1 or v in dom f )
assume v in rng s1 ; ::_thesis: v in dom f
then ex n being set st
( n in NAT & v = s1 . n ) by A8, FUNCT_1:def_3;
hence v in dom f by A6; ::_thesis: verum
end;
A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_not_|.(((f_/*_s1)_._n)_-_(f_/._x0)).|_<_r
let n be Element of NAT ; ::_thesis: not |.(((f /* s1) . n) - (f /. x0)).| < r
not |.((f /. (s1 . n)) - (f /. x0)).| < r by A6;
hence not |.(((f /* s1) . n) - (f /. x0)).| < r by A7, FUNCT_2:109; ::_thesis: verum
end;
A10: now__::_thesis:_for_s_being_Real_st_0_<_s_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
||.((s1_._m)_-_x0).||_<_s
let s be Real; ::_thesis: ( 0 < s implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s )
consider n being Element of NAT such that
A11: s " < n by SEQ_4:3;
assume 0 < s ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
then A12: 0 < s " by XREAL_1:122;
(s ") + 0 < n + 1 by A11, XREAL_1:8;
then 1 / (n + 1) < 1 / (s ") by A12, XREAL_1:76;
then A13: 1 / (n + 1) < s by XCMPLX_1:216;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
||.((s1 . m) - x0).|| < s
let m be Element of NAT ; ::_thesis: ( k <= m implies ||.((s1 . m) - x0).|| < s )
assume k <= m ; ::_thesis: ||.((s1 . m) - x0).|| < s
then k + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (k + 1) by NAT_1:3, XREAL_1:118;
then 1 / (m + 1) < s by A13, XXREAL_0:2;
hence ||.((s1 . m) - x0).|| < s by A6, XXREAL_0:2; ::_thesis: verum
end;
then A14: s1 is convergent by NORMSP_1:def_6;
then lim s1 = x0 by A10, NORMSP_1:def_7;
then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A14, Def10;
then consider n being Element of NAT such that
A15: for m being Element of NAT st n <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < r by A2, COMSEQ_2:def_6;
|.(((f /* s1) . n) - (f /. x0)).| < r by A15;
hence contradiction by A9; ::_thesis: verum
end;
assume that
A16: x0 in dom f and
A17: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) )
assume that
A18: rng s1 c= dom f and
A19: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) )
A20: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_
ex_k_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_k_<=_m_holds_
|.(((f_/*_s1)_._m)_-_(f_/._x0)).|_<_p
let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p )
assume 0 < p ; ::_thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p
then consider s being Real such that
A21: 0 < s and
A22: for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < p by A17;
consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
||.((s1 . m) - x0).|| < s by A19, A21, NORMSP_1:def_7;
take k = n; ::_thesis: for m being Element of NAT st k <= m holds
|.(((f /* s1) . m) - (f /. x0)).| < p
let m be Element of NAT ; ::_thesis: ( k <= m implies |.(((f /* s1) . m) - (f /. x0)).| < p )
assume k <= m ; ::_thesis: |.(((f /* s1) . m) - (f /. x0)).| < p
then A24: ||.((s1 . m) - x0).|| < s by A23;
dom s1 = NAT by FUNCT_2:def_1;
then s1 . m in rng s1 by FUNCT_1:3;
then |.((f /. (s1 . m)) - (f /. x0)).| < p by A18, A22, A24;
hence |.(((f /* s1) . m) - (f /. x0)).| < p by A18, FUNCT_2:109; ::_thesis: verum
end;
then f /* s1 is convergent by COMSEQ_2:def_5;
hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A20, COMSEQ_2:def_6; ::_thesis: verum
end;
hence f is_continuous_in x0 by A16, Def10; ::_thesis: verum
end;
theorem Th14: :: NCFCONT1:14
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let x0 be Point of CNS1; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ) )
hence x0 in dom f by Def5; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1
consider r being Real such that
A2: 0 < r and
A3: { y where y is Point of CNS2 : ||.(y - (f /. x0)).|| < r } c= N1 by Def1;
consider s being Real such that
A4: 0 < s and
A5: for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th8;
reconsider N = { z where z is Point of CNS1 : ||.(z - x0).|| < s } as Neighbourhood of x0 by A4, Th2;
take N ; ::_thesis: for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of CNS1; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A6: x1 in dom f and
A7: x1 in N ; ::_thesis: f /. x1 in N1
ex z being Point of CNS1 st
( x1 = z & ||.(z - x0).|| < s ) by A7;
then ||.((f /. x1) - (f /. x0)).|| < r by A5, A6;
then f /. x1 in { y where y is Point of CNS2 : ||.(y - (f /. x0)).|| < r } ;
hence f /. x1 in N1 by A3; ::_thesis: verum
end;
assume that
A8: x0 in dom f and
A9: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Point_of_CNS1_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 Point of CNS1 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 Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
then reconsider N1 = { y where y is Point of CNS2 : ||.(y - (f /. x0)).|| < r } as Neighbourhood of f /. x0 by Th2;
consider N2 being Neighbourhood of x0 such that
A10: for x1 being Point of CNS1 st x1 in dom f & x1 in N2 holds
f /. x1 in N1 by A9;
consider s being Real such that
A11: 0 < s and
A12: { z where z is Point of CNS1 : ||.(z - x0).|| < s } c= N2 by Def1;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
proof
let x1 be Point of CNS1; ::_thesis: ( x1 in dom f & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A13: x1 in dom f and
A14: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
x1 in { z where z is Point of CNS1 : ||.(z - x0).|| < s } by A14;
then f /. x1 in N1 by A10, A12, A13;
then ex y being Point of CNS2 st
( f /. x1 = y & ||.(y - (f /. x0)).|| < r ) ;
hence ||.((f /. x1) - (f /. x0)).|| < r ; ::_thesis: verum
end;
hence ( 0 < s & ( for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) by A11; ::_thesis: verum
end;
hence f is_continuous_in x0 by A8, Th8; ::_thesis: verum
end;
theorem Th15: :: NCFCONT1:15
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let x0 be Point of CNS; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) )
hence x0 in dom f by Def6; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1
consider r being Real such that
A2: 0 < r and
A3: { y where y is Point of RNS : ||.(y - (f /. x0)).|| < r } c= N1 by NFCONT_1:def_1;
consider s being Real such that
A4: 0 < s and
A5: for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th9;
reconsider N = { z where z is Point of CNS : ||.(z - x0).|| < s } as Neighbourhood of x0 by A4, Th2;
take N ; ::_thesis: for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of CNS; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A6: x1 in dom f and
A7: x1 in N ; ::_thesis: f /. x1 in N1
ex z being Point of CNS st
( x1 = z & ||.(z - x0).|| < s ) by A7;
then ||.((f /. x1) - (f /. x0)).|| < r by A5, A6;
then f /. x1 in { y where y is Point of RNS : ||.(y - (f /. x0)).|| < r } ;
hence f /. x1 in N1 by A3; ::_thesis: verum
end;
assume that
A8: x0 in dom f and
A9: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Point_of_CNS_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 Point of CNS 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 Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
then reconsider N1 = { y where y is Point of RNS : ||.(y - (f /. x0)).|| < r } as Neighbourhood of f /. x0 by NFCONT_1:3;
consider N2 being Neighbourhood of x0 such that
A10: for x1 being Point of CNS st x1 in dom f & x1 in N2 holds
f /. x1 in N1 by A9;
consider s being Real such that
A11: 0 < s and
A12: { z where z is Point of CNS : ||.(z - x0).|| < s } c= N2 by Def1;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
proof
let x1 be Point of CNS; ::_thesis: ( x1 in dom f & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A13: x1 in dom f and
A14: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
x1 in { z where z is Point of CNS : ||.(z - x0).|| < s } by A14;
then f /. x1 in N1 by A10, A12, A13;
then ex y being Point of RNS st
( f /. x1 = y & ||.(y - (f /. x0)).|| < r ) ;
hence ||.((f /. x1) - (f /. x0)).|| < r ; ::_thesis: verum
end;
hence ( 0 < s & ( for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) by A11; ::_thesis: verum
end;
hence f is_continuous_in x0 by A8, Th9; ::_thesis: verum
end;
theorem Th16: :: NCFCONT1:16
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
let x0 be Point of RNS; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ) )
hence x0 in dom f by Def7; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1
consider r being Real such that
A2: 0 < r and
A3: { y where y is Point of CNS : ||.(y - (f /. x0)).|| < r } c= N1 by Def1;
consider s being Real such that
A4: 0 < s and
A5: for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th10;
reconsider N = { z where z is Point of RNS : ||.(z - x0).|| < s } as Neighbourhood of x0 by A4, NFCONT_1:3;
take N ; ::_thesis: for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of RNS; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A6: x1 in dom f and
A7: x1 in N ; ::_thesis: f /. x1 in N1
ex z being Point of RNS st
( x1 = z & ||.(z - x0).|| < s ) by A7;
then ||.((f /. x1) - (f /. x0)).|| < r by A5, A6;
then f /. x1 in { y where y is Point of CNS : ||.(y - (f /. x0)).|| < r } ;
hence f /. x1 in N1 by A3; ::_thesis: verum
end;
assume that
A8: x0 in dom f and
A9: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_r_being_Real_st_0_<_r_holds_
ex_s_being_Real_st_
(_0_<_s_&_(_for_x1_being_Point_of_RNS_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 Point of RNS 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 Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
then reconsider N1 = { y where y is Point of CNS : ||.(y - (f /. x0)).|| < r } as Neighbourhood of f /. x0 by Th2;
consider N2 being Neighbourhood of x0 such that
A10: for x1 being Point of RNS st x1 in dom f & x1 in N2 holds
f /. x1 in N1 by A9;
consider s being Real such that
A11: 0 < s and
A12: { z where z is Point of RNS : ||.(z - x0).|| < s } c= N2 by NFCONT_1:def_1;
take s = s; ::_thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
proof
let x1 be Point of RNS; ::_thesis: ( x1 in dom f & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A13: x1 in dom f and
A14: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
x1 in { z where z is Point of RNS : ||.(z - x0).|| < s } by A14;
then f /. x1 in N1 by A10, A12, A13;
then ex y being Point of CNS st
( f /. x1 = y & ||.(y - (f /. x0)).|| < r ) ;
hence ||.((f /. x1) - (f /. x0)).|| < r ; ::_thesis: verum
end;
hence ( 0 < s & ( for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) by A11; ::_thesis: verum
end;
hence f is_continuous_in x0 by A8, Th10; ::_thesis: verum
end;
theorem Th17: :: NCFCONT1:17
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let x0 be Point of CNS1; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) )
hence x0 in dom f by Def5; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
consider N being Neighbourhood of x0 such that
A2: for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1 by A1, Th14;
take N ; ::_thesis: f .: N c= N1
now__::_thesis:_for_r_being_set_st_r_in_f_.:_N_holds_
r_in_N1
let r be set ; ::_thesis: ( r in f .: N implies r in N1 )
assume r in f .: N ; ::_thesis: r in N1
then consider x being Point of CNS1 such that
A3: x in dom f and
A4: x in N and
A5: r = f . x by PARTFUN2:59;
r = f /. x by A3, A5, PARTFUN1:def_6;
hence r in N1 by A2, A3, A4; ::_thesis: verum
end;
hence f .: N c= N1 by TARSKI:def_3; ::_thesis: verum
end;
assume that
A6: x0 in dom f and
A7: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_
for_x1_being_Point_of_CNS1_st_x1_in_dom_f_&_x1_in_N_holds_
f_/._x1_in_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1
consider N being Neighbourhood of x0 such that
A8: f .: N c= N1 by A7;
take N = N; ::_thesis: for x1 being Point of CNS1 st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of CNS1; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A9: x1 in dom f and
A10: x1 in N ; ::_thesis: f /. x1 in N1
f . x1 in f .: N by A9, A10, FUNCT_1:def_6;
then f . x1 in N1 by A8;
hence f /. x1 in N1 by A9, PARTFUN1:def_6; ::_thesis: verum
end;
hence f is_continuous_in x0 by A6, Th14; ::_thesis: verum
end;
theorem Th18: :: NCFCONT1:18
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let x0 be Point of CNS; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) )
hence x0 in dom f by Def6; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
consider N being Neighbourhood of x0 such that
A2: for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1 by A1, Th15;
take N ; ::_thesis: f .: N c= N1
now__::_thesis:_for_r_being_set_st_r_in_f_.:_N_holds_
r_in_N1
let r be set ; ::_thesis: ( r in f .: N implies r in N1 )
assume r in f .: N ; ::_thesis: r in N1
then consider x being Point of CNS such that
A3: x in dom f and
A4: x in N and
A5: r = f . x by PARTFUN2:59;
r = f /. x by A3, A5, PARTFUN1:def_6;
hence r in N1 by A2, A3, A4; ::_thesis: verum
end;
hence f .: N c= N1 by TARSKI:def_3; ::_thesis: verum
end;
assume that
A6: x0 in dom f and
A7: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_
for_x1_being_Point_of_CNS_st_x1_in_dom_f_&_x1_in_N_holds_
f_/._x1_in_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1
consider N being Neighbourhood of x0 such that
A8: f .: N c= N1 by A7;
take N = N; ::_thesis: for x1 being Point of CNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of CNS; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A9: x1 in dom f and
A10: x1 in N ; ::_thesis: f /. x1 in N1
f . x1 in f .: N by A9, A10, FUNCT_1:def_6;
then f . x1 in N1 by A8;
hence f /. x1 in N1 by A9, PARTFUN1:def_6; ::_thesis: verum
end;
hence f is_continuous_in x0 by A6, Th15; ::_thesis: verum
end;
theorem Th19: :: NCFCONT1:19
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS holds
( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
let x0 be Point of RNS; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) )
thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) implies f is_continuous_in x0 )
proof
assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) )
hence x0 in dom f by Def7; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
consider N being Neighbourhood of x0 such that
A2: for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1 by A1, Th16;
take N ; ::_thesis: f .: N c= N1
now__::_thesis:_for_r_being_set_st_r_in_f_.:_N_holds_
r_in_N1
let r be set ; ::_thesis: ( r in f .: N implies r in N1 )
assume r in f .: N ; ::_thesis: r in N1
then consider x being Point of RNS such that
A3: x in dom f and
A4: x in N and
A5: r = f . x by PARTFUN2:59;
r = f /. x by A3, A5, PARTFUN1:def_6;
hence r in N1 by A2, A3, A4; ::_thesis: verum
end;
hence f .: N c= N1 by TARSKI:def_3; ::_thesis: verum
end;
assume that
A6: x0 in dom f and
A7: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_
for_x1_being_Point_of_RNS_st_x1_in_dom_f_&_x1_in_N_holds_
f_/._x1_in_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st
for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1
consider N being Neighbourhood of x0 such that
A8: f .: N c= N1 by A7;
take N = N; ::_thesis: for x1 being Point of RNS st x1 in dom f & x1 in N holds
f /. x1 in N1
let x1 be Point of RNS; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 )
assume that
A9: x1 in dom f and
A10: x1 in N ; ::_thesis: f /. x1 in N1
f . x1 in f .: N by A9, A10, FUNCT_1:def_6;
then f . x1 in N1 by A8;
hence f /. x1 in N1 by A9, PARTFUN1:def_6; ::_thesis: verum
end;
hence f is_continuous_in x0 by A6, Th16; ::_thesis: verum
end;
theorem :: NCFCONT1:20
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let x0 be Point of CNS1; ::_thesis: ( x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 )
assume A1: x0 in dom f ; ::_thesis: ( for N being Neighbourhood of x0 holds not (dom f) /\ N = {x0} or f is_continuous_in x0 )
given N being Neighbourhood of x0 such that A2: (dom f) /\ N = {x0} ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_f_.:_N_c=_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
take N = N; ::_thesis: f .: N c= N1
A3: f /. x0 in N1 by Th3;
f .: N = Im (f,x0) by A2, RELAT_1:112
.= {(f . x0)} by A1, FUNCT_1:59
.= {(f /. x0)} by A1, PARTFUN1:def_6 ;
hence f .: N c= N1 by A3, ZFMISC_1:31; ::_thesis: verum
end;
hence f is_continuous_in x0 by A1, Th17; ::_thesis: verum
end;
theorem :: NCFCONT1:21
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let x0 be Point of CNS; ::_thesis: ( x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 )
assume A1: x0 in dom f ; ::_thesis: ( for N being Neighbourhood of x0 holds not (dom f) /\ N = {x0} or f is_continuous_in x0 )
given N being Neighbourhood of x0 such that A2: (dom f) /\ N = {x0} ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_f_.:_N_c=_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
take N = N; ::_thesis: f .: N c= N1
A3: f /. x0 in N1 by NFCONT_1:4;
f .: N = Im (f,x0) by A2, RELAT_1:112
.= {(f . x0)} by A1, FUNCT_1:59
.= {(f /. x0)} by A1, PARTFUN1:def_6 ;
hence f .: N c= N1 by A3, ZFMISC_1:31; ::_thesis: verum
end;
hence f is_continuous_in x0 by A1, Th18; ::_thesis: verum
end;
theorem :: NCFCONT1:22
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS st x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds
f is_continuous_in x0
let x0 be Point of RNS; ::_thesis: ( x0 in dom f & ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 )
assume A1: x0 in dom f ; ::_thesis: ( for N being Neighbourhood of x0 holds not (dom f) /\ N = {x0} or f is_continuous_in x0 )
given N being Neighbourhood of x0 such that A2: (dom f) /\ N = {x0} ; ::_thesis: f is_continuous_in x0
now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_f_.:_N_c=_N1
let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1
take N = N; ::_thesis: f .: N c= N1
A3: f /. x0 in N1 by Th3;
f .: N = Im (f,x0) by A2, RELAT_1:112
.= {(f . x0)} by A1, FUNCT_1:59
.= {(f /. x0)} by A1, PARTFUN1:def_6 ;
hence f .: N c= N1 by A3, ZFMISC_1:31; ::_thesis: verum
end;
hence f is_continuous_in x0 by A1, Th19; ::_thesis: verum
end;
theorem Th23: :: NCFCONT1:23
for CNS1, CNS2 being ComplexNormSpace
for h1, h2 being PartFunc of CNS1,CNS2
for seq being sequence of CNS1 st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for h1, h2 being PartFunc of CNS1,CNS2
for seq being sequence of CNS1 st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let h1, h2 be PartFunc of CNS1,CNS2; ::_thesis: for seq being sequence of CNS1 st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let seq be sequence of CNS1; ::_thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) )
A1: (dom h1) /\ (dom h2) c= dom h1 by XBOOLE_1:17;
A2: (dom h1) /\ (dom h2) c= dom h2 by XBOOLE_1:17;
assume A3: rng seq c= (dom h1) /\ (dom h2) ; ::_thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_+_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_+_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)
A5: seq . n in dom (h1 + h2) by A4, Th4;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A5, VFUNCT_1:def_1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def_2; ::_thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq)
A6: rng seq c= dom (h1 - h2) by A3, VFUNCT_1:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_-_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_-_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)
A7: seq . n in dom (h1 - h2) by A6, Th4;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A6, FUNCT_2:109
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A7, VFUNCT_1:def_2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def_3; ::_thesis: verum
end;
theorem Th24: :: NCFCONT1:24
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h1, h2 being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h1, h2 being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let RNS be RealNormSpace; ::_thesis: for h1, h2 being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let h1, h2 be PartFunc of CNS,RNS; ::_thesis: for seq being sequence of CNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let seq be sequence of CNS; ::_thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) )
A1: (dom h1) /\ (dom h2) c= dom h1 by XBOOLE_1:17;
A2: (dom h1) /\ (dom h2) c= dom h2 by XBOOLE_1:17;
assume A3: rng seq c= (dom h1) /\ (dom h2) ; ::_thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_+_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_+_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)
A5: seq . n in dom (h1 + h2) by A4, Th5;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A5, VFUNCT_1:def_1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def_2; ::_thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq)
A6: rng seq c= dom (h1 - h2) by A3, VFUNCT_1:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_-_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_-_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)
A7: seq . n in dom (h1 - h2) by A6, Th5;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A6, FUNCT_2:109
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A7, VFUNCT_1:def_2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def_3; ::_thesis: verum
end;
theorem Th25: :: NCFCONT1:25
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h1, h2 being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h1, h2 being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let RNS be RealNormSpace; ::_thesis: for h1, h2 being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let h1, h2 be PartFunc of RNS,CNS; ::_thesis: for seq being sequence of RNS st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
let seq be sequence of RNS; ::_thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) )
A1: (dom h1) /\ (dom h2) c= dom h1 by XBOOLE_1:17;
A2: (dom h1) /\ (dom h2) c= dom h2 by XBOOLE_1:17;
assume A3: rng seq c= (dom h1) /\ (dom h2) ; ::_thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )
then A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_+_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_+_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n)
A5: seq . n in dom (h1 + h2) by A4, Th6;
thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109
.= (h1 /. (seq . n)) + (h2 /. (seq . n)) by A5, VFUNCT_1:def_1
.= ((h1 /* seq) . n) + (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def_2; ::_thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq)
A6: rng seq c= dom (h1 - h2) by A3, VFUNCT_1:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_-_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_-_((h2_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n)
A7: seq . n in dom (h1 - h2) by A6, Th6;
thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A6, FUNCT_2:109
.= (h1 /. (seq . n)) - (h2 /. (seq . n)) by A7, VFUNCT_1:def_2
.= ((h1 /* seq) . n) - (h2 /. (seq . n)) by A3, A1, FUNCT_2:109, XBOOLE_1:1
.= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A3, A2, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum
end;
hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def_3; ::_thesis: verum
end;
theorem Th26: :: NCFCONT1:26
for CNS1, CNS2 being ComplexNormSpace
for h being PartFunc of CNS1,CNS2
for seq being sequence of CNS1
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for h being PartFunc of CNS1,CNS2
for seq being sequence of CNS1
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let h be PartFunc of CNS1,CNS2; ::_thesis: for seq being sequence of CNS1
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let seq be sequence of CNS1; ::_thesis: for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let z be Complex; ::_thesis: ( rng seq c= dom h implies (z (#) h) /* seq = z * (h /* seq) )
assume A1: rng seq c= dom h ; ::_thesis: (z (#) h) /* seq = z * (h /* seq)
then A2: rng seq c= dom (z (#) h) by VFUNCT_2:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((z_(#)_h)_/*_seq)_._n_=_z_*_((h_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((z (#) h) /* seq) . n = z * ((h /* seq) . n)
A3: seq . n in dom (z (#) h) by A2, Th4;
thus ((z (#) h) /* seq) . n = (z (#) h) /. (seq . n) by A2, FUNCT_2:109
.= z * (h /. (seq . n)) by A3, VFUNCT_2:def_2
.= z * ((h /* seq) . n) by A1, FUNCT_2:109 ; ::_thesis: verum
end;
hence (z (#) h) /* seq = z * (h /* seq) by CLVECT_1:def_14; ::_thesis: verum
end;
theorem Th27: :: NCFCONT1:27
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h being PartFunc of CNS,RNS
for seq being sequence of CNS
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h being PartFunc of CNS,RNS
for seq being sequence of CNS
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let RNS be RealNormSpace; ::_thesis: for h being PartFunc of CNS,RNS
for seq being sequence of CNS
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let h be PartFunc of CNS,RNS; ::_thesis: for seq being sequence of CNS
for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let seq be sequence of CNS; ::_thesis: for r being Real st rng seq c= dom h holds
(r (#) h) /* seq = r * (h /* seq)
let r be Real; ::_thesis: ( rng seq c= dom h implies (r (#) h) /* seq = r * (h /* seq) )
assume A1: rng seq c= dom h ; ::_thesis: (r (#) h) /* seq = r * (h /* seq)
then A2: rng seq c= dom (r (#) h) by VFUNCT_1:def_4;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((r_(#)_h)_/*_seq)_._n_=_r_*_((h_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((r (#) h) /* seq) . n = r * ((h /* seq) . n)
A3: seq . n in dom (r (#) h) by A2, Th5;
thus ((r (#) h) /* seq) . n = (r (#) h) /. (seq . n) by A2, FUNCT_2:109
.= r * (h /. (seq . n)) by A3, VFUNCT_1:def_4
.= r * ((h /* seq) . n) by A1, FUNCT_2:109 ; ::_thesis: verum
end;
hence (r (#) h) /* seq = r * (h /* seq) by NORMSP_1:def_5; ::_thesis: verum
end;
theorem Th28: :: NCFCONT1:28
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h being PartFunc of RNS,CNS
for seq being sequence of RNS
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h being PartFunc of RNS,CNS
for seq being sequence of RNS
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let RNS be RealNormSpace; ::_thesis: for h being PartFunc of RNS,CNS
for seq being sequence of RNS
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let h be PartFunc of RNS,CNS; ::_thesis: for seq being sequence of RNS
for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let seq be sequence of RNS; ::_thesis: for z being Complex st rng seq c= dom h holds
(z (#) h) /* seq = z * (h /* seq)
let z be Complex; ::_thesis: ( rng seq c= dom h implies (z (#) h) /* seq = z * (h /* seq) )
assume A1: rng seq c= dom h ; ::_thesis: (z (#) h) /* seq = z * (h /* seq)
then A2: rng seq c= dom (z (#) h) by VFUNCT_2:def_2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((z_(#)_h)_/*_seq)_._n_=_z_*_((h_/*_seq)_._n)
let n be Element of NAT ; ::_thesis: ((z (#) h) /* seq) . n = z * ((h /* seq) . n)
A3: seq . n in dom (z (#) h) by A2, Th6;
thus ((z (#) h) /* seq) . n = (z (#) h) /. (seq . n) by A2, FUNCT_2:109
.= z * (h /. (seq . n)) by A3, VFUNCT_2:def_2
.= z * ((h /* seq) . n) by A1, FUNCT_2:109 ; ::_thesis: verum
end;
hence (z (#) h) /* seq = z * (h /* seq) by CLVECT_1:def_14; ::_thesis: verum
end;
theorem Th29: :: NCFCONT1:29
for CNS1, CNS2 being ComplexNormSpace
for h being PartFunc of CNS1,CNS2
for seq being sequence of CNS1 st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for h being PartFunc of CNS1,CNS2
for seq being sequence of CNS1 st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let h be PartFunc of CNS1,CNS2; ::_thesis: for seq being sequence of CNS1 st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let seq be sequence of CNS1; ::_thesis: ( rng seq c= dom h implies ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) )
assume A1: rng seq c= dom h ; ::_thesis: ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
then A2: rng seq c= dom ||.h.|| by NORMSP_0:def_3;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.(h_/*_seq).||_._n_=_(||.h.||_/*_seq)_._n
let n be Element of NAT ; ::_thesis: ||.(h /* seq).|| . n = (||.h.|| /* seq) . n
seq . n in rng seq by Th7;
then seq . n in dom h by A1;
then A3: seq . n in dom ||.h.|| by NORMSP_0:def_3;
thus ||.(h /* seq).|| . n = ||.((h /* seq) . n).|| by NORMSP_0:def_4
.= ||.(h /. (seq . n)).|| by A1, FUNCT_2:109
.= ||.h.|| . (seq . n) by A3, NORMSP_0:def_3
.= ||.h.|| /. (seq . n) by A3, PARTFUN1:def_6
.= (||.h.|| /* seq) . n by A2, FUNCT_2:109 ; ::_thesis: verum
end;
hence ||.(h /* seq).|| = ||.h.|| /* seq by FUNCT_2:63; ::_thesis: - (h /* seq) = (- h) /* seq
thus - (h /* seq) = (- 1r) * (h /* seq) by Th1
.= ((- 1r) (#) h) /* seq by A1, Th26
.= (- h) /* seq by VFUNCT_2:23 ; ::_thesis: verum
end;
theorem Th30: :: NCFCONT1:30
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let RNS be RealNormSpace; ::_thesis: for h being PartFunc of CNS,RNS
for seq being sequence of CNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let h be PartFunc of CNS,RNS; ::_thesis: for seq being sequence of CNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let seq be sequence of CNS; ::_thesis: ( rng seq c= dom h implies ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) )
assume A1: rng seq c= dom h ; ::_thesis: ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
then A2: rng seq c= dom ||.h.|| by NORMSP_0:def_3;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.(h_/*_seq).||_._n_=_(||.h.||_/*_seq)_._n
let n be Element of NAT ; ::_thesis: ||.(h /* seq).|| . n = (||.h.|| /* seq) . n
seq . n in rng seq by Th7;
then seq . n in dom h by A1;
then A3: seq . n in dom ||.h.|| by NORMSP_0:def_3;
thus ||.(h /* seq).|| . n = ||.((h /* seq) . n).|| by NORMSP_0:def_4
.= ||.(h /. (seq . n)).|| by A1, FUNCT_2:109
.= ||.h.|| . (seq . n) by A3, NORMSP_0:def_3
.= ||.h.|| /. (seq . n) by A3, PARTFUN1:def_6
.= (||.h.|| /* seq) . n by A2, FUNCT_2:109 ; ::_thesis: verum
end;
hence ||.(h /* seq).|| = ||.h.|| /* seq by FUNCT_2:63; ::_thesis: - (h /* seq) = (- h) /* seq
thus - (h /* seq) = (- 1) * (h /* seq) by NFCONT_1:2
.= ((- 1) (#) h) /* seq by A1, Th27
.= (- h) /* seq by VFUNCT_1:23 ; ::_thesis: verum
end;
theorem Th31: :: NCFCONT1:31
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for h being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for h being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let RNS be RealNormSpace; ::_thesis: for h being PartFunc of RNS,CNS
for seq being sequence of RNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let h be PartFunc of RNS,CNS; ::_thesis: for seq being sequence of RNS st rng seq c= dom h holds
( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
let seq be sequence of RNS; ::_thesis: ( rng seq c= dom h implies ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) )
assume A1: rng seq c= dom h ; ::_thesis: ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq )
then A2: rng seq c= dom ||.h.|| by NORMSP_0:def_3;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.(h_/*_seq).||_._n_=_(||.h.||_/*_seq)_._n
let n be Element of NAT ; ::_thesis: ||.(h /* seq).|| . n = (||.h.|| /* seq) . n
seq . n in rng seq by NFCONT_1:6;
then seq . n in dom h by A1;
then A3: seq . n in dom ||.h.|| by NORMSP_0:def_3;
thus ||.(h /* seq).|| . n = ||.((h /* seq) . n).|| by NORMSP_0:def_4
.= ||.(h /. (seq . n)).|| by A1, FUNCT_2:109
.= ||.h.|| . (seq . n) by A3, NORMSP_0:def_3
.= ||.h.|| /. (seq . n) by A3, PARTFUN1:def_6
.= (||.h.|| /* seq) . n by A2, FUNCT_2:109 ; ::_thesis: verum
end;
hence ||.(h /* seq).|| = ||.h.|| /* seq by FUNCT_2:63; ::_thesis: - (h /* seq) = (- h) /* seq
thus - (h /* seq) = (- 1r) * (h /* seq) by Th1
.= ((- 1r) (#) h) /* seq by A1, Th28
.= (- h) /* seq by VFUNCT_2:23 ; ::_thesis: verum
end;
theorem :: NCFCONT1:32
for CNS1, CNS2 being ComplexNormSpace
for f1, f2 being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f1, f2 being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let f1, f2 be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let x0 be Point of CNS1; ::_thesis: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) )
assume that
A1: f1 is_continuous_in x0 and
A2: f2 is_continuous_in x0 ; ::_thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
A3: ( x0 in dom f1 & x0 in dom f2 ) by A1, A2, Def5;
now__::_thesis:_(_x0_in_dom_(f1_+_f2)_&_(_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._x0_=_lim_((f1_+_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A4: x0 in dom (f1 + f2) by VFUNCT_1:def_1; ::_thesis: for s1 being sequence of CNS1 st rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) )
assume that
A5: rng s1 c= dom (f1 + f2) and
A6: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def_1;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then A8: rng s1 c= dom f2 by A5, XBOOLE_1:1;
then A9: f2 /* s1 is convergent by A2, A6, Def5;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then A10: rng s1 c= dom f1 by A5, XBOOLE_1:1;
then A11: f1 /* s1 is convergent by A1, A6, Def5;
then (f1 /* s1) + (f2 /* s1) is convergent by A9, CLVECT_1:113;
hence (f1 + f2) /* s1 is convergent by A7, Th23; ::_thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
A12: f1 /. x0 = lim (f1 /* s1) by A1, A6, A10, Def5;
A13: f2 /. x0 = lim (f2 /* s1) by A2, A6, A8, Def5;
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A11, A12, A9, A13, CLVECT_1:119
.= lim ((f1 + f2) /* s1) by A7, Th23 ; ::_thesis: verum
end;
hence f1 + f2 is_continuous_in x0 by Def5; ::_thesis: f1 - f2 is_continuous_in x0
now__::_thesis:_(_x0_in_dom_(f1_-_f2)_&_(_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._x0_=_lim_((f1_-_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A14: x0 in dom (f1 - f2) by VFUNCT_1:def_2; ::_thesis: for s1 being sequence of CNS1 st rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) )
assume that
A15: rng s1 c= dom (f1 - f2) and
A16: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
A17: rng s1 c= (dom f1) /\ (dom f2) by A15, VFUNCT_1:def_2;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then A18: rng s1 c= dom f2 by A15, XBOOLE_1:1;
then A19: f2 /* s1 is convergent by A2, A16, Def5;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then A20: rng s1 c= dom f1 by A15, XBOOLE_1:1;
then A21: f1 /* s1 is convergent by A1, A16, Def5;
then (f1 /* s1) - (f2 /* s1) is convergent by A19, CLVECT_1:114;
hence (f1 - f2) /* s1 is convergent by A17, Th23; ::_thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
A22: f1 /. x0 = lim (f1 /* s1) by A1, A16, A20, Def5;
A23: f2 /. x0 = lim (f2 /* s1) by A2, A16, A18, Def5;
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A14, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, A22, A19, A23, CLVECT_1:120
.= lim ((f1 - f2) /* s1) by A17, Th23 ; ::_thesis: verum
end;
hence f1 - f2 is_continuous_in x0 by Def5; ::_thesis: verum
end;
theorem :: NCFCONT1:33
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f1, f2 being PartFunc of CNS,RNS
for x0 being Point of CNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f1, f2 being PartFunc of CNS,RNS
for x0 being Point of CNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let RNS be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of CNS,RNS
for x0 being Point of CNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let f1, f2 be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let x0 be Point of CNS; ::_thesis: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) )
assume that
A1: f1 is_continuous_in x0 and
A2: f2 is_continuous_in x0 ; ::_thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
A3: ( x0 in dom f1 & x0 in dom f2 ) by A1, A2, Def6;
now__::_thesis:_(_x0_in_dom_(f1_+_f2)_&_(_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._x0_=_lim_((f1_+_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A4: x0 in dom (f1 + f2) by VFUNCT_1:def_1; ::_thesis: for s1 being sequence of CNS st rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) )
assume that
A5: rng s1 c= dom (f1 + f2) and
A6: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def_1;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then A8: rng s1 c= dom f2 by A5, XBOOLE_1:1;
then A9: f2 /* s1 is convergent by A2, A6, Def6;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then A10: rng s1 c= dom f1 by A5, XBOOLE_1:1;
then A11: f1 /* s1 is convergent by A1, A6, Def6;
then (f1 /* s1) + (f2 /* s1) is convergent by A9, NORMSP_1:19;
hence (f1 + f2) /* s1 is convergent by A7, Th24; ::_thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
A12: f1 /. x0 = lim (f1 /* s1) by A1, A6, A10, Def6;
A13: f2 /. x0 = lim (f2 /* s1) by A2, A6, A8, Def6;
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A11, A12, A9, A13, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A7, Th24 ; ::_thesis: verum
end;
hence f1 + f2 is_continuous_in x0 by Def6; ::_thesis: f1 - f2 is_continuous_in x0
now__::_thesis:_(_x0_in_dom_(f1_-_f2)_&_(_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._x0_=_lim_((f1_-_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A14: x0 in dom (f1 - f2) by VFUNCT_1:def_2; ::_thesis: for s1 being sequence of CNS st rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) )
assume that
A15: rng s1 c= dom (f1 - f2) and
A16: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
A17: rng s1 c= (dom f1) /\ (dom f2) by A15, VFUNCT_1:def_2;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then A18: rng s1 c= dom f2 by A15, XBOOLE_1:1;
then A19: f2 /* s1 is convergent by A2, A16, Def6;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then A20: rng s1 c= dom f1 by A15, XBOOLE_1:1;
then A21: f1 /* s1 is convergent by A1, A16, Def6;
then (f1 /* s1) - (f2 /* s1) is convergent by A19, NORMSP_1:20;
hence (f1 - f2) /* s1 is convergent by A17, Th24; ::_thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
A22: f1 /. x0 = lim (f1 /* s1) by A1, A16, A20, Def6;
A23: f2 /. x0 = lim (f2 /* s1) by A2, A16, A18, Def6;
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A14, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, A22, A19, A23, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A17, Th24 ; ::_thesis: verum
end;
hence f1 - f2 is_continuous_in x0 by Def6; ::_thesis: verum
end;
theorem :: NCFCONT1:34
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f1, f2 being PartFunc of RNS,CNS
for x0 being Point of RNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f1, f2 being PartFunc of RNS,CNS
for x0 being Point of RNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let RNS be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of RNS,CNS
for x0 being Point of RNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let f1, f2 be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let x0 be Point of RNS; ::_thesis: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) )
assume that
A1: f1 is_continuous_in x0 and
A2: f2 is_continuous_in x0 ; ::_thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
A3: ( x0 in dom f1 & x0 in dom f2 ) by A1, A2, Def7;
now__::_thesis:_(_x0_in_dom_(f1_+_f2)_&_(_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._x0_=_lim_((f1_+_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A4: x0 in dom (f1 + f2) by VFUNCT_1:def_1; ::_thesis: for s1 being sequence of RNS st rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) )
assume that
A5: rng s1 c= dom (f1 + f2) and
A6: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def_1;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then A8: rng s1 c= dom f2 by A5, XBOOLE_1:1;
then A9: f2 /* s1 is convergent by A2, A6, Def7;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1;
then dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then A10: rng s1 c= dom f1 by A5, XBOOLE_1:1;
then A11: f1 /* s1 is convergent by A1, A6, Def7;
then (f1 /* s1) + (f2 /* s1) is convergent by A9, CLVECT_1:113;
hence (f1 + f2) /* s1 is convergent by A7, Th25; ::_thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
A12: f1 /. x0 = lim (f1 /* s1) by A1, A6, A10, Def7;
A13: f2 /. x0 = lim (f2 /* s1) by A2, A6, A8, Def7;
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A11, A12, A9, A13, CLVECT_1:119
.= lim ((f1 + f2) /* s1) by A7, Th25 ; ::_thesis: verum
end;
hence f1 + f2 is_continuous_in x0 by Def7; ::_thesis: f1 - f2 is_continuous_in x0
now__::_thesis:_(_x0_in_dom_(f1_-_f2)_&_(_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._x0_=_lim_((f1_-_f2)_/*_s1)_)_)_)
x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def_4;
hence A14: x0 in dom (f1 - f2) by VFUNCT_1:def_2; ::_thesis: for s1 being sequence of RNS st rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) )
assume that
A15: rng s1 c= dom (f1 - f2) and
A16: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
A17: rng s1 c= (dom f1) /\ (dom f2) by A15, VFUNCT_1:def_2;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then A18: rng s1 c= dom f2 by A15, XBOOLE_1:1;
then A19: f2 /* s1 is convergent by A2, A16, Def7;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2;
then dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then A20: rng s1 c= dom f1 by A15, XBOOLE_1:1;
then A21: f1 /* s1 is convergent by A1, A16, Def7;
then (f1 /* s1) - (f2 /* s1) is convergent by A19, CLVECT_1:114;
hence (f1 - f2) /* s1 is convergent by A17, Th25; ::_thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
A22: f1 /. x0 = lim (f1 /* s1) by A1, A16, A20, Def7;
A23: f2 /. x0 = lim (f2 /* s1) by A2, A16, A18, Def7;
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A14, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, A22, A19, A23, CLVECT_1:120
.= lim ((f1 - f2) /* s1) by A17, Th25 ; ::_thesis: verum
end;
hence f1 - f2 is_continuous_in x0 by Def7; ::_thesis: verum
end;
theorem Th35: :: NCFCONT1:35
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let x0 be Point of CNS1; ::_thesis: for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let z be Complex; ::_thesis: ( f is_continuous_in x0 implies z (#) f is_continuous_in x0 )
assume A1: f is_continuous_in x0 ; ::_thesis: z (#) f is_continuous_in x0
then x0 in dom f by Def5;
hence A2: x0 in dom (z (#) f) by VFUNCT_2:def_2; :: according to NCFCONT1:def_5 ::_thesis: for seq being sequence of CNS1 st rng seq c= dom (z (#) f) & seq is convergent & lim seq = x0 holds
( (z (#) f) /* seq is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* seq) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (z (#) f) & s1 is convergent & lim s1 = x0 implies ( (z (#) f) /* s1 is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* s1) ) )
assume that
A3: rng s1 c= dom (z (#) f) and
A4: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (z (#) f) /* s1 is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* s1) )
A5: rng s1 c= dom f by A3, VFUNCT_2:def_2;
then A6: f /. x0 = lim (f /* s1) by A1, A4, Def5;
A7: f /* s1 is convergent by A1, A4, A5, Def5;
then z * (f /* s1) is convergent by CLVECT_1:116;
hence (z (#) f) /* s1 is convergent by A5, Th26; ::_thesis: (z (#) f) /. x0 = lim ((z (#) f) /* s1)
thus (z (#) f) /. x0 = z * (f /. x0) by A2, VFUNCT_2:def_2
.= lim (z * (f /* s1)) by A7, A6, CLVECT_1:122
.= lim ((z (#) f) /* s1) by A5, Th26 ; ::_thesis: verum
end;
theorem Th36: :: NCFCONT1:36
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS
for r being Real st f is_continuous_in x0 holds
r (#) f is_continuous_in x0
let x0 be Point of CNS; ::_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 A1: f is_continuous_in x0 ; ::_thesis: r (#) f is_continuous_in x0
then x0 in dom f by Def6;
hence A2: x0 in dom (r (#) f) by VFUNCT_1:def_4; :: according to NCFCONT1:def_6 ::_thesis: for seq being sequence of CNS st rng seq c= dom (r (#) f) & seq is convergent & lim seq = x0 holds
( (r (#) f) /* seq is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* seq) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (r (#) f) & s1 is convergent & lim s1 = x0 implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) ) )
assume that
A3: rng s1 c= dom (r (#) f) and
A4: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) )
A5: rng s1 c= dom f by A3, VFUNCT_1:def_4;
then A6: f /. x0 = lim (f /* s1) by A1, A4, Def6;
A7: f /* s1 is convergent by A1, A4, A5, Def6;
then r * (f /* s1) is convergent by NORMSP_1:22;
hence (r (#) f) /* s1 is convergent by A5, Th27; ::_thesis: (r (#) f) /. x0 = lim ((r (#) f) /* s1)
thus (r (#) f) /. x0 = r * (f /. x0) by A2, VFUNCT_1:def_4
.= lim (r * (f /* s1)) by A7, A6, NORMSP_1:28
.= lim ((r (#) f) /* s1) by A5, Th27 ; ::_thesis: verum
end;
theorem Th37: :: NCFCONT1:37
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS
for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let x0 be Point of RNS; ::_thesis: for z being Complex st f is_continuous_in x0 holds
z (#) f is_continuous_in x0
let z be Complex; ::_thesis: ( f is_continuous_in x0 implies z (#) f is_continuous_in x0 )
assume A1: f is_continuous_in x0 ; ::_thesis: z (#) f is_continuous_in x0
then x0 in dom f by Def7;
hence A2: x0 in dom (z (#) f) by VFUNCT_2:def_2; :: according to NCFCONT1:def_7 ::_thesis: for seq being sequence of RNS st rng seq c= dom (z (#) f) & seq is convergent & lim seq = x0 holds
( (z (#) f) /* seq is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* seq) )
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (z (#) f) & s1 is convergent & lim s1 = x0 implies ( (z (#) f) /* s1 is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* s1) ) )
assume that
A3: rng s1 c= dom (z (#) f) and
A4: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (z (#) f) /* s1 is convergent & (z (#) f) /. x0 = lim ((z (#) f) /* s1) )
A5: rng s1 c= dom f by A3, VFUNCT_2:def_2;
then A6: f /. x0 = lim (f /* s1) by A1, A4, Def7;
A7: f /* s1 is convergent by A1, A4, A5, Def7;
then z * (f /* s1) is convergent by CLVECT_1:116;
hence (z (#) f) /* s1 is convergent by A5, Th28; ::_thesis: (z (#) f) /. x0 = lim ((z (#) f) /* s1)
thus (z (#) f) /. x0 = z * (f /. x0) by A2, VFUNCT_2:def_2
.= lim (z * (f /* s1)) by A7, A6, CLVECT_1:122
.= lim ((z (#) f) /* s1) by A5, Th28 ; ::_thesis: verum
end;
theorem :: NCFCONT1:38
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let x0 be Point of CNS1; ::_thesis: ( f is_continuous_in x0 implies ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) )
assume A1: f is_continuous_in x0 ; ::_thesis: ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
then A2: x0 in dom f by Def5;
now__::_thesis:_(_x0_in_dom_||.f.||_&_(_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_dom_||.f.||_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_||.f.||_/*_s1_is_convergent_&_||.f.||_/._x0_=_lim_(||.f.||_/*_s1)_)_)_)
thus A3: x0 in dom ||.f.|| by A2, NORMSP_0:def_3; ::_thesis: for s1 being sequence of CNS1 st rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 holds
( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 implies ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) ) )
assume that
A4: rng s1 c= dom ||.f.|| and
A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
A6: rng s1 c= dom f by A4, NORMSP_0:def_3;
then A7: f /. x0 = lim (f /* s1) by A1, A5, Def5;
A8: f /* s1 is convergent by A1, A5, A6, Def5;
then ||.(f /* s1).|| is convergent by CLVECT_1:117;
hence ||.f.|| /* s1 is convergent by A6, Th29; ::_thesis: ||.f.|| /. x0 = lim (||.f.|| /* s1)
thus ||.f.|| /. x0 = ||.f.|| . x0 by A3, PARTFUN1:def_6
.= ||.(f /. x0).|| by A3, NORMSP_0:def_3
.= lim ||.(f /* s1).|| by A8, A7, CLOPBAN1:40
.= lim (||.f.|| /* s1) by A6, Th29 ; ::_thesis: verum
end;
hence ||.f.|| is_continuous_in x0 by Def9; ::_thesis: - f is_continuous_in x0
- f = (- 1r) (#) f by VFUNCT_2:23;
hence - f is_continuous_in x0 by A1, Th35; ::_thesis: verum
end;
theorem :: NCFCONT1:39
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let x0 be Point of CNS; ::_thesis: ( f is_continuous_in x0 implies ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) )
assume A1: f is_continuous_in x0 ; ::_thesis: ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
then A2: x0 in dom f by Def6;
now__::_thesis:_(_x0_in_dom_||.f.||_&_(_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_||.f.||_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_||.f.||_/*_s1_is_convergent_&_||.f.||_/._x0_=_lim_(||.f.||_/*_s1)_)_)_)
thus A3: x0 in dom ||.f.|| by A2, NORMSP_0:def_3; ::_thesis: for s1 being sequence of CNS st rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 holds
( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 implies ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) ) )
assume that
A4: rng s1 c= dom ||.f.|| and
A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
A6: rng s1 c= dom f by A4, NORMSP_0:def_3;
then A7: f /. x0 = lim (f /* s1) by A1, A5, Def6;
A8: f /* s1 is convergent by A1, A5, A6, Def6;
then ||.(f /* s1).|| is convergent by NORMSP_1:23;
hence ||.f.|| /* s1 is convergent by A6, Th30; ::_thesis: ||.f.|| /. x0 = lim (||.f.|| /* s1)
thus ||.f.|| /. x0 = ||.f.|| . x0 by A3, PARTFUN1:def_6
.= ||.(f /. x0).|| by A3, NORMSP_0:def_3
.= lim ||.(f /* s1).|| by A8, A7, LOPBAN_1:41
.= lim (||.f.|| /* s1) by A6, Th30 ; ::_thesis: verum
end;
hence ||.f.|| is_continuous_in x0 by Def9; ::_thesis: - f is_continuous_in x0
- f = (- 1) (#) f by VFUNCT_1:23;
hence - f is_continuous_in x0 by A1, Th36; ::_thesis: verum
end;
theorem :: NCFCONT1:40
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS st f is_continuous_in x0 holds
( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
let x0 be Point of RNS; ::_thesis: ( f is_continuous_in x0 implies ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) )
assume A1: f is_continuous_in x0 ; ::_thesis: ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 )
then A2: x0 in dom f by Def7;
now__::_thesis:_(_x0_in_dom_||.f.||_&_(_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_||.f.||_&_s1_is_convergent_&_lim_s1_=_x0_holds_
(_||.f.||_/*_s1_is_convergent_&_||.f.||_/._x0_=_lim_(||.f.||_/*_s1)_)_)_)
thus A3: x0 in dom ||.f.|| by A2, NORMSP_0:def_3; ::_thesis: for s1 being sequence of RNS st rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 holds
( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 implies ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) ) )
assume that
A4: rng s1 c= dom ||.f.|| and
A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( ||.f.|| /* s1 is convergent & ||.f.|| /. x0 = lim (||.f.|| /* s1) )
A6: rng s1 c= dom f by A4, NORMSP_0:def_3;
then A7: f /. x0 = lim (f /* s1) by A1, A5, Def7;
A8: f /* s1 is convergent by A1, A5, A6, Def7;
then ||.(f /* s1).|| is convergent by CLVECT_1:117;
hence ||.f.|| /* s1 is convergent by A6, Th31; ::_thesis: ||.f.|| /. x0 = lim (||.f.|| /* s1)
thus ||.f.|| /. x0 = ||.f.|| . x0 by A3, PARTFUN1:def_6
.= ||.(f /. x0).|| by A3, NORMSP_0:def_3
.= lim ||.(f /* s1).|| by A8, A7, CLOPBAN1:40
.= lim (||.f.|| /* s1) by A6, Th31 ; ::_thesis: verum
end;
hence ||.f.|| is_continuous_in x0 by NFCONT_1:def_6; ::_thesis: - f is_continuous_in x0
- f = (- 1r) (#) f by VFUNCT_2:23;
hence - f is_continuous_in x0 by A1, Th37; ::_thesis: verum
end;
definition
let CNS1, CNS2 be ComplexNormSpace;
let f be PartFunc of CNS1,CNS2;
let X be set ;
predf is_continuous_on X means :Def11: :: NCFCONT1:def 11
( X c= dom f & ( for x0 being Point of CNS1 st x0 in X holds
f | X is_continuous_in x0 ) );
end;
:: deftheorem Def11 defines is_continuous_on NCFCONT1:def_11_:_
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS1 st x0 in X holds
f | X is_continuous_in x0 ) ) );
definition
let CNS be ComplexNormSpace;
let RNS be RealNormSpace;
let f be PartFunc of CNS,RNS;
let X be set ;
predf is_continuous_on X means :Def12: :: NCFCONT1:def 12
( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) );
end;
:: deftheorem Def12 defines is_continuous_on NCFCONT1:def_12_:_
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) ) );
definition
let RNS be RealNormSpace;
let CNS be ComplexNormSpace;
let g be PartFunc of RNS,CNS;
let X be set ;
predg is_continuous_on X means :Def13: :: NCFCONT1:def 13
( X c= dom g & ( for x0 being Point of RNS st x0 in X holds
g | X is_continuous_in x0 ) );
end;
:: deftheorem Def13 defines is_continuous_on NCFCONT1:def_13_:_
for RNS being RealNormSpace
for CNS being ComplexNormSpace
for g being PartFunc of RNS,CNS
for X being set holds
( g is_continuous_on X iff ( X c= dom g & ( for x0 being Point of RNS st x0 in X holds
g | X is_continuous_in x0 ) ) );
definition
let CNS be ComplexNormSpace;
let f be PartFunc of the carrier of CNS,COMPLEX;
let X be set ;
predf is_continuous_on X means :Def14: :: NCFCONT1:def 14
( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) );
end;
:: deftheorem Def14 defines is_continuous_on NCFCONT1:def_14_:_
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,COMPLEX
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) ) );
definition
let CNS be ComplexNormSpace;
let f be PartFunc of the carrier of CNS,REAL;
let X be set ;
predf is_continuous_on X means :Def15: :: NCFCONT1:def 15
( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) );
end;
:: deftheorem Def15 defines is_continuous_on NCFCONT1:def_15_:_
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0 ) ) );
definition
let RNS be RealNormSpace;
let f be PartFunc of the carrier of RNS,COMPLEX;
let X be set ;
predf is_continuous_on X means :Def16: :: NCFCONT1:def 16
( X c= dom f & ( for x0 being Point of RNS st x0 in X holds
f | X is_continuous_in x0 ) );
end;
:: deftheorem Def16 defines is_continuous_on NCFCONT1:def_16_:_
for RNS being RealNormSpace
for f being PartFunc of the carrier of RNS,COMPLEX
for X being set holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS st x0 in X holds
f | X is_continuous_in x0 ) ) );
theorem Th41: :: NCFCONT1:41
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) )
then A2: X c= dom f by Def11;
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )
assume that
A3: rng s1 c= X and
A4: s1 is convergent and
A5: lim s1 in X ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) )
A6: f | X is_continuous_in lim s1 by A1, A5, Def11;
A7: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
f /. (lim s1) = (f | X) /. (lim s1) by A5, A7, PARTFUN2:15
.= lim (f /* s1) by A3, A4, A7, A6, A9, Def5 ;
hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A7, A6, A9, Def5; ::_thesis: verum
end;
hence ( X c= dom f & ( for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, Def11; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for s1 being sequence of CNS1 st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on X
now__::_thesis:_for_x1_being_Point_of_CNS1_st_x1_in_X_holds_
f_|_X_is_continuous_in_x1
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
let x1 be Point of CNS1; ::_thesis: ( x1 in X implies f | X is_continuous_in x1 )
assume A13: x1 in X ; ::_thesis: f | X is_continuous_in x1
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_dom_(f_|_X)_&_s1_is_convergent_&_lim_s1_=_x1_holds_
(_(f_|_X)_/*_s1_is_convergent_&_(f_|_X)_/._x1_=_lim_((f_|_X)_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x1 implies ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) )
assume that
A14: rng s1 c= dom (f | X) and
A15: s1 is convergent and
A16: lim s1 = x1 ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
(f | X) /. (lim s1) = f /. (lim s1) by A13, A12, A16, PARTFUN2:15
.= lim ((f | X) /* s1) by A11, A13, A12, A14, A15, A16, A18 ;
hence ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) by A11, A13, A12, A14, A15, A16, A18; ::_thesis: verum
end;
hence f | X is_continuous_in x1 by A13, A12, Def5; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def11; ::_thesis: verum
end;
theorem Th42: :: NCFCONT1:42
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) )
then A2: X c= dom f by Def12;
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )
assume that
A3: rng s1 c= X and
A4: s1 is convergent and
A5: lim s1 in X ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) )
A6: f | X is_continuous_in lim s1 by A1, A5, Def12;
A7: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
f /. (lim s1) = (f | X) /. (lim s1) by A5, A7, PARTFUN2:15
.= lim (f /* s1) by A3, A4, A7, A6, A9, Def6 ;
hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A7, A6, A9, Def6; ::_thesis: verum
end;
hence ( X c= dom f & ( for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, Def12; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for s1 being sequence of CNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on X
now__::_thesis:_for_x1_being_Point_of_CNS_st_x1_in_X_holds_
f_|_X_is_continuous_in_x1
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
let x1 be Point of CNS; ::_thesis: ( x1 in X implies f | X is_continuous_in x1 )
assume A13: x1 in X ; ::_thesis: f | X is_continuous_in x1
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_dom_(f_|_X)_&_s1_is_convergent_&_lim_s1_=_x1_holds_
(_(f_|_X)_/*_s1_is_convergent_&_(f_|_X)_/._x1_=_lim_((f_|_X)_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x1 implies ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) )
assume that
A14: rng s1 c= dom (f | X) and
A15: s1 is convergent and
A16: lim s1 = x1 ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
(f | X) /. (lim s1) = f /. (lim s1) by A13, A12, A16, PARTFUN2:15
.= lim ((f | X) /* s1) by A11, A13, A12, A14, A15, A16, A18 ;
hence ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) by A11, A13, A12, A14, A15, A16, A18; ::_thesis: verum
end;
hence f | X is_continuous_in x1 by A13, A12, Def6; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def12; ::_thesis: verum
end;
theorem Th43: :: NCFCONT1:43
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) )
then A2: X c= dom f by Def13;
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) )
assume that
A3: rng s1 c= X and
A4: s1 is convergent and
A5: lim s1 in X ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) )
A6: f | X is_continuous_in lim s1 by A1, A5, Def13;
A7: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
f /. (lim s1) = (f | X) /. (lim s1) by A5, A7, PARTFUN2:15
.= lim (f /* s1) by A3, A4, A7, A6, A9, Def7 ;
hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A7, A6, A9, Def7; ::_thesis: verum
end;
hence ( X c= dom f & ( for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, Def13; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on X
now__::_thesis:_for_x1_being_Point_of_RNS_st_x1_in_X_holds_
f_|_X_is_continuous_in_x1
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
let x1 be Point of RNS; ::_thesis: ( x1 in X implies f | X is_continuous_in x1 )
assume A13: x1 in X ; ::_thesis: f | X is_continuous_in x1
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_dom_(f_|_X)_&_s1_is_convergent_&_lim_s1_=_x1_holds_
(_(f_|_X)_/*_s1_is_convergent_&_(f_|_X)_/._x1_=_lim_((f_|_X)_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x1 implies ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) )
assume that
A14: rng s1 c= dom (f | X) and
A15: s1 is convergent and
A16: lim s1 = x1 ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* 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 | X) /* s1) . n = (f | X) /. (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 | X) /* s1 = f /* s1 by FUNCT_2:63;
(f | X) /. (lim s1) = f /. (lim s1) by A13, A12, A16, PARTFUN2:15
.= lim ((f | X) /* s1) by A11, A13, A12, A14, A15, A16, A18 ;
hence ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) by A11, A13, A12, A14, A15, A16, A18; ::_thesis: verum
end;
hence f | X is_continuous_in x1 by A13, A12, Def7; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def13; ::_thesis: verum
end;
theorem Th44: :: NCFCONT1:44
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence X c= dom f by Def11; ::_thesis: for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
A2: X c= dom f by A1, Def11;
let x0 be Point of CNS1; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
f | X is_continuous_in x0 by A1, A3, Def11;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th8;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be Point of CNS1; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15
.= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A9, PARTFUN2:15 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of CNS1
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_CNS1_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS1; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of CNS1 st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
let x1 be Point of CNS1; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A12, A13, PARTFUN2:15
.= ||.((f /. x1) - (f /. x0)).|| by A16, PARTFUN2:15 ;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th8; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def11; ::_thesis: verum
end;
theorem Th45: :: NCFCONT1:45
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence X c= dom f by Def12; ::_thesis: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
A2: X c= dom f by A1, Def12;
let x0 be Point of CNS; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
f | X is_continuous_in x0 by A1, A3, Def12;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th9;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be Point of CNS; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15
.= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A9, PARTFUN2:15 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A12, A13, PARTFUN2:15
.= ||.((f /. x1) - (f /. x0)).|| by A16, PARTFUN2:15 ;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th9; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def12; ::_thesis: verum
end;
theorem Th46: :: NCFCONT1:46
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ) )
hence X c= dom f by Def13; ::_thesis: for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
A2: X c= dom f by A1, Def13;
let x0 be Point of RNS; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
f | X is_continuous_in x0 by A1, A3, Def13;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th10;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r
let x1 be Point of RNS; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies ||.((f /. x1) - (f /. x0)).|| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15
.= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A9, PARTFUN2:15 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_RNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of RNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
let x1 be Point of RNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r
||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A12, A13, PARTFUN2:15
.= ||.((f /. x1) - (f /. x0)).|| by A16, PARTFUN2:15 ;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th10; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def13; ::_thesis: verum
end;
theorem :: NCFCONT1:47
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of the carrier of CNS,COMPLEX; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
hence X c= dom f by Def14; ::_thesis: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
A2: X c= dom f by A1, Def14;
let x0 be Point of CNS; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
f | X is_continuous_in x0 by A1, A3, Def14;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r by A4, Th12;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r
let x1 be Point of CNS; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then |.((f /. x1) - (f /. x0)).| = |.(((f | X) /. x1) - (f /. x0)).| by A7, PARTFUN2:15
.= |.(((f | X) /. x1) - ((f | X) /. x0)).| by A3, A9, PARTFUN2:15 ;
hence |.((f /. x1) - (f /. x0)).| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies |.(((f | X) /. x1) - ((f | X) /. x0)).| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: |.(((f | X) /. x1) - ((f | X) /. x0)).| < r
|.(((f | X) /. x1) - ((f | X) /. x0)).| = |.(((f | X) /. x1) - (f /. x0)).| by A12, A13, PARTFUN2:15
.= |.((f /. x1) - (f /. x0)).| by A16, PARTFUN2:15 ;
hence |.(((f | X) /. x1) - ((f | X) /. x0)).| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th12; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def14; ::_thesis: verum
end;
theorem :: NCFCONT1:48
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ) )
hence X c= dom f by Def15; ::_thesis: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
A2: X c= dom f by A1, Def15;
let x0 be Point of CNS; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
f | X is_continuous_in x0 by A1, A3, Def15;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r by A4, Th11;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r
let x1 be Point of CNS; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies abs ((f /. x1) - (f /. x0)) < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: abs ((f /. x1) - (f /. x0)) < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then abs ((f /. x1) - (f /. x0)) = abs (((f | X) /. x1) - (f /. x0)) by A7, PARTFUN2:15
.= abs (((f | X) /. x1) - ((f | X) /. x0)) by A3, A9, PARTFUN2:15 ;
hence abs ((f /. x1) - (f /. x0)) < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of CNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of CNS st x1 in X & ||.(x1 - x0).|| < s holds
abs ((f /. x1) - (f /. x0)) < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < r
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies abs (((f | X) /. x1) - ((f | X) /. x0)) < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: abs (((f | X) /. x1) - ((f | X) /. x0)) < r
abs (((f | X) /. x1) - ((f | X) /. x0)) = abs (((f | X) /. x1) - (f /. x0)) by A12, A13, PARTFUN2:15
.= abs ((f /. x1) - (f /. x0)) by A16, PARTFUN2:15 ;
hence abs (((f | X) /. x1) - ((f | X) /. x0)) < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th11; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def15; ::_thesis: verum
end;
theorem :: NCFCONT1:49
for RNS being RealNormSpace
for X being set
for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
proof
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let X be set ; ::_thesis: for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
let f be PartFunc of the carrier of RNS,COMPLEX; ::_thesis: ( f is_continuous_on X iff ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) )
thus ( f is_continuous_on X implies ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) ) ) ::_thesis: ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: ( X c= dom f & ( for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ) )
hence X c= dom f by Def16; ::_thesis: for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
A2: X c= dom f by A1, Def16;
let x0 be Point of RNS; ::_thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
f | X is_continuous_in x0 by A1, A3, Def16;
then consider s being Real such that
A5: 0 < s and
A6: for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r by A4, Th13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A5; ::_thesis: for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r
let x1 be Point of RNS; ::_thesis: ( x1 in X & ||.(x1 - x0).|| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume that
A7: x1 in X and
A8: ||.(x1 - x0).|| < s ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r
A9: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A2, XBOOLE_1:28 ;
then |.((f /. x1) - (f /. x0)).| = |.(((f | X) /. x1) - (f /. x0)).| by A7, PARTFUN2:15
.= |.(((f | X) /. x1) - ((f | X) /. x0)).| by A3, A9, PARTFUN2:15 ;
hence |.((f /. x1) - (f /. x0)).| < r by A6, A9, A7, A8; ::_thesis: verum
end;
assume that
A10: X c= dom f and
A11: for x0 being Point of RNS
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_on X
A12: dom (f | X) = (dom f) /\ X by PARTFUN2:15
.= X by A10, XBOOLE_1:28 ;
now__::_thesis:_for_x0_being_Point_of_RNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of RNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A13: x0 in X ; ::_thesis: f | X is_continuous_in x0
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
proof
let r be Real; ::_thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) ) )
assume 0 < r ; ::_thesis: ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
then consider s being Real such that
A14: 0 < s and
A15: for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r by A11, A13;
take s ; ::_thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r ) )
thus 0 < s by A14; ::_thesis: for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < r
let x1 be Point of RNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < s implies |.(((f | X) /. x1) - ((f | X) /. x0)).| < r )
assume that
A16: x1 in dom (f | X) and
A17: ||.(x1 - x0).|| < s ; ::_thesis: |.(((f | X) /. x1) - ((f | X) /. x0)).| < r
|.(((f | X) /. x1) - ((f | X) /. x0)).| = |.(((f | X) /. x1) - (f /. x0)).| by A12, A13, PARTFUN2:15
.= |.((f /. x1) - (f /. x0)).| by A16, PARTFUN2:15 ;
hence |.(((f | X) /. x1) - ((f | X) /. x0)).| < r by A12, A15, A16, A17; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A12, A13, Th13; ::_thesis: verum
end;
hence f is_continuous_on X by A10, Def16; ::_thesis: verum
end;
theorem :: NCFCONT1:50
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def11;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def11;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def11;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in X holds
f | X is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def11;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem :: NCFCONT1:51
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def12;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def12;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def12;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def12;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem :: NCFCONT1:52
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def13;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def13;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def13;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in X holds
f | X is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def13;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem :: NCFCONT1:53
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of the carrier of CNS,COMPLEX; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def14;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_14 ::_thesis: for x0 being Point of CNS st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def14;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def14;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_14 ::_thesis: for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def14;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem Th54: :: NCFCONT1:54
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,REAL holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def15;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_15 ::_thesis: for x0 being Point of CNS st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def15;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def15;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_15 ::_thesis: for x0 being Point of CNS st x0 in X holds
f | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def15;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem :: NCFCONT1:55
for RNS being RealNormSpace
for X being set
for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
proof
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_continuous_on X iff f | X is_continuous_on X )
let f be PartFunc of the carrier of RNS,COMPLEX; ::_thesis: ( f is_continuous_on X iff f | X is_continuous_on X )
thus ( f is_continuous_on X implies f | X is_continuous_on X ) ::_thesis: ( f | X is_continuous_on X implies f is_continuous_on X )
proof
assume A1: f is_continuous_on X ; ::_thesis: f | X is_continuous_on X
then X c= dom f by Def16;
then X c= (dom f) /\ X by XBOOLE_1:28;
hence X c= dom (f | X) by RELAT_1:61; :: according to NCFCONT1:def_16 ::_thesis: for x0 being Point of RNS st x0 in X holds
(f | X) | X is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in X implies (f | X) | X is_continuous_in r )
assume r in X ; ::_thesis: (f | X) | X is_continuous_in r
then f | X is_continuous_in r by A1, Def16;
hence (f | X) | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
assume A2: f | X is_continuous_on X ; ::_thesis: f is_continuous_on X
then X c= dom (f | X) by Def16;
then ( (dom f) /\ X c= dom f & X c= (dom f) /\ X ) by RELAT_1:61, XBOOLE_1:17;
hence X c= dom f by XBOOLE_1:1; :: according to NCFCONT1:def_16 ::_thesis: for x0 being Point of RNS st x0 in X holds
f | X is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in X implies f | X is_continuous_in r )
assume r in X ; ::_thesis: f | X is_continuous_in r
then (f | X) | X is_continuous_in r by A2, Def16;
hence f | X is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
theorem Th56: :: NCFCONT1:56
for CNS1, CNS2 being ComplexNormSpace
for X, X1 being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X & X1 c= X implies f is_continuous_on X1 )
assume that
A1: f is_continuous_on X and
A2: X1 c= X ; ::_thesis: f is_continuous_on X1
X c= dom f by A1, Def11;
hence A3: X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in X1 holds
f | X1 is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in X1 implies f | X1 is_continuous_in r )
assume A4: r in X1 ; ::_thesis: f | X1 is_continuous_in r
then A5: f | X is_continuous_in r by A1, A2, Def11;
thus f | X1 is_continuous_in r ::_thesis: verum
proof
(dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
then dom (f | X1) c= (dom f) /\ X by RELAT_1:61;
then A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
r in (dom f) /\ X1 by A3, A4, XBOOLE_0:def_4;
hence A7: r in dom (f | X1) by RELAT_1:61; :: according to NCFCONT1:def_5 ::_thesis: for seq being sequence of CNS1 st rng seq c= dom (f | X1) & seq is convergent & lim seq = r holds
( (f | X1) /* seq is convergent & (f | X1) /. r = lim ((f | X1) /* seq) )
then A8: (f | X) /. r = f /. r by A6, PARTFUN2:15
.= (f | X1) /. r by A7, PARTFUN2:15 ;
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = r implies ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) )
assume that
A9: rng s1 c= dom (f | X1) and
A10: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) )
A11: rng s1 c= dom (f | X) by A9, A6, XBOOLE_1:1;
A12: now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_((f_|_X1)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = ((f | X1) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A13: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
.= f /. (s1 . n) by A11, A13, PARTFUN2:15
.= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
.= ((f | X1) /* s1) . n by A9, FUNCT_2:109 ; ::_thesis: verum
end;
( (f | X) /* s1 is convergent & (f | X) /. r = lim ((f | X) /* s1) ) by A5, A10, A11, Def5;
hence ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem Th57: :: NCFCONT1:57
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of CNS,RNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of CNS,RNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of CNS,RNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X & X1 c= X implies f is_continuous_on X1 )
assume that
A1: f is_continuous_on X and
A2: X1 c= X ; ::_thesis: f is_continuous_on X1
X c= dom f by A1, Def12;
hence A3: X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in X1 holds
f | X1 is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X1 implies f | X1 is_continuous_in r )
assume A4: r in X1 ; ::_thesis: f | X1 is_continuous_in r
then A5: f | X is_continuous_in r by A1, A2, Def12;
thus f | X1 is_continuous_in r ::_thesis: verum
proof
(dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
then dom (f | X1) c= (dom f) /\ X by RELAT_1:61;
then A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
r in (dom f) /\ X1 by A3, A4, XBOOLE_0:def_4;
hence A7: r in dom (f | X1) by RELAT_1:61; :: according to NCFCONT1:def_6 ::_thesis: for seq being sequence of CNS st rng seq c= dom (f | X1) & seq is convergent & lim seq = r holds
( (f | X1) /* seq is convergent & (f | X1) /. r = lim ((f | X1) /* seq) )
then A8: (f | X) /. r = f /. r by A6, PARTFUN2:15
.= (f | X1) /. r by A7, PARTFUN2:15 ;
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = r implies ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) )
assume that
A9: rng s1 c= dom (f | X1) and
A10: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) )
A11: rng s1 c= dom (f | X) by A9, A6, XBOOLE_1:1;
A12: now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_((f_|_X1)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = ((f | X1) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A13: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
.= f /. (s1 . n) by A11, A13, PARTFUN2:15
.= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
.= ((f | X1) /* s1) . n by A9, FUNCT_2:109 ; ::_thesis: verum
end;
( (f | X) /* s1 is convergent & (f | X) /. r = lim ((f | X) /* s1) ) by A5, A10, A11, Def6;
hence ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem Th58: :: NCFCONT1:58
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of RNS,CNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of RNS,CNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of RNS,CNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X & X1 c= X implies f is_continuous_on X1 )
assume that
A1: f is_continuous_on X and
A2: X1 c= X ; ::_thesis: f is_continuous_on X1
X c= dom f by A1, Def13;
hence A3: X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in X1 holds
f | X1 is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in X1 implies f | X1 is_continuous_in r )
assume A4: r in X1 ; ::_thesis: f | X1 is_continuous_in r
then A5: f | X is_continuous_in r by A1, A2, Def13;
thus f | X1 is_continuous_in r ::_thesis: verum
proof
(dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
then dom (f | X1) c= (dom f) /\ X by RELAT_1:61;
then A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
r in (dom f) /\ X1 by A3, A4, XBOOLE_0:def_4;
hence A7: r in dom (f | X1) by RELAT_1:61; :: according to NCFCONT1:def_7 ::_thesis: for seq being sequence of RNS st rng seq c= dom (f | X1) & seq is convergent & lim seq = r holds
( (f | X1) /* seq is convergent & (f | X1) /. r = lim ((f | X1) /* seq) )
then A8: (f | X) /. r = f /. r by A6, PARTFUN2:15
.= (f | X1) /. r by A7, PARTFUN2:15 ;
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = r implies ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) )
assume that
A9: rng s1 c= dom (f | X1) and
A10: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) )
A11: rng s1 c= dom (f | X) by A9, A6, XBOOLE_1:1;
A12: now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_((f_|_X1)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = ((f | X1) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A13: s1 . n in rng s1 by FUNCT_1:3;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
.= f /. (s1 . n) by A11, A13, PARTFUN2:15
.= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
.= ((f | X1) /* s1) . n by A9, FUNCT_2:109 ; ::_thesis: verum
end;
( (f | X) /* s1 is convergent & (f | X) /. r = lim ((f | X) /* s1) ) by A5, A10, A11, Def7;
hence ( (f | X1) /* s1 is convergent & (f | X1) /. r = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem :: NCFCONT1:59
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st x0 in dom f holds
f is_continuous_on {x0}
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for x0 being Point of CNS1 st x0 in dom f holds
f is_continuous_on {x0}
let f be PartFunc of CNS1,CNS2; ::_thesis: for x0 being Point of CNS1 st x0 in dom f holds
f is_continuous_on {x0}
let x0 be Point of CNS1; ::_thesis: ( x0 in dom f implies f is_continuous_on {x0} )
assume A1: x0 in dom f ; ::_thesis: f is_continuous_on {x0}
thus {x0} c= dom f :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in {x0} holds
f | {x0} is_continuous_in x0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {x0} or x in dom f )
assume x in {x0} ; ::_thesis: x in dom f
hence x in dom f by A1, TARSKI:def_1; ::_thesis: verum
end;
let p be Point of CNS1; ::_thesis: ( p in {x0} implies f | {x0} is_continuous_in p )
assume A2: p in {x0} ; ::_thesis: f | {x0} is_continuous_in p
thus f | {x0} is_continuous_in p ::_thesis: verum
proof
p in dom f by A1, A2, TARSKI:def_1;
then p in (dom f) /\ {x0} by A2, XBOOLE_0:def_4;
hence p in dom (f | {x0}) by RELAT_1:61; :: according to NCFCONT1:def_5 ::_thesis: for seq being sequence of CNS1 st rng seq c= dom (f | {x0}) & seq is convergent & lim seq = p holds
( (f | {x0}) /* seq is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* seq) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (f | {x0}) & s1 is convergent & lim s1 = p implies ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) ) )
assume that
A3: rng s1 c= dom (f | {x0}) and
s1 is convergent and
lim s1 = p ; ::_thesis: ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) )
A4: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17;
rng s1 c= (dom f) /\ {x0} by A3, RELAT_1:61;
then A5: rng s1 c= {x0} by A4, XBOOLE_1:1;
A6: now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_=_x0
let n be Element of NAT ; ::_thesis: s1 . n = x0
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
hence s1 . n = x0 by A5, TARSKI:def_1; ::_thesis: verum
end;
A7: p = x0 by A2, TARSKI:def_1;
A8: now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
||.((((f_|_{x0})_/*_s1)_._m)_-_((f_|_{x0})_/._p)).||_<_g
let g be Real; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume A9: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume n <= m ; ::_thesis: ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| = ||.(((f | {x0}) /. (s1 . m)) - ((f | {x0}) /. x0)).|| by A7, A3, FUNCT_2:109
.= ||.(((f | {x0}) /. x0) - ((f | {x0}) /. x0)).|| by A6
.= ||.(0. CNS2).|| by RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g by A9; ::_thesis: verum
end;
hence (f | {x0}) /* s1 is convergent by CLVECT_1:def_15; ::_thesis: (f | {x0}) /. p = lim ((f | {x0}) /* s1)
hence (f | {x0}) /. p = lim ((f | {x0}) /* s1) by A8, CLVECT_1:def_16; ::_thesis: verum
end;
end;
theorem :: NCFCONT1:60
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f holds
f is_continuous_on {x0}
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f holds
f is_continuous_on {x0}
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for x0 being Point of CNS st x0 in dom f holds
f is_continuous_on {x0}
let f be PartFunc of CNS,RNS; ::_thesis: for x0 being Point of CNS st x0 in dom f holds
f is_continuous_on {x0}
let x0 be Point of CNS; ::_thesis: ( x0 in dom f implies f is_continuous_on {x0} )
assume A1: x0 in dom f ; ::_thesis: f is_continuous_on {x0}
thus {x0} c= dom f :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in {x0} holds
f | {x0} is_continuous_in x0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {x0} or x in dom f )
assume x in {x0} ; ::_thesis: x in dom f
hence x in dom f by A1, TARSKI:def_1; ::_thesis: verum
end;
let p be Point of CNS; ::_thesis: ( p in {x0} implies f | {x0} is_continuous_in p )
assume A2: p in {x0} ; ::_thesis: f | {x0} is_continuous_in p
thus f | {x0} is_continuous_in p ::_thesis: verum
proof
p in dom f by A1, A2, TARSKI:def_1;
then p in (dom f) /\ {x0} by A2, XBOOLE_0:def_4;
hence p in dom (f | {x0}) by RELAT_1:61; :: according to NCFCONT1:def_6 ::_thesis: for seq being sequence of CNS st rng seq c= dom (f | {x0}) & seq is convergent & lim seq = p holds
( (f | {x0}) /* seq is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* seq) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (f | {x0}) & s1 is convergent & lim s1 = p implies ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) ) )
assume that
A3: rng s1 c= dom (f | {x0}) and
s1 is convergent and
lim s1 = p ; ::_thesis: ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) )
A4: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17;
rng s1 c= (dom f) /\ {x0} by A3, RELAT_1:61;
then A5: rng s1 c= {x0} by A4, XBOOLE_1:1;
A6: now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_=_x0
let n be Element of NAT ; ::_thesis: s1 . n = x0
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
hence s1 . n = x0 by A5, TARSKI:def_1; ::_thesis: verum
end;
A7: p = x0 by A2, TARSKI:def_1;
A8: now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
||.((((f_|_{x0})_/*_s1)_._m)_-_((f_|_{x0})_/._p)).||_<_g
let g be Real; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume A9: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume n <= m ; ::_thesis: ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| = ||.(((f | {x0}) /. (s1 . m)) - ((f | {x0}) /. x0)).|| by A7, A3, FUNCT_2:109
.= ||.(((f | {x0}) /. x0) - ((f | {x0}) /. x0)).|| by A6
.= ||.(0. RNS).|| by RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g by A9; ::_thesis: verum
end;
hence (f | {x0}) /* s1 is convergent by NORMSP_1:def_6; ::_thesis: (f | {x0}) /. p = lim ((f | {x0}) /* s1)
hence (f | {x0}) /. p = lim ((f | {x0}) /* s1) by A8, NORMSP_1:def_7; ::_thesis: verum
end;
end;
theorem :: NCFCONT1:61
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f holds
f is_continuous_on {x0}
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f holds
f is_continuous_on {x0}
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for x0 being Point of RNS st x0 in dom f holds
f is_continuous_on {x0}
let f be PartFunc of RNS,CNS; ::_thesis: for x0 being Point of RNS st x0 in dom f holds
f is_continuous_on {x0}
let x0 be Point of RNS; ::_thesis: ( x0 in dom f implies f is_continuous_on {x0} )
assume A1: x0 in dom f ; ::_thesis: f is_continuous_on {x0}
thus {x0} c= dom f :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in {x0} holds
f | {x0} is_continuous_in x0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {x0} or x in dom f )
assume x in {x0} ; ::_thesis: x in dom f
hence x in dom f by A1, TARSKI:def_1; ::_thesis: verum
end;
let p be Point of RNS; ::_thesis: ( p in {x0} implies f | {x0} is_continuous_in p )
assume A2: p in {x0} ; ::_thesis: f | {x0} is_continuous_in p
thus f | {x0} is_continuous_in p ::_thesis: verum
proof
p in dom f by A1, A2, TARSKI:def_1;
then p in (dom f) /\ {x0} by A2, XBOOLE_0:def_4;
hence p in dom (f | {x0}) by RELAT_1:61; :: according to NCFCONT1:def_7 ::_thesis: for seq being sequence of RNS st rng seq c= dom (f | {x0}) & seq is convergent & lim seq = p holds
( (f | {x0}) /* seq is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* seq) )
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= dom (f | {x0}) & s1 is convergent & lim s1 = p implies ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) ) )
assume that
A3: rng s1 c= dom (f | {x0}) and
s1 is convergent and
lim s1 = p ; ::_thesis: ( (f | {x0}) /* s1 is convergent & (f | {x0}) /. p = lim ((f | {x0}) /* s1) )
A4: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17;
rng s1 c= (dom f) /\ {x0} by A3, RELAT_1:61;
then A5: rng s1 c= {x0} by A4, XBOOLE_1:1;
A6: now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_=_x0
let n be Element of NAT ; ::_thesis: s1 . n = x0
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
hence s1 . n = x0 by A5, TARSKI:def_1; ::_thesis: verum
end;
A7: p = x0 by A2, TARSKI:def_1;
A8: now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
||.((((f_|_{x0})_/*_s1)_._m)_-_((f_|_{x0})_/._p)).||_<_g
let g be Real; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume A9: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g )
assume n <= m ; ::_thesis: ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g
||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| = ||.(((f | {x0}) /. (s1 . m)) - ((f | {x0}) /. x0)).|| by A7, A3, FUNCT_2:109
.= ||.(((f | {x0}) /. x0) - ((f | {x0}) /. x0)).|| by A6
.= ||.(0. CNS).|| by RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g by A9; ::_thesis: verum
end;
hence (f | {x0}) /* s1 is convergent by CLVECT_1:def_15; ::_thesis: (f | {x0}) /. p = lim ((f | {x0}) /* s1)
hence (f | {x0}) /. p = lim ((f | {x0}) /* s1) by A8, CLVECT_1:def_16; ::_thesis: verum
end;
end;
theorem Th62: :: NCFCONT1:62
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let X be set ; ::_thesis: for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let f1, f2 be PartFunc of CNS1,CNS2; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X implies ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X ) )
assume A1: ( f1 is_continuous_on X & f2 is_continuous_on X ) ; ::_thesis: ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
then ( X c= dom f1 & X c= dom f2 ) by Th41;
then A2: X c= (dom f1) /\ (dom f2) by XBOOLE_1:19;
then A3: X c= dom (f1 + f2) by VFUNCT_1:def_1;
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) )
A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A4, A5, A6, Th41;
then A8: (f1 /* s1) + (f2 /* s1) is convergent by CLVECT_1:113;
A9: rng s1 c= (dom f1) /\ (dom f2) by A2, A4, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A4, A5, A6, Th41;
then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A3, A6, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, CLVECT_1:119
.= lim ((f1 + f2) /* s1) by A9, Th23 ;
hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th23; ::_thesis: verum
end;
hence f1 + f2 is_continuous_on X by A3, Th41; ::_thesis: f1 - f2 is_continuous_on X
A10: X c= dom (f1 - f2) by A2, VFUNCT_1:def_2;
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) )
assume that
A11: rng s1 c= X and
A12: s1 is convergent and
A13: lim s1 in X ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) )
A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A11, A12, A13, Th41;
then A15: (f1 /* s1) - (f2 /* s1) is convergent by CLVECT_1:114;
A16: rng s1 c= (dom f1) /\ (dom f2) by A2, A11, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A11, A12, A13, Th41;
then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A10, A13, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A14, CLVECT_1:120
.= lim ((f1 - f2) /* s1) by A16, Th23 ;
hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th23; ::_thesis: verum
end;
hence f1 - f2 is_continuous_on X by A10, Th41; ::_thesis: verum
end;
theorem Th63: :: NCFCONT1:63
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let X be set ; ::_thesis: for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let f1, f2 be PartFunc of CNS,RNS; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X implies ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X ) )
assume A1: ( f1 is_continuous_on X & f2 is_continuous_on X ) ; ::_thesis: ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
then ( X c= dom f1 & X c= dom f2 ) by Th42;
then A2: X c= (dom f1) /\ (dom f2) by XBOOLE_1:19;
then A3: X c= dom (f1 + f2) by VFUNCT_1:def_1;
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) )
A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A4, A5, A6, Th42;
then A8: (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19;
A9: rng s1 c= (dom f1) /\ (dom f2) by A2, A4, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A4, A5, A6, Th42;
then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A3, A6, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A9, Th24 ;
hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th24; ::_thesis: verum
end;
hence f1 + f2 is_continuous_on X by A3, Th42; ::_thesis: f1 - f2 is_continuous_on X
A10: X c= dom (f1 - f2) by A2, VFUNCT_1:def_2;
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) )
assume that
A11: rng s1 c= X and
A12: s1 is convergent and
A13: lim s1 in X ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) )
A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A11, A12, A13, Th42;
then A15: (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20;
A16: rng s1 c= (dom f1) /\ (dom f2) by A2, A11, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A11, A12, A13, Th42;
then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A10, A13, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A14, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A16, Th24 ;
hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th24; ::_thesis: verum
end;
hence f1 - f2 is_continuous_on X by A10, Th42; ::_thesis: verum
end;
theorem Th64: :: NCFCONT1:64
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let X be set ; ::_thesis: for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X holds
( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
let f1, f2 be PartFunc of RNS,CNS; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X implies ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X ) )
assume A1: ( f1 is_continuous_on X & f2 is_continuous_on X ) ; ::_thesis: ( f1 + f2 is_continuous_on X & f1 - f2 is_continuous_on X )
then ( X c= dom f1 & X c= dom f2 ) by Th43;
then A2: X c= (dom f1) /\ (dom f2) by XBOOLE_1:19;
then A3: X c= dom (f1 + f2) by VFUNCT_1:def_1;
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) )
A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A4, A5, A6, Th43;
then A8: (f1 /* s1) + (f2 /* s1) is convergent by CLVECT_1:113;
A9: rng s1 c= (dom f1) /\ (dom f2) by A2, A4, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A4, A5, A6, Th43;
then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A3, A6, VFUNCT_1:def_1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, CLVECT_1:119
.= lim ((f1 + f2) /* s1) by A9, Th25 ;
hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th25; ::_thesis: verum
end;
hence f1 + f2 is_continuous_on X by A3, Th43; ::_thesis: f1 - f2 is_continuous_on X
A10: X c= dom (f1 - f2) by A2, VFUNCT_1:def_2;
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) )
assume that
A11: rng s1 c= X and
A12: s1 is convergent and
A13: lim s1 in X ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) )
A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A11, A12, A13, Th43;
then A15: (f1 /* s1) - (f2 /* s1) is convergent by CLVECT_1:114;
A16: rng s1 c= (dom f1) /\ (dom f2) by A2, A11, XBOOLE_1:1;
( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A11, A12, A13, Th43;
then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A10, A13, VFUNCT_1:def_2
.= lim ((f1 /* s1) - (f2 /* s1)) by A14, CLVECT_1:120
.= lim ((f1 - f2) /* s1) by A16, Th25 ;
hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th25; ::_thesis: verum
end;
hence f1 - f2 is_continuous_on X by A10, Th43; ::_thesis: verum
end;
theorem :: NCFCONT1:65
for CNS1, CNS2 being ComplexNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let f1, f2 be PartFunc of CNS1,CNS2; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X1 implies ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) )
assume ( f1 is_continuous_on X & f2 is_continuous_on X1 ) ; ::_thesis: ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
then ( f1 is_continuous_on X /\ X1 & f2 is_continuous_on X /\ X1 ) by Th56, XBOOLE_1:17;
hence ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) by Th62; ::_thesis: verum
end;
theorem :: NCFCONT1:66
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS,RNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let f1, f2 be PartFunc of CNS,RNS; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X1 implies ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) )
assume ( f1 is_continuous_on X & f2 is_continuous_on X1 ) ; ::_thesis: ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
then ( f1 is_continuous_on X /\ X1 & f2 is_continuous_on X /\ X1 ) by Th57, XBOOLE_1:17;
hence ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) by Th63; ::_thesis: verum
end;
theorem :: NCFCONT1:67
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of RNS,CNS st f1 is_continuous_on X & f2 is_continuous_on X1 holds
( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
let f1, f2 be PartFunc of RNS,CNS; ::_thesis: ( f1 is_continuous_on X & f2 is_continuous_on X1 implies ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) )
assume ( f1 is_continuous_on X & f2 is_continuous_on X1 ) ; ::_thesis: ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 )
then ( f1 is_continuous_on X /\ X1 & f2 is_continuous_on X /\ X1 ) by Th58, XBOOLE_1:17;
hence ( f1 + f2 is_continuous_on X /\ X1 & f1 - f2 is_continuous_on X /\ X1 ) by Th64; ::_thesis: verum
end;
theorem Th68: :: NCFCONT1:68
for z being Complex
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
z (#) f is_continuous_on X
proof
let z be Complex; ::_thesis: for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
z (#) f is_continuous_on X
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
z (#) f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
z (#) f is_continuous_on X
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X implies z (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; ::_thesis: z (#) f is_continuous_on X
then A2: X c= dom f by Th41;
then A3: X c= dom (z (#) f) by VFUNCT_2:def_2;
now__::_thesis:_for_s1_being_sequence_of_CNS1_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(z_(#)_f)_/*_s1_is_convergent_&_(z_(#)_f)_/._(lim_s1)_=_lim_((z_(#)_f)_/*_s1)_)
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A4, A5, A6, Th41;
then A8: z * (f /* s1) is convergent by CLVECT_1:116;
f /. (lim s1) = lim (f /* s1) by A1, A4, A5, A6, Th41;
then (z (#) f) /. (lim s1) = z * (lim (f /* s1)) by A3, A6, VFUNCT_2:def_2
.= lim (z * (f /* s1)) by A7, CLVECT_1:122
.= lim ((z (#) f) /* s1) by A2, A4, Th26, XBOOLE_1:1 ;
hence ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) by A2, A4, A8, Th26, XBOOLE_1:1; ::_thesis: verum
end;
hence z (#) f is_continuous_on X by A3, Th41; ::_thesis: verum
end;
theorem Th69: :: NCFCONT1:69
for r being Real
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
proof
let r be Real; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_continuous_on X holds
r (#) f is_continuous_on X
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X implies r (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; ::_thesis: r (#) f is_continuous_on X
then A2: X c= dom f by Th42;
then A3: X c= dom (r (#) f) by VFUNCT_1:def_4;
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(r_(#)_f)_/*_s1_is_convergent_&_(r_(#)_f)_/._(lim_s1)_=_lim_((r_(#)_f)_/*_s1)_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A4, A5, A6, Th42;
then A8: r * (f /* s1) is convergent by NORMSP_1:22;
f /. (lim s1) = lim (f /* s1) by A1, A4, A5, A6, Th42;
then (r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A3, A6, VFUNCT_1:def_4
.= lim (r * (f /* s1)) by A7, NORMSP_1:28
.= lim ((r (#) f) /* s1) by A2, A4, Th27, XBOOLE_1:1 ;
hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A2, A4, A8, Th27, XBOOLE_1:1; ::_thesis: verum
end;
hence r (#) f is_continuous_on X by A3, Th42; ::_thesis: verum
end;
theorem Th70: :: NCFCONT1:70
for z being Complex
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
proof
let z be Complex; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X implies z (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; ::_thesis: z (#) f is_continuous_on X
then A2: X c= dom f by Th43;
then A3: X c= dom (z (#) f) by VFUNCT_2:def_2;
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_
(_(z_(#)_f)_/*_s1_is_convergent_&_(z_(#)_f)_/._(lim_s1)_=_lim_((z_(#)_f)_/*_s1)_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; ::_thesis: ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A4, A5, A6, Th43;
then A8: z * (f /* s1) is convergent by CLVECT_1:116;
f /. (lim s1) = lim (f /* s1) by A1, A4, A5, A6, Th43;
then (z (#) f) /. (lim s1) = z * (lim (f /* s1)) by A3, A6, VFUNCT_2:def_2
.= lim (z * (f /* s1)) by A7, CLVECT_1:122
.= lim ((z (#) f) /* s1) by A2, A4, Th28, XBOOLE_1:1 ;
hence ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) by A2, A4, A8, Th28, XBOOLE_1:1; ::_thesis: verum
end;
hence z (#) f is_continuous_on X by A3, Th43; ::_thesis: verum
end;
theorem Th71: :: NCFCONT1:71
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_continuous_on X implies ( ||.f.|| is_continuous_on X & - f is_continuous_on X ) )
assume A1: f is_continuous_on X ; ::_thesis: ( ||.f.|| is_continuous_on X & - f is_continuous_on X )
thus ||.f.|| is_continuous_on X ::_thesis: - f is_continuous_on X
proof
A2: X c= dom f by A1, Def11;
hence A3: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NCFCONT1:def_15 ::_thesis: for x0 being Point of CNS1 st x0 in X holds
||.f.|| | X is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in X implies ||.f.|| | X is_continuous_in r )
assume A4: r in X ; ::_thesis: ||.f.|| | X is_continuous_in r
then A5: f | X is_continuous_in r by A1, Def11;
thus ||.f.|| | X is_continuous_in r ::_thesis: verum
proof
A6: r in (dom ||.f.||) /\ X by A3, A4, XBOOLE_0:def_4;
hence r in dom (||.f.|| | X) by RELAT_1:61; :: according to NCFCONT1:def_9 ::_thesis: for seq being sequence of CNS1 st rng seq c= dom (||.f.|| | X) & seq is convergent & lim seq = r holds
( (||.f.|| | X) /* seq is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* seq) )
let s1 be sequence of CNS1; ::_thesis: ( rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r implies ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) ) )
assume that
A7: rng s1 c= dom (||.f.|| | X) and
A8: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61;
then rng s1 c= (dom f) /\ X by NORMSP_0:def_3;
then A9: rng s1 c= dom (f | X) by RELAT_1:61;
then A10: (f | X) /. r = lim ((f | X) /* s1) by A5, A8, Def5;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.((f_|_X)_/*_s1).||_._n_=_((||.f.||_|_X)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A11: s1 . n in rng s1 by FUNCT_1:3;
then s1 . n in dom (f | X) by A9;
then A12: s1 . n in (dom f) /\ X by RELAT_1:61;
then A13: s1 . n in X by XBOOLE_0:def_4;
s1 . n in dom f by A12, XBOOLE_0:def_4;
then A14: s1 . n in dom ||.f.|| by NORMSP_0:def_3;
thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def_4
.= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109
.= ||.(f /. (s1 . n)).|| by A9, A11, PARTFUN2:15
.= ||.f.|| . (s1 . n) by A14, NORMSP_0:def_3
.= (||.f.|| | X) . (s1 . n) by A13, FUNCT_1:49
.= (||.f.|| | X) /. (s1 . n) by A7, A11, PARTFUN1:def_6
.= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:109 ; ::_thesis: verum
end;
then A15: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63;
A16: (f | X) /* s1 is convergent by A5, A8, A9, Def5;
hence (||.f.|| | X) /* s1 is convergent by A15, CLVECT_1:117; ::_thesis: (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1)
||.((f | X) /. r).|| = ||.(f /. r).|| by A2, A4, PARTFUN2:17
.= ||.f.|| . r by A3, A4, NORMSP_0:def_3
.= ||.f.|| /. r by A3, A4, PARTFUN1:def_6
.= (||.f.|| | X) /. r by A6, PARTFUN2:16 ;
hence (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) by A16, A10, A15, CLOPBAN1:19; ::_thesis: verum
end;
end;
(- 1r) (#) f is_continuous_on X by A1, Th68;
hence - f is_continuous_on X by VFUNCT_2:23; ::_thesis: verum
end;
theorem Th72: :: NCFCONT1:72
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_continuous_on X implies ( ||.f.|| is_continuous_on X & - f is_continuous_on X ) )
assume A1: f is_continuous_on X ; ::_thesis: ( ||.f.|| is_continuous_on X & - f is_continuous_on X )
thus ||.f.|| is_continuous_on X ::_thesis: - f is_continuous_on X
proof
A2: X c= dom f by A1, Def12;
hence A3: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NCFCONT1:def_15 ::_thesis: for x0 being Point of CNS st x0 in X holds
||.f.|| | X is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in X implies ||.f.|| | X is_continuous_in r )
assume A4: r in X ; ::_thesis: ||.f.|| | X is_continuous_in r
then A5: f | X is_continuous_in r by A1, Def12;
thus ||.f.|| | X is_continuous_in r ::_thesis: verum
proof
A6: r in (dom ||.f.||) /\ X by A3, A4, XBOOLE_0:def_4;
hence r in dom (||.f.|| | X) by RELAT_1:61; :: according to NCFCONT1:def_9 ::_thesis: for seq being sequence of CNS st rng seq c= dom (||.f.|| | X) & seq is convergent & lim seq = r holds
( (||.f.|| | X) /* seq is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* seq) )
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= dom (||.f.|| | X) & s1 is convergent & lim s1 = r implies ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) ) )
assume that
A7: rng s1 c= dom (||.f.|| | X) and
A8: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61;
then rng s1 c= (dom f) /\ X by NORMSP_0:def_3;
then A9: rng s1 c= dom (f | X) by RELAT_1:61;
then A10: (f | X) /. r = lim ((f | X) /* s1) by A5, A8, Def6;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.((f_|_X)_/*_s1).||_._n_=_((||.f.||_|_X)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A11: s1 . n in rng s1 by FUNCT_1:3;
then s1 . n in dom (f | X) by A9;
then A12: s1 . n in (dom f) /\ X by RELAT_1:61;
then A13: s1 . n in X by XBOOLE_0:def_4;
s1 . n in dom f by A12, XBOOLE_0:def_4;
then A14: s1 . n in dom ||.f.|| by NORMSP_0:def_3;
thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def_4
.= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109
.= ||.(f /. (s1 . n)).|| by A9, A11, PARTFUN2:15
.= ||.f.|| . (s1 . n) by A14, NORMSP_0:def_3
.= (||.f.|| | X) . (s1 . n) by A13, FUNCT_1:49
.= (||.f.|| | X) /. (s1 . n) by A7, A11, PARTFUN1:def_6
.= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:109 ; ::_thesis: verum
end;
then A15: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63;
A16: (f | X) /* s1 is convergent by A5, A8, A9, Def6;
hence (||.f.|| | X) /* s1 is convergent by A15, NORMSP_1:23; ::_thesis: (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1)
||.((f | X) /. r).|| = ||.(f /. r).|| by A2, A4, PARTFUN2:17
.= ||.f.|| . r by A3, A4, NORMSP_0:def_3
.= ||.f.|| /. r by A3, A4, PARTFUN1:def_6
.= (||.f.|| | X) /. r by A6, PARTFUN2:16 ;
hence (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) by A16, A10, A15, LOPBAN_1:20; ::_thesis: verum
end;
end;
(- 1) (#) f is_continuous_on X by A1, Th69;
hence - f is_continuous_on X by VFUNCT_1:23; ::_thesis: verum
end;
theorem Th73: :: NCFCONT1:73
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_continuous_on X holds
( ||.f.|| is_continuous_on X & - f is_continuous_on X )
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_continuous_on X implies ( ||.f.|| is_continuous_on X & - f is_continuous_on X ) )
assume A1: f is_continuous_on X ; ::_thesis: ( ||.f.|| is_continuous_on X & - f is_continuous_on X )
thus ||.f.|| is_continuous_on X ::_thesis: - f is_continuous_on X
proof
A2: X c= dom f by A1, Def13;
hence A3: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NFCONT_1:def_8 ::_thesis: for b1 being Element of the carrier of RNS holds
( not b1 in X or ||.f.|| | X is_continuous_in b1 )
let r be Point of RNS; ::_thesis: ( not r in X or ||.f.|| | X is_continuous_in r )
assume A4: r in X ; ::_thesis: ||.f.|| | X is_continuous_in r
then A5: f | X is_continuous_in r by A1, Def13;
thus ||.f.|| | X is_continuous_in r ::_thesis: verum
proof
A6: r in (dom ||.f.||) /\ X by A3, A4, XBOOLE_0:def_4;
hence r in dom (||.f.|| | X) by RELAT_1:61; :: according to NFCONT_1:def_6 ::_thesis: for b1 being Element of K6(K7(NAT, the carrier of RNS)) holds
( not rng b1 c= dom (||.f.|| | X) or not b1 is convergent or not lim b1 = r or ( (||.f.|| | X) /* b1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* b1) ) )
let s1 be sequence of RNS; ::_thesis: ( not rng s1 c= dom (||.f.|| | X) or not s1 is convergent or not lim s1 = r or ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) ) )
assume that
A7: rng s1 c= dom (||.f.|| | X) and
A8: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) )
rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61;
then rng s1 c= (dom f) /\ X by NORMSP_0:def_3;
then A9: rng s1 c= dom (f | X) by RELAT_1:61;
then A10: (f | X) /. r = lim ((f | X) /* s1) by A5, A8, Def7;
now__::_thesis:_for_n_being_Element_of_NAT_holds_||.((f_|_X)_/*_s1).||_._n_=_((||.f.||_|_X)_/*_s1)_._n
let n be Element of NAT ; ::_thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n
dom s1 = NAT by FUNCT_2:def_1;
then A11: s1 . n in rng s1 by FUNCT_1:3;
then s1 . n in dom (f | X) by A9;
then A12: s1 . n in (dom f) /\ X by RELAT_1:61;
then A13: s1 . n in X by XBOOLE_0:def_4;
s1 . n in dom f by A12, XBOOLE_0:def_4;
then A14: s1 . n in dom ||.f.|| by NORMSP_0:def_3;
thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def_4
.= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109
.= ||.(f /. (s1 . n)).|| by A9, A11, PARTFUN2:15
.= ||.f.|| . (s1 . n) by A14, NORMSP_0:def_3
.= (||.f.|| | X) . (s1 . n) by A13, FUNCT_1:49
.= (||.f.|| | X) /. (s1 . n) by A7, A11, PARTFUN1:def_6
.= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:109 ; ::_thesis: verum
end;
then A15: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63;
A16: (f | X) /* s1 is convergent by A5, A8, A9, Def7;
hence (||.f.|| | X) /* s1 is convergent by A15, CLVECT_1:117; ::_thesis: (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1)
||.((f | X) /. r).|| = ||.(f /. r).|| by A2, A4, PARTFUN2:17
.= ||.f.|| . r by A3, A4, NORMSP_0:def_3
.= ||.f.|| /. r by A3, A4, PARTFUN1:def_6
.= (||.f.|| | X) /. r by A6, PARTFUN2:16 ;
hence (||.f.|| | X) /. r = lim ((||.f.|| | X) /* s1) by A16, A10, A15, CLOPBAN1:19; ::_thesis: verum
end;
end;
(- 1r) (#) f is_continuous_on X by A1, Th70;
hence - f is_continuous_on X by VFUNCT_2:23; ::_thesis: verum
end;
theorem :: NCFCONT1:74
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st f is total & ( for x1, x2 being Point of CNS1 holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS1 st f is_continuous_in x0 holds
f is_continuous_on the carrier of CNS1
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is total & ( for x1, x2 being Point of CNS1 holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS1 st f is_continuous_in x0 holds
f is_continuous_on the carrier of CNS1
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is total & ( for x1, x2 being Point of CNS1 holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS1 st f is_continuous_in x0 implies f is_continuous_on the carrier of CNS1 )
assume that
A1: f is total and
A2: for x1, x2 being Point of CNS1 holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; ::_thesis: ( for x0 being Point of CNS1 holds not f is_continuous_in x0 or f is_continuous_on the carrier of CNS1 )
A3: dom f = the carrier of CNS1 by A1, PARTFUN1:def_2;
given x0 being Point of CNS1 such that A4: f is_continuous_in x0 ; ::_thesis: f is_continuous_on the carrier of CNS1
(f /. x0) + (0. CNS2) = f /. x0 by RLVECT_1:4
.= f /. (x0 + (0. CNS1)) by RLVECT_1:4
.= (f /. x0) + (f /. (0. CNS1)) by A2 ;
then A5: f /. (0. CNS1) = 0. CNS2 by RLVECT_1:8;
A6: now__::_thesis:_for_x1_being_Point_of_CNS1_holds_-_(f_/._x1)_=_f_/._(-_x1)
let x1 be Point of CNS1; ::_thesis: - (f /. x1) = f /. (- x1)
0. CNS2 = f /. (x1 + (- x1)) by A5, RLVECT_1:5
.= (f /. x1) + (f /. (- x1)) by A2 ;
hence - (f /. x1) = f /. (- x1) by RLVECT_1:6; ::_thesis: verum
end;
A7: now__::_thesis:_for_x1,_x2_being_Point_of_CNS1_holds_f_/._(x1_-_x2)_=_(f_/._x1)_-_(f_/._x2)
let x1, x2 be Point of CNS1; ::_thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2)
thus f /. (x1 - x2) = f /. (x1 + (- x2)) by RLVECT_1:def_11
.= (f /. x1) + (f /. (- x2)) by A2
.= (f /. x1) + (- (f /. x2)) by A6
.= (f /. x1) - (f /. x2) by RLVECT_1:def_11 ; ::_thesis: verum
end;
now__::_thesis:_for_x1_being_Point_of_CNS1
for_r_being_Real_st_x1_in_the_carrier_of_CNS1_&_r_>_0_holds_
ex_s_being_Real_st_
(_s_>_0_&_(_for_x2_being_Point_of_CNS1_st_x2_in_the_carrier_of_CNS1_&_||.(x2_-_x1).||_<_s_holds_
||.((f_/._x2)_-_(f_/._x1)).||_<_r_)_)
let x1 be Point of CNS1; ::_thesis: for r being Real st x1 in the carrier of CNS1 & r > 0 holds
ex s being Real st
( s > 0 & ( for x2 being Point of CNS1 st x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
let r be Real; ::_thesis: ( x1 in the carrier of CNS1 & r > 0 implies ex s being Real st
( s > 0 & ( for x2 being Point of CNS1 st x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) ) )
assume that
x1 in the carrier of CNS1 and
A8: r > 0 ; ::_thesis: ex s being Real st
( s > 0 & ( for x2 being Point of CNS1 st x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
set y = x1 - x0;
consider s being Real such that
A9: 0 < s and
A10: for x1 being Point of CNS1 st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th8;
take s = s; ::_thesis: ( s > 0 & ( for x2 being Point of CNS1 st x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
thus s > 0 by A9; ::_thesis: for x2 being Point of CNS1 st x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r
let x2 be Point of CNS1; ::_thesis: ( x2 in the carrier of CNS1 & ||.(x2 - x1).|| < s implies ||.((f /. x2) - (f /. x1)).|| < r )
assume that
x2 in the carrier of CNS1 and
A11: ||.(x2 - x1).|| < s ; ::_thesis: ||.((f /. x2) - (f /. x1)).|| < r
A12: (x1 - x0) + x0 = x1 - (x0 - x0) by RLVECT_1:29
.= x1 - (0. CNS1) by RLVECT_1:15
.= x1 by RLVECT_1:13 ;
then A13: ||.((x2 - (x1 - x0)) - x0).|| = ||.(x2 - x1).|| by RLVECT_1:27;
||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2, A12
.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27
.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;
hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A13; ::_thesis: verum
end;
hence f is_continuous_on the carrier of CNS1 by A3, Th44; ::_thesis: verum
end;
theorem :: NCFCONT1:75
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st f is total & ( for x1, x2 being Point of CNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of CNS
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st f is total & ( for x1, x2 being Point of CNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of CNS
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS st f is total & ( for x1, x2 being Point of CNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of CNS
let f be PartFunc of CNS,RNS; ::_thesis: ( f is total & ( for x1, x2 being Point of CNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of CNS st f is_continuous_in x0 implies f is_continuous_on the carrier of CNS )
assume that
A1: f is total and
A2: for x1, x2 being Point of CNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; ::_thesis: ( for x0 being Point of CNS holds not f is_continuous_in x0 or f is_continuous_on the carrier of CNS )
A3: dom f = the carrier of CNS by A1, PARTFUN1:def_2;
given x0 being Point of CNS such that A4: f is_continuous_in x0 ; ::_thesis: f is_continuous_on the carrier of CNS
(f /. x0) + (0. RNS) = f /. x0 by RLVECT_1:4
.= f /. (x0 + (0. CNS)) by RLVECT_1:4
.= (f /. x0) + (f /. (0. CNS)) by A2 ;
then A5: f /. (0. CNS) = 0. RNS by RLVECT_1:8;
A6: now__::_thesis:_for_x1_being_Point_of_CNS_holds_-_(f_/._x1)_=_f_/._(-_x1)
let x1 be Point of CNS; ::_thesis: - (f /. x1) = f /. (- x1)
0. RNS = f /. (x1 + (- x1)) by A5, RLVECT_1:5
.= (f /. x1) + (f /. (- x1)) by A2 ;
hence - (f /. x1) = f /. (- x1) by RLVECT_1:6; ::_thesis: verum
end;
A7: now__::_thesis:_for_x1,_x2_being_Point_of_CNS_holds_f_/._(x1_-_x2)_=_(f_/._x1)_-_(f_/._x2)
let x1, x2 be Point of CNS; ::_thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2)
thus f /. (x1 - x2) = f /. (x1 + (- x2)) by RLVECT_1:def_11
.= (f /. x1) + (f /. (- x2)) by A2
.= (f /. x1) + (- (f /. x2)) by A6
.= (f /. x1) - (f /. x2) by RLVECT_1:def_11 ; ::_thesis: verum
end;
now__::_thesis:_for_x1_being_Point_of_CNS
for_r_being_Real_st_x1_in_the_carrier_of_CNS_&_r_>_0_holds_
ex_s_being_Real_st_
(_s_>_0_&_(_for_x2_being_Point_of_CNS_st_x2_in_the_carrier_of_CNS_&_||.(x2_-_x1).||_<_s_holds_
||.((f_/._x2)_-_(f_/._x1)).||_<_r_)_)
let x1 be Point of CNS; ::_thesis: for r being Real st x1 in the carrier of CNS & r > 0 holds
ex s being Real st
( s > 0 & ( for x2 being Point of CNS st x2 in the carrier of CNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
let r be Real; ::_thesis: ( x1 in the carrier of CNS & r > 0 implies ex s being Real st
( s > 0 & ( for x2 being Point of CNS st x2 in the carrier of CNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) ) )
assume that
x1 in the carrier of CNS and
A8: r > 0 ; ::_thesis: ex s being Real st
( s > 0 & ( for x2 being Point of CNS st x2 in the carrier of CNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
set y = x1 - x0;
consider s being Real such that
A9: 0 < s and
A10: for x1 being Point of CNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th9;
take s = s; ::_thesis: ( s > 0 & ( for x2 being Point of CNS st x2 in the carrier of CNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
thus s > 0 by A9; ::_thesis: for x2 being Point of CNS st x2 in the carrier of CNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r
let x2 be Point of CNS; ::_thesis: ( x2 in the carrier of CNS & ||.(x2 - x1).|| < s implies ||.((f /. x2) - (f /. x1)).|| < r )
assume that
x2 in the carrier of CNS and
A11: ||.(x2 - x1).|| < s ; ::_thesis: ||.((f /. x2) - (f /. x1)).|| < r
A12: (x1 - x0) + x0 = x1 - (x0 - x0) by RLVECT_1:29
.= x1 - (0. CNS) by RLVECT_1:15
.= x1 by RLVECT_1:13 ;
then A13: ||.((x2 - (x1 - x0)) - x0).|| = ||.(x2 - x1).|| by RLVECT_1:27;
||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2, A12
.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27
.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;
hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A13; ::_thesis: verum
end;
hence f is_continuous_on the carrier of CNS by A3, Th45; ::_thesis: verum
end;
theorem :: NCFCONT1:76
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st f is total & ( for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of RNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of RNS
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st f is total & ( for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of RNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of RNS
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS st f is total & ( for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of RNS st f is_continuous_in x0 holds
f is_continuous_on the carrier of RNS
let f be PartFunc of RNS,CNS; ::_thesis: ( f is total & ( for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being Point of RNS st f is_continuous_in x0 implies f is_continuous_on the carrier of RNS )
assume that
A1: f is total and
A2: for x1, x2 being Point of RNS holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; ::_thesis: ( for x0 being Point of RNS holds not f is_continuous_in x0 or f is_continuous_on the carrier of RNS )
A3: dom f = the carrier of RNS by A1, PARTFUN1:def_2;
given x0 being Point of RNS such that A4: f is_continuous_in x0 ; ::_thesis: f is_continuous_on the carrier of RNS
(f /. x0) + (0. CNS) = f /. x0 by RLVECT_1:4
.= f /. (x0 + (0. RNS)) by RLVECT_1:4
.= (f /. x0) + (f /. (0. RNS)) by A2 ;
then A5: f /. (0. RNS) = 0. CNS by RLVECT_1:8;
A6: now__::_thesis:_for_x1_being_Point_of_RNS_holds_-_(f_/._x1)_=_f_/._(-_x1)
let x1 be Point of RNS; ::_thesis: - (f /. x1) = f /. (- x1)
0. CNS = f /. (x1 + (- x1)) by A5, RLVECT_1:5
.= (f /. x1) + (f /. (- x1)) by A2 ;
hence - (f /. x1) = f /. (- x1) by RLVECT_1:6; ::_thesis: verum
end;
A7: now__::_thesis:_for_x1,_x2_being_Point_of_RNS_holds_f_/._(x1_-_x2)_=_(f_/._x1)_-_(f_/._x2)
let x1, x2 be Point of RNS; ::_thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2)
thus f /. (x1 - x2) = f /. (x1 + (- x2)) by RLVECT_1:def_11
.= (f /. x1) + (f /. (- x2)) by A2
.= (f /. x1) + (- (f /. x2)) by A6
.= (f /. x1) - (f /. x2) by RLVECT_1:def_11 ; ::_thesis: verum
end;
now__::_thesis:_for_x1_being_Point_of_RNS
for_r_being_Real_st_x1_in_the_carrier_of_RNS_&_r_>_0_holds_
ex_s_being_Real_st_
(_s_>_0_&_(_for_x2_being_Point_of_RNS_st_x2_in_the_carrier_of_RNS_&_||.(x2_-_x1).||_<_s_holds_
||.((f_/._x2)_-_(f_/._x1)).||_<_r_)_)
let x1 be Point of RNS; ::_thesis: for r being Real st x1 in the carrier of RNS & r > 0 holds
ex s being Real st
( s > 0 & ( for x2 being Point of RNS st x2 in the carrier of RNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
let r be Real; ::_thesis: ( x1 in the carrier of RNS & r > 0 implies ex s being Real st
( s > 0 & ( for x2 being Point of RNS st x2 in the carrier of RNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) ) )
assume that
x1 in the carrier of RNS and
A8: r > 0 ; ::_thesis: ex s being Real st
( s > 0 & ( for x2 being Point of RNS st x2 in the carrier of RNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
set y = x1 - x0;
consider s being Real such that
A9: 0 < s and
A10: for x1 being Point of RNS st x1 in dom f & ||.(x1 - x0).|| < s holds
||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th10;
take s = s; ::_thesis: ( s > 0 & ( for x2 being Point of RNS st x2 in the carrier of RNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r ) )
thus s > 0 by A9; ::_thesis: for x2 being Point of RNS st x2 in the carrier of RNS & ||.(x2 - x1).|| < s holds
||.((f /. x2) - (f /. x1)).|| < r
let x2 be Point of RNS; ::_thesis: ( x2 in the carrier of RNS & ||.(x2 - x1).|| < s implies ||.((f /. x2) - (f /. x1)).|| < r )
assume that
x2 in the carrier of RNS and
A11: ||.(x2 - x1).|| < s ; ::_thesis: ||.((f /. x2) - (f /. x1)).|| < r
A12: (x1 - x0) + x0 = x1 - (x0 - x0) by RLVECT_1:29
.= x1 - (0. RNS) by RLVECT_1:15
.= x1 by RLVECT_1:13 ;
then A13: ||.((x2 - (x1 - x0)) - x0).|| = ||.(x2 - x1).|| by RLVECT_1:27;
||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2, A12
.= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27
.= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ;
hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A13; ::_thesis: verum
end;
hence f is_continuous_on the carrier of RNS by A3, Th46; ::_thesis: verum
end;
theorem Th77: :: NCFCONT1:77
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2 st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of CNS1,CNS2; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_sequence_of_CNS2_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,_the_carrier_of_CNS2))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be sequence of CNS2; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT, the carrier of CNS2)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT, the carrier of CNS2)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of CNS1 st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS1 st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of CNS1 such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of CNS1 such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by Th7;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of CNS1 such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def11;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def5;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def5, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by Def2; ::_thesis: verum
end;
theorem Th78: :: NCFCONT1:78
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of CNS,RNS; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_sequence_of_RNS_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,_the_carrier_of_RNS))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be sequence of RNS; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT, the carrier of RNS)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT, the carrier of RNS)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of CNS such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of CNS such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by Th7;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of CNS such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def12;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def6;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def6, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by NFCONT_1:def_2; ::_thesis: verum
end;
theorem Th79: :: NCFCONT1:79
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of RNS,CNS; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_sequence_of_CNS_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,_the_carrier_of_CNS))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be sequence of CNS; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT, the carrier of CNS)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT, the carrier of CNS)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of RNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of RNS st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of RNS such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of RNS such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by NFCONT_1:6;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of RNS such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, NFCONT_1:def_2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def13;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def7;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def7, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by Def2; ::_thesis: verum
end;
theorem :: NCFCONT1:80
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,COMPLEX st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,COMPLEX st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of the carrier of CNS,COMPLEX; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_Complex_Sequence_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,COMPLEX))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be Complex_Sequence; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT,COMPLEX)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT,COMPLEX)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of CNS such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of CNS such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by Th7;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of CNS such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def14;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def8;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def8, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by CFCONT_1:def_3; ::_thesis: verum
end;
theorem Th81: :: NCFCONT1:81
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,REAL st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,REAL))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be Real_Sequence; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT,REAL)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT,REAL)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of CNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of CNS st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of CNS such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of CNS such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by Th7;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of CNS such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, Def2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def15;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def9;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def9, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by RCOMP_1:def_3; ::_thesis: verum
end;
theorem :: NCFCONT1:82
for RNS being RealNormSpace
for f being PartFunc of the carrier of RNS,COMPLEX st dom f is compact & f is_continuous_on dom f holds
rng f is compact
proof
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of the carrier of RNS,COMPLEX st dom f is compact & f is_continuous_on dom f holds
rng f is compact
let f be PartFunc of the carrier of RNS,COMPLEX; ::_thesis: ( dom f is compact & f is_continuous_on dom f implies rng f is compact )
assume that
A1: dom f is compact and
A2: f is_continuous_on dom f ; ::_thesis: rng f is compact
now__::_thesis:_for_s1_being_Complex_Sequence_st_rng_s1_c=_rng_f_holds_
ex_q2_being_Element_of_K6(K7(NAT,COMPLEX))_st_
(_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_)
let s1 be Complex_Sequence; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of K6(K7(NAT,COMPLEX)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) )
assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of K6(K7(NAT,COMPLEX)) st
( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
defpred S1[ set , set ] means ( $2 in dom f & f /. $2 = s1 . $1 );
A4: for n being Element of NAT ex p being Point of RNS st S1[n,p]
proof
let n be Element of NAT ; ::_thesis: ex p being Point of RNS st S1[n,p]
dom s1 = NAT by FUNCT_2:def_1;
then s1 . n in rng s1 by FUNCT_1:3;
then consider p being Point of RNS such that
A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3;
take p ; ::_thesis: S1[n,p]
thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum
end;
consider q1 being sequence of RNS such that
A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4);
now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_
x_in_dom_f
let x be set ; ::_thesis: ( x in rng q1 implies x in dom f )
assume x in rng q1 ; ::_thesis: x in dom f
then ex n being Element of NAT st x = q1 . n by NFCONT_1:6;
hence x in dom f by A6; ::_thesis: verum
end;
then A7: rng q1 c= dom f by TARSKI:def_3;
then consider s2 being sequence of RNS such that
A8: s2 is subsequence of q1 and
A9: s2 is convergent and
A10: lim s2 in dom f by A1, NFCONT_1:def_2;
take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f )
rng s2 c= rng q1 by A8, VALUED_0:21;
then A11: rng s2 c= dom f by A7, XBOOLE_1:1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n
let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n
f /. (q1 . n) = s1 . n by A6;
hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum
end;
then A12: f /* q1 = s1 by FUNCT_2:63;
f | (dom f) is_continuous_in lim s2 by A2, A10, Def16;
then A13: f is_continuous_in lim s2 by RELAT_1:68;
then f /. (lim s2) = lim (f /* s2) by A9, A11, Def10;
hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def10, PARTFUN2:2, VALUED_0:22; ::_thesis: verum
end;
hence rng f is compact by CFCONT_1:def_3; ::_thesis: verum
end;
theorem :: NCFCONT1:83
for CNS1, CNS2 being ComplexNormSpace
for Y being Subset of CNS1
for f being PartFunc of CNS1,CNS2 st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for Y being Subset of CNS1
for f being PartFunc of CNS1,CNS2 st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let Y be Subset of CNS1; ::_thesis: for f being PartFunc of CNS1,CNS2 st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let f be PartFunc of CNS1,CNS2; ::_thesis: ( Y c= dom f & Y is compact & f is_continuous_on Y implies f .: Y is compact )
assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y ; ::_thesis: f .: Y is compact
A4: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A1, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A4; :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A3, Def11;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then rng (f | Y) is compact by A2, A4, Th77;
hence f .: Y is compact by RELAT_1:115; ::_thesis: verum
end;
theorem :: NCFCONT1:84
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for Y being Subset of CNS
for f being PartFunc of CNS,RNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for Y being Subset of CNS
for f being PartFunc of CNS,RNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let RNS be RealNormSpace; ::_thesis: for Y being Subset of CNS
for f being PartFunc of CNS,RNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let Y be Subset of CNS; ::_thesis: for f being PartFunc of CNS,RNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let f be PartFunc of CNS,RNS; ::_thesis: ( Y c= dom f & Y is compact & f is_continuous_on Y implies f .: Y is compact )
assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y ; ::_thesis: f .: Y is compact
A4: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A1, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A4; :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A3, Def12;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then rng (f | Y) is compact by A2, A4, Th78;
hence f .: Y is compact by RELAT_1:115; ::_thesis: verum
end;
theorem :: NCFCONT1:85
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for Y being Subset of RNS
for f being PartFunc of RNS,CNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for Y being Subset of RNS
for f being PartFunc of RNS,CNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let RNS be RealNormSpace; ::_thesis: for Y being Subset of RNS
for f being PartFunc of RNS,CNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let Y be Subset of RNS; ::_thesis: for f being PartFunc of RNS,CNS st Y c= dom f & Y is compact & f is_continuous_on Y holds
f .: Y is compact
let f be PartFunc of RNS,CNS; ::_thesis: ( Y c= dom f & Y is compact & f is_continuous_on Y implies f .: Y is compact )
assume that
A1: Y c= dom f and
A2: Y is compact and
A3: f is_continuous_on Y ; ::_thesis: f .: Y is compact
A4: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A1, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A4; :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A3, Def13;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then rng (f | Y) is compact by A2, A4, Th79;
hence f .: Y is compact by RELAT_1:115; ::_thesis: verum
end;
theorem Th86: :: NCFCONT1:86
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,REAL st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) ) )
assume ( dom f <> {} & dom f is compact & f is_continuous_on dom f ) ; ::_thesis: ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
then A1: ( rng f <> {} & rng f is compact ) by Th81, RELAT_1:42;
then consider x being Element of CNS such that
A2: ( x in dom f & upper_bound (rng f) = f . x ) by PARTFUN1:3, RCOMP_1:14;
take x ; ::_thesis: ex x2 being Point of CNS st
( x in dom f & x2 in dom f & f /. x = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
consider y being Element of CNS such that
A3: ( y in dom f & lower_bound (rng f) = f . y ) by A1, PARTFUN1:3, RCOMP_1:14;
take y ; ::_thesis: ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) )
thus ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) ) by A2, A3, PARTFUN1:def_6; ::_thesis: verum
end;
theorem Th87: :: NCFCONT1:87
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS1 st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2 st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS1 st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of CNS1 st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) ) )
assume that
A1: dom f <> {} and
A2: dom f is compact and
A3: f is_continuous_on dom f ; ::_thesis: ex x1, x2 being Point of CNS1 st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
A4: dom f = dom ||.f.|| by NORMSP_0:def_3;
dom ||.f.|| is compact by A2, NORMSP_0:def_3;
then A5: rng ||.f.|| is compact by A3, A4, Th71, Th81;
A6: rng ||.f.|| <> {} by A1, A4, RELAT_1:42;
then consider x being Element of CNS1 such that
A7: ( x in dom ||.f.|| & upper_bound (rng ||.f.||) = ||.f.|| . x ) by A5, PARTFUN1:3, RCOMP_1:14;
consider y being Element of CNS1 such that
A8: ( y in dom ||.f.|| & lower_bound (rng ||.f.||) = ||.f.|| . y ) by A6, A5, PARTFUN1:3, RCOMP_1:14;
take x ; ::_thesis: ex x2 being Point of CNS1 st
( x in dom f & x2 in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
take y ; ::_thesis: ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) )
thus ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) ) by A7, A8, NORMSP_0:def_3, PARTFUN1:def_6; ::_thesis: verum
end;
theorem Th88: :: NCFCONT1:88
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let f be PartFunc of CNS,RNS; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) ) )
assume that
A1: dom f <> {} and
A2: dom f is compact and
A3: f is_continuous_on dom f ; ::_thesis: ex x1, x2 being Point of CNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
A4: dom f = dom ||.f.|| by NORMSP_0:def_3;
dom ||.f.|| is compact by A2, NORMSP_0:def_3;
then A5: rng ||.f.|| is compact by A3, A4, Th72, Th81;
A6: rng ||.f.|| <> {} by A1, A4, RELAT_1:42;
then consider x being Element of CNS such that
A7: ( x in dom ||.f.|| & upper_bound (rng ||.f.||) = ||.f.|| . x ) by A5, PARTFUN1:3, RCOMP_1:14;
consider y being Element of CNS such that
A8: ( y in dom ||.f.|| & lower_bound (rng ||.f.||) = ||.f.|| . y ) by A6, A5, PARTFUN1:3, RCOMP_1:14;
take x ; ::_thesis: ex x2 being Point of CNS st
( x in dom f & x2 in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
take y ; ::_thesis: ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) )
thus ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) ) by A7, A8, NORMSP_0:def_3, PARTFUN1:def_6; ::_thesis: verum
end;
theorem Th89: :: NCFCONT1:89
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of RNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of RNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of RNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
let f be PartFunc of RNS,CNS; ::_thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of RNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) ) )
assume that
A1: dom f <> {} and
A2: dom f is compact and
A3: f is_continuous_on dom f ; ::_thesis: ex x1, x2 being Point of RNS st
( x1 in dom f & x2 in dom f & ||.f.|| /. x1 = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
A4: dom f = dom ||.f.|| by NORMSP_0:def_3;
dom ||.f.|| is compact by A2, NORMSP_0:def_3;
then A5: rng ||.f.|| is compact by A3, A4, Th73, NFCONT_1:31;
A6: rng ||.f.|| <> {} by A1, A4, RELAT_1:42;
then consider x being Element of RNS such that
A7: ( x in dom ||.f.|| & upper_bound (rng ||.f.||) = ||.f.|| . x ) by A5, PARTFUN1:3, RCOMP_1:14;
consider y being Element of RNS such that
A8: ( y in dom ||.f.|| & lower_bound (rng ||.f.||) = ||.f.|| . y ) by A6, A5, PARTFUN1:3, RCOMP_1:14;
take x ; ::_thesis: ex x2 being Point of RNS st
( x in dom f & x2 in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. x2 = lower_bound (rng ||.f.||) )
take y ; ::_thesis: ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) )
thus ( x in dom f & y in dom f & ||.f.|| /. x = upper_bound (rng ||.f.||) & ||.f.|| /. y = lower_bound (rng ||.f.||) ) by A7, A8, NORMSP_0:def_3, PARTFUN1:def_6; ::_thesis: verum
end;
theorem Th90: :: NCFCONT1:90
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 holds ||.f.|| | X = ||.(f | X).||
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 holds ||.f.|| | X = ||.(f | X).||
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 holds ||.f.|| | X = ||.(f | X).||
let f be PartFunc of CNS1,CNS2; ::_thesis: ||.f.|| | X = ||.(f | X).||
A1: dom (||.f.|| | X) = (dom ||.f.||) /\ X by RELAT_1:61
.= (dom f) /\ X by NORMSP_0:def_3
.= dom (f | X) by RELAT_1:61
.= dom ||.(f | X).|| by NORMSP_0:def_3 ;
now__::_thesis:_for_c_being_Point_of_CNS1_st_c_in_dom_(||.f.||_|_X)_holds_
(||.f.||_|_X)_._c_=_||.(f_|_X).||_._c
let c be Point of CNS1; ::_thesis: ( c in dom (||.f.|| | X) implies (||.f.|| | X) . c = ||.(f | X).|| . c )
assume A2: c in dom (||.f.|| | X) ; ::_thesis: (||.f.|| | X) . c = ||.(f | X).|| . c
then A3: c in dom (f | X) by A1, NORMSP_0:def_3;
c in (dom ||.f.||) /\ X by A2, RELAT_1:61;
then A4: c in dom ||.f.|| by XBOOLE_0:def_4;
thus (||.f.|| | X) . c = ||.f.|| . c by A2, FUNCT_1:47
.= ||.(f /. c).|| by A4, NORMSP_0:def_3
.= ||.((f | X) /. c).|| by A3, PARTFUN2:15
.= ||.(f | X).|| . c by A1, A2, NORMSP_0:def_3 ; ::_thesis: verum
end;
hence ||.f.|| | X = ||.(f | X).|| by A1, PARTFUN1:5; ::_thesis: verum
end;
theorem Th91: :: NCFCONT1:91
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds ||.f.|| | X = ||.(f | X).||
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds ||.f.|| | X = ||.(f | X).||
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS holds ||.f.|| | X = ||.(f | X).||
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS holds ||.f.|| | X = ||.(f | X).||
let f be PartFunc of CNS,RNS; ::_thesis: ||.f.|| | X = ||.(f | X).||
A1: dom (||.f.|| | X) = (dom ||.f.||) /\ X by RELAT_1:61
.= (dom f) /\ X by NORMSP_0:def_3
.= dom (f | X) by RELAT_1:61
.= dom ||.(f | X).|| by NORMSP_0:def_3 ;
now__::_thesis:_for_c_being_Point_of_CNS_st_c_in_dom_(||.f.||_|_X)_holds_
(||.f.||_|_X)_._c_=_||.(f_|_X).||_._c
let c be Point of CNS; ::_thesis: ( c in dom (||.f.|| | X) implies (||.f.|| | X) . c = ||.(f | X).|| . c )
assume A2: c in dom (||.f.|| | X) ; ::_thesis: (||.f.|| | X) . c = ||.(f | X).|| . c
then A3: c in dom (f | X) by A1, NORMSP_0:def_3;
c in (dom ||.f.||) /\ X by A2, RELAT_1:61;
then A4: c in dom ||.f.|| by XBOOLE_0:def_4;
thus (||.f.|| | X) . c = ||.f.|| . c by A2, FUNCT_1:47
.= ||.(f /. c).|| by A4, NORMSP_0:def_3
.= ||.((f | X) /. c).|| by A3, PARTFUN2:15
.= ||.(f | X).|| . c by A1, A2, NORMSP_0:def_3 ; ::_thesis: verum
end;
hence ||.f.|| | X = ||.(f | X).|| by A1, PARTFUN1:5; ::_thesis: verum
end;
theorem Th92: :: NCFCONT1:92
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds ||.f.|| | X = ||.(f | X).||
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS holds ||.f.|| | X = ||.(f | X).||
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS holds ||.f.|| | X = ||.(f | X).||
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS holds ||.f.|| | X = ||.(f | X).||
let f be PartFunc of RNS,CNS; ::_thesis: ||.f.|| | X = ||.(f | X).||
A1: dom (||.f.|| | X) = (dom ||.f.||) /\ X by RELAT_1:61
.= (dom f) /\ X by NORMSP_0:def_3
.= dom (f | X) by RELAT_1:61
.= dom ||.(f | X).|| by NORMSP_0:def_3 ;
now__::_thesis:_for_c_being_Point_of_RNS_st_c_in_dom_(||.f.||_|_X)_holds_
(||.f.||_|_X)_._c_=_||.(f_|_X).||_._c
let c be Point of RNS; ::_thesis: ( c in dom (||.f.|| | X) implies (||.f.|| | X) . c = ||.(f | X).|| . c )
assume A2: c in dom (||.f.|| | X) ; ::_thesis: (||.f.|| | X) . c = ||.(f | X).|| . c
then A3: c in dom (f | X) by A1, NORMSP_0:def_3;
c in (dom ||.f.||) /\ X by A2, RELAT_1:61;
then A4: c in dom ||.f.|| by XBOOLE_0:def_4;
thus (||.f.|| | X) . c = ||.f.|| . c by A2, FUNCT_1:47
.= ||.(f /. c).|| by A4, NORMSP_0:def_3
.= ||.((f | X) /. c).|| by A3, PARTFUN2:15
.= ||.(f | X).|| . c by A1, A2, NORMSP_0:def_3 ; ::_thesis: verum
end;
hence ||.f.|| | X = ||.(f | X).|| by A1, PARTFUN1:5; ::_thesis: verum
end;
theorem :: NCFCONT1:93
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2
for Y being Subset of CNS1 st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2
for Y being Subset of CNS1 st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let f be PartFunc of CNS1,CNS2; ::_thesis: for Y being Subset of CNS1 st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let Y be Subset of CNS1; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A5; :: according to NCFCONT1:def_11 ::_thesis: for x0 being Point of CNS1 st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of CNS1; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A4, Def11;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of CNS1 such that
A6: x1 in dom (f | Y) and
A7: x2 in dom (f | Y) and
A8: ( ||.(f | Y).|| /. x1 = upper_bound (rng ||.(f | Y).||) & ||.(f | Y).|| /. x2 = lower_bound (rng ||.(f | Y).||) ) by A1, A3, A5, Th87;
A9: dom f = dom ||.f.|| by NORMSP_0:def_3;
take x1 ; ::_thesis: ex x2 being Point of CNS1 st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
take x2 ; ::_thesis: ( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
thus ( x1 in Y & x2 in Y ) by A5, A6, A7; ::_thesis: ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A10: ||.f.|| .: Y = rng (||.f.|| | Y) by RELAT_1:115
.= rng ||.(f | Y).|| by Th90 ;
A11: x2 in dom ||.(f | Y).|| by A7, NORMSP_0:def_3;
then A12: ||.(f | Y).|| /. x2 = ||.(f | Y).|| . x2 by PARTFUN1:def_6
.= ||.((f | Y) /. x2).|| by A11, NORMSP_0:def_3
.= ||.(f /. x2).|| by A7, PARTFUN2:15
.= ||.f.|| . x2 by A2, A5, A7, A9, NORMSP_0:def_3
.= ||.f.|| /. x2 by A2, A5, A7, A9, PARTFUN1:def_6 ;
A13: x1 in dom ||.(f | Y).|| by A6, NORMSP_0:def_3;
then ||.(f | Y).|| /. x1 = ||.(f | Y).|| . x1 by PARTFUN1:def_6
.= ||.((f | Y) /. x1).|| by A13, NORMSP_0:def_3
.= ||.(f /. x1).|| by A6, PARTFUN2:15
.= ||.f.|| . x1 by A2, A5, A6, A9, NORMSP_0:def_3
.= ||.f.|| /. x1 by A2, A5, A6, A9, PARTFUN1:def_6 ;
hence ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) by A8, A12, A10; ::_thesis: verum
end;
theorem :: NCFCONT1:94
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS
for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS
for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let f be PartFunc of CNS,RNS; ::_thesis: for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let Y be Subset of CNS; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A5; :: according to NCFCONT1:def_12 ::_thesis: for x0 being Point of CNS st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A4, Def12;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of CNS such that
A6: x1 in dom (f | Y) and
A7: x2 in dom (f | Y) and
A8: ( ||.(f | Y).|| /. x1 = upper_bound (rng ||.(f | Y).||) & ||.(f | Y).|| /. x2 = lower_bound (rng ||.(f | Y).||) ) by A1, A3, A5, Th88;
A9: dom f = dom ||.f.|| by NORMSP_0:def_3;
take x1 ; ::_thesis: ex x2 being Point of CNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
take x2 ; ::_thesis: ( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
thus ( x1 in Y & x2 in Y ) by A5, A6, A7; ::_thesis: ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A10: ||.f.|| .: Y = rng (||.f.|| | Y) by RELAT_1:115
.= rng ||.(f | Y).|| by Th91 ;
A11: x2 in dom ||.(f | Y).|| by A7, NORMSP_0:def_3;
then A12: ||.(f | Y).|| /. x2 = ||.(f | Y).|| . x2 by PARTFUN1:def_6
.= ||.((f | Y) /. x2).|| by A11, NORMSP_0:def_3
.= ||.(f /. x2).|| by A7, PARTFUN2:15
.= ||.f.|| . x2 by A2, A5, A7, A9, NORMSP_0:def_3
.= ||.f.|| /. x2 by A2, A5, A7, A9, PARTFUN1:def_6 ;
A13: x1 in dom ||.(f | Y).|| by A6, NORMSP_0:def_3;
then ||.(f | Y).|| /. x1 = ||.(f | Y).|| . x1 by PARTFUN1:def_6
.= ||.((f | Y) /. x1).|| by A13, NORMSP_0:def_3
.= ||.(f /. x1).|| by A6, PARTFUN2:15
.= ||.f.|| . x1 by A2, A5, A6, A9, NORMSP_0:def_3
.= ||.f.|| /. x1 by A2, A5, A6, A9, PARTFUN1:def_6 ;
hence ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) by A8, A12, A10; ::_thesis: verum
end;
theorem :: NCFCONT1:95
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for Y being Subset of RNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS
for Y being Subset of RNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS
for Y being Subset of RNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let f be PartFunc of RNS,CNS; ::_thesis: for Y being Subset of RNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let Y be Subset of RNS; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A5; :: according to NCFCONT1:def_13 ::_thesis: for x0 being Point of RNS st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of RNS; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A4, Def13;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of RNS such that
A6: x1 in dom (f | Y) and
A7: x2 in dom (f | Y) and
A8: ( ||.(f | Y).|| /. x1 = upper_bound (rng ||.(f | Y).||) & ||.(f | Y).|| /. x2 = lower_bound (rng ||.(f | Y).||) ) by A1, A3, A5, Th89;
A9: dom f = dom ||.f.|| by NORMSP_0:def_3;
take x1 ; ::_thesis: ex x2 being Point of RNS st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
take x2 ; ::_thesis: ( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
thus ( x1 in Y & x2 in Y ) by A5, A6, A7; ::_thesis: ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A10: ||.f.|| .: Y = rng (||.f.|| | Y) by RELAT_1:115
.= rng ||.(f | Y).|| by Th92 ;
A11: x2 in dom ||.(f | Y).|| by A7, NORMSP_0:def_3;
then A12: ||.(f | Y).|| /. x2 = ||.(f | Y).|| . x2 by PARTFUN1:def_6
.= ||.((f | Y) /. x2).|| by A11, NORMSP_0:def_3
.= ||.(f /. x2).|| by A7, PARTFUN2:15
.= ||.f.|| . x2 by A2, A5, A7, A9, NORMSP_0:def_3
.= ||.f.|| /. x2 by A2, A5, A7, A9, PARTFUN1:def_6 ;
A13: x1 in dom ||.(f | Y).|| by A6, NORMSP_0:def_3;
then ||.(f | Y).|| /. x1 = ||.(f | Y).|| . x1 by PARTFUN1:def_6
.= ||.((f | Y) /. x1).|| by A13, NORMSP_0:def_3
.= ||.(f /. x1).|| by A6, PARTFUN2:15
.= ||.f.|| . x1 by A2, A5, A6, A9, NORMSP_0:def_3
.= ||.f.|| /. x1 by A2, A5, A6, A9, PARTFUN1:def_6 ;
hence ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) by A8, A12, A10; ::_thesis: verum
end;
theorem :: NCFCONT1:96
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL
for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,REAL
for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: for Y being Subset of CNS st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
let Y be Subset of CNS; ::_thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; ::_thesis: ex x1, x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:61
.= Y by A2, XBOOLE_1:28 ;
f | Y is_continuous_on Y
proof
thus Y c= dom (f | Y) by A5; :: according to NCFCONT1:def_15 ::_thesis: for x0 being Point of CNS st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of CNS; ::_thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; ::_thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A4, Def15;
hence (f | Y) | Y is_continuous_in r by RELAT_1:72; ::_thesis: verum
end;
then consider x1, x2 being Point of CNS such that
A6: ( x1 in dom (f | Y) & x2 in dom (f | Y) ) and
A7: ( (f | Y) /. x1 = upper_bound (rng (f | Y)) & (f | Y) /. x2 = lower_bound (rng (f | Y)) ) by A1, A3, A5, Th86;
take x1 ; ::_thesis: ex x2 being Point of CNS st
( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
take x2 ; ::_thesis: ( x1 in Y & x2 in Y & f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
thus ( x1 in Y & x2 in Y ) by A5, A6; ::_thesis: ( f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) )
( (f | Y) /. x1 = f /. x1 & (f | Y) /. x2 = f /. x2 ) by A6, PARTFUN2:15;
hence ( f /. x1 = upper_bound (f .: Y) & f /. x2 = lower_bound (f .: Y) ) by A7, RELAT_1:115; ::_thesis: verum
end;
definition
let CNS1, CNS2 be ComplexNormSpace;
let X be set ;
let f be PartFunc of CNS1,CNS2;
predf is_Lipschitzian_on X means :Def17: :: NCFCONT1:def 17
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def17 defines is_Lipschitzian_on NCFCONT1:def_17_:_
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ) );
definition
let CNS be ComplexNormSpace;
let RNS be RealNormSpace;
let X be set ;
let f be PartFunc of CNS,RNS;
predf is_Lipschitzian_on X means :Def18: :: NCFCONT1:def 18
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def18 defines is_Lipschitzian_on NCFCONT1:def_18_:_
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ) );
definition
let RNS be RealNormSpace;
let CNS be ComplexNormSpace;
let X be set ;
let f be PartFunc of RNS,CNS;
predf is_Lipschitzian_on X means :Def19: :: NCFCONT1:def 19
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def19 defines is_Lipschitzian_on NCFCONT1:def_19_:_
for RNS being RealNormSpace
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of RNS,CNS holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ) );
definition
let CNS be ComplexNormSpace;
let X be set ;
let f be PartFunc of the carrier of CNS,COMPLEX;
predf is_Lipschitzian_on X means :Def20: :: NCFCONT1:def 20
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def20 defines is_Lipschitzian_on NCFCONT1:def_20_:_
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,COMPLEX holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| ) ) ) );
definition
let CNS be ComplexNormSpace;
let X be set ;
let f be PartFunc of the carrier of CNS,REAL;
predf is_Lipschitzian_on X means :Def21: :: NCFCONT1:def 21
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def21 defines is_Lipschitzian_on NCFCONT1:def_21_:_
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,REAL holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| ) ) ) );
definition
let RNS be RealNormSpace;
let X be set ;
let f be PartFunc of the carrier of RNS,COMPLEX;
predf is_Lipschitzian_on X means :Def22: :: NCFCONT1:def 22
( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| ) ) );
end;
:: deftheorem Def22 defines is_Lipschitzian_on NCFCONT1:def_22_:_
for RNS being RealNormSpace
for X being set
for f being PartFunc of the carrier of RNS,COMPLEX holds
( f is_Lipschitzian_on X iff ( X c= dom f & ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| ) ) ) );
theorem Th97: :: NCFCONT1:97
for CNS1, CNS2 being ComplexNormSpace
for X, X1 being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1 )
assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X ; ::_thesis: f is_Lipschitzian_on X1
X c= dom f by A1, Def17;
hence X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_17 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider s being Real such that
A3: 0 < s and
A4: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A1, Def17;
take s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A3; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume ( x1 in X1 & x2 in X1 ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A2, A4; ::_thesis: verum
end;
theorem Th98: :: NCFCONT1:98
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1 )
assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X ; ::_thesis: f is_Lipschitzian_on X1
X c= dom f by A1, Def18;
hence X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_18 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider s being Real such that
A3: 0 < s and
A4: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A1, Def18;
take s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A3; ::_thesis: for x1, x2 being Point of CNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume ( x1 in X1 & x2 in X1 ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A2, A4; ::_thesis: verum
end;
theorem Th99: :: NCFCONT1:99
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let X, X1 be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1 )
assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X ; ::_thesis: f is_Lipschitzian_on X1
X c= dom f by A1, Def19;
hence X1 c= dom f by A2, XBOOLE_1:1; :: according to NCFCONT1:def_19 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider s being Real such that
A3: 0 < s and
A4: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A1, Def19;
take s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A3; ::_thesis: for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume ( x1 in X1 & x2 in X1 ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A2, A4; ::_thesis: verum
end;
theorem :: NCFCONT1:100
for CNS1, CNS2 being ComplexNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of CNS1,CNS2; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 + f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 + f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th97, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def17;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th97, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def17;
X /\ X1 c= dom f1 by A3, Def17;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 + f2) by VFUNCT_1:def_1; :: according to NCFCONT1:def_17 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def17;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A8, A11, VFUNCT_1:def_1
.= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_1
.= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:28
.= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27
.= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:def_11
.= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) + (- (f1 /. x2))) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_3
.= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_11 ;
then ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by CLVECT_1:def_13;
hence ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:101
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of CNS,RNS; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 + f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 + f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th98, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def18;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th98, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def18;
X /\ X1 c= dom f1 by A3, Def18;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 + f2) by VFUNCT_1:def_1; :: according to NCFCONT1:def_18 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def18;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A8, A11, VFUNCT_1:def_1
.= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_1
.= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:28
.= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27
.= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:def_11
.= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) + (- (f1 /. x2))) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_3
.= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_11 ;
then ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:def_1;
hence ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:102
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 + f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of RNS,CNS; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 + f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 + f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th99, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def19;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th99, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def19;
X /\ X1 c= dom f1 by A3, Def19;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 + f2) by VFUNCT_1:def_1; :: according to NCFCONT1:def_19 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def19;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A8, A11, VFUNCT_1:def_1
.= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_1
.= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:28
.= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27
.= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:def_11
.= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) + (- (f1 /. x2))) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_3
.= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_11 ;
then ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by CLVECT_1:def_13;
hence ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:103
for CNS1, CNS2 being ComplexNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS1,CNS2 st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of CNS1,CNS2; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 - f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 - f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th97, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def17;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th97, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def17;
X /\ X1 c= dom f1 by A3, Def17;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 - f2) by VFUNCT_1:def_2; :: according to NCFCONT1:def_17 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def17;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A8, A11, VFUNCT_1:def_2
.= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_2
.= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:27
.= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:28
.= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:27 ;
then ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by CLVECT_1:104;
hence ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:104
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of CNS,RNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of CNS,RNS; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 - f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 - f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th98, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def18;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th98, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def18;
X /\ X1 c= dom f1 by A3, Def18;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 - f2) by VFUNCT_1:def_2; :: according to NCFCONT1:def_18 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def18;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A8, A11, VFUNCT_1:def_2
.= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_2
.= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:27
.= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:28
.= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:27 ;
then ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:3;
hence ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:105
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let RNS be RealNormSpace; ::_thesis: for X, X1 being set
for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of RNS,CNS st f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 holds
f1 - f2 is_Lipschitzian_on X /\ X1
let f1, f2 be PartFunc of RNS,CNS; ::_thesis: ( f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1 - f2 is_Lipschitzian_on X /\ X1 )
assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1 ; ::_thesis: f1 - f2 is_Lipschitzian_on X /\ X1
A3: f1 is_Lipschitzian_on X /\ X1 by A1, Th99, XBOOLE_1:17;
then consider s being Real such that
A4: 0 < s and
A5: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by Def19;
A6: f2 is_Lipschitzian_on X /\ X1 by A2, Th99, XBOOLE_1:17;
then A7: X /\ X1 c= dom f2 by Def19;
X /\ X1 c= dom f1 by A3, Def19;
then X /\ X1 c= (dom f1) /\ (dom f2) by A7, XBOOLE_1:19;
hence A8: X /\ X1 c= dom (f1 - f2) by VFUNCT_1:def_2; :: according to NCFCONT1:def_19 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider g being Real such that
A9: 0 < g and
A10: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A6, Def19;
take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| ) )
0 + 0 < s + g by A4, A9, XREAL_1:8;
hence 0 < p ; ::_thesis: for x1, x2 being Point of RNS st x1 in X /\ X1 & x2 in X /\ X1 holds
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X /\ X1 & x2 in X /\ X1 implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| )
assume that
A11: x1 in X /\ X1 and
A12: x2 in X /\ X1 ; ::_thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).||
A13: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * ||.(x1 - x2).|| by A10, A11, A12;
||.((f1 /. x1) - (f1 /. x2)).|| <= s * ||.(x1 - x2).|| by A5, A11, A12;
then A14: ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * ||.(x1 - x2).||) + (g * ||.(x1 - x2).||) by A13, XREAL_1:7;
||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A8, A11, VFUNCT_1:def_2
.= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A8, A12, VFUNCT_1:def_2
.= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:27
.= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:28
.= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28
.= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:27 ;
then ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by CLVECT_1:104;
hence ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * ||.(x1 - x2).|| by A14, XXREAL_0:2; ::_thesis: verum
end;
theorem Th106: :: NCFCONT1:106
for z being Complex
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
proof
let z be Complex; ::_thesis: for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_Lipschitzian_on X implies z (#) f is_Lipschitzian_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: z (#) f is_Lipschitzian_on X
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def17;
X c= dom f by A1, Def17;
hence A4: X c= dom (z (#) f) by VFUNCT_2:def_2; :: according to NCFCONT1:def_17 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
now__::_thesis:_ex_s_being_Real_st_
(_0_<_s_&_(_for_x1,_x2_being_Point_of_CNS1_st_x1_in_X_&_x2_in_X_holds_
||.(((z_(#)_f)_/._x1)_-_((z_(#)_f)_/._x2)).||_<=_s_*_||.(x1_-_x2).||_)_)
percases ( z = 0 or z <> 0 ) ;
supposeA5: z = 0 ; ::_thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
take s = s; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X & x2 in X implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| )
assume that
A6: x1 in X and
A7: x2 in X ; ::_thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
0 <= ||.(x1 - x2).|| by CLVECT_1:105;
then A8: s * 0 <= s * ||.(x1 - x2).|| by A2, XREAL_1:64;
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A4, A6, VFUNCT_2:def_2
.= ||.((0. CNS2) - ((z (#) f) /. x2)).|| by A5, CLVECT_1:1
.= ||.((0. CNS2) - (z * (f /. x2))).|| by A4, A7, VFUNCT_2:def_2
.= ||.((0. CNS2) - (0. CNS2)).|| by A5, CLVECT_1:1
.= ||.(0. CNS2).|| by RLVECT_1:13
.= 0 by CLVECT_1:102 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A8; ::_thesis: verum
end;
supposeA9: z <> 0 ; ::_thesis: ex g being Element of REAL st
( 0 < g & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
take g = |.z.| * s; ::_thesis: ( 0 < g & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
0 < |.z.| by A9, COMPLEX1:47;
then 0 * s < |.z.| * s by A2, XREAL_1:68;
hence 0 < g ; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X & x2 in X implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| )
assume that
A10: x1 in X and
A11: x2 in X ; ::_thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
0 <= |.z.| by COMPLEX1:46;
then A12: |.z.| * ||.((f /. x1) - (f /. x2)).|| <= |.z.| * (s * ||.(x1 - x2).||) by A3, A10, A11, XREAL_1:64;
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A4, A10, VFUNCT_2:def_2
.= ||.((z * (f /. x1)) - (z * (f /. x2))).|| by A4, A11, VFUNCT_2:def_2
.= ||.(z * ((f /. x1) - (f /. x2))).|| by CLVECT_1:9
.= |.z.| * ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:def_13 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A12; ::_thesis: verum
end;
end;
end;
hence ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ; ::_thesis: verum
end;
theorem Th107: :: NCFCONT1:107
for r being Real
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
proof
let r be Real; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
r (#) f is_Lipschitzian_on X
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_Lipschitzian_on X implies r (#) f is_Lipschitzian_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: r (#) f is_Lipschitzian_on X
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def18;
X c= dom f by A1, Def18;
hence A4: X c= dom (r (#) f) by VFUNCT_1:def_4; :: according to NCFCONT1:def_18 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
now__::_thesis:_ex_s_being_Real_st_
(_0_<_s_&_(_for_x1,_x2_being_Point_of_CNS_st_x1_in_X_&_x2_in_X_holds_
||.(((r_(#)_f)_/._x1)_-_((r_(#)_f)_/._x2)).||_<=_s_*_||.(x1_-_x2).||_)_)
percases ( r = 0 or r <> 0 ) ;
supposeA5: r = 0 ; ::_thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
take s = s; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; ::_thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X & x2 in X implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| )
assume that
A6: x1 in X and
A7: x2 in X ; ::_thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
0 <= ||.(x1 - x2).|| by CLVECT_1:105;
then A8: s * 0 <= s * ||.(x1 - x2).|| by A2, XREAL_1:64;
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A4, A6, VFUNCT_1:def_4
.= ||.((0. RNS) - ((r (#) f) /. x2)).|| by A5, RLVECT_1:10
.= ||.((0. RNS) - (r * (f /. x2))).|| by A4, A7, VFUNCT_1:def_4
.= ||.((0. RNS) - (0. RNS)).|| by A5, RLVECT_1:10
.= ||.(0. RNS).|| by RLVECT_1:13
.= 0 by NORMSP_1:1 ;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A8; ::_thesis: verum
end;
supposeA9: r <> 0 ; ::_thesis: ex g being Element of REAL st
( 0 < g & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
take g = (abs r) * s; ::_thesis: ( 0 < g & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
0 < abs r by A9, COMPLEX1:47;
then 0 * s < (abs r) * s by A2, XREAL_1:68;
hence 0 < g ; ::_thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X & x2 in X implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| )
assume that
A10: x1 in X and
A11: x2 in X ; ::_thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
0 <= abs r by COMPLEX1:46;
then A12: (abs r) * ||.((f /. x1) - (f /. x2)).|| <= (abs r) * (s * ||.(x1 - x2).||) by A3, A10, A11, XREAL_1:64;
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A4, A10, VFUNCT_1:def_4
.= ||.((r * (f /. x1)) - (r * (f /. x2))).|| by A4, A11, VFUNCT_1:def_4
.= ||.(r * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= (abs r) * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def_1 ;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A12; ::_thesis: verum
end;
end;
end;
hence ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ; ::_thesis: verum
end;
theorem Th108: :: NCFCONT1:108
for z being Complex
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
proof
let z be Complex; ::_thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
z (#) f is_Lipschitzian_on X
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_Lipschitzian_on X implies z (#) f is_Lipschitzian_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: z (#) f is_Lipschitzian_on X
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def19;
X c= dom f by A1, Def19;
hence A4: X c= dom (z (#) f) by VFUNCT_2:def_2; :: according to NCFCONT1:def_19 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
now__::_thesis:_ex_s_being_Real_st_
(_0_<_s_&_(_for_x1,_x2_being_Point_of_RNS_st_x1_in_X_&_x2_in_X_holds_
||.(((z_(#)_f)_/._x1)_-_((z_(#)_f)_/._x2)).||_<=_s_*_||.(x1_-_x2).||_)_)
percases ( z = 0 or z <> 0 ) ;
supposeA5: z = 0 ; ::_thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
take s = s; ::_thesis: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; ::_thesis: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X & x2 in X implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| )
assume that
A6: x1 in X and
A7: x2 in X ; ::_thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).||
0 <= ||.(x1 - x2).|| by NORMSP_1:4;
then A8: s * 0 <= s * ||.(x1 - x2).|| by A2, XREAL_1:64;
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A4, A6, VFUNCT_2:def_2
.= ||.((0. CNS) - ((z (#) f) /. x2)).|| by A5, CLVECT_1:1
.= ||.((0. CNS) - (z * (f /. x2))).|| by A4, A7, VFUNCT_2:def_2
.= ||.((0. CNS) - (0. CNS)).|| by A5, CLVECT_1:1
.= ||.(0. CNS).|| by RLVECT_1:13
.= 0 by CLVECT_1:102 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= s * ||.(x1 - x2).|| by A8; ::_thesis: verum
end;
supposeA9: z <> 0 ; ::_thesis: ex g being Element of REAL st
( 0 < g & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
take g = |.z.| * s; ::_thesis: ( 0 < g & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| ) )
0 < |.z.| by A9, COMPLEX1:47;
then 0 * s < |.z.| * s by A2, XREAL_1:68;
hence 0 < g ; ::_thesis: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X & x2 in X implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| )
assume that
A10: x1 in X and
A11: x2 in X ; ::_thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).||
0 <= |.z.| by COMPLEX1:46;
then A12: |.z.| * ||.((f /. x1) - (f /. x2)).|| <= |.z.| * (s * ||.(x1 - x2).||) by A3, A10, A11, XREAL_1:64;
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A4, A10, VFUNCT_2:def_2
.= ||.((z * (f /. x1)) - (z * (f /. x2))).|| by A4, A11, VFUNCT_2:def_2
.= ||.(z * ((f /. x1) - (f /. x2))).|| by CLVECT_1:9
.= |.z.| * ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:def_13 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= g * ||.(x1 - x2).|| by A12; ::_thesis: verum
end;
end;
end;
hence ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| <= r * ||.(x1 - x2).|| ) ) ; ::_thesis: verum
end;
theorem :: NCFCONT1:109
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_Lipschitzian_on X implies ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X ) )
assume A1: f is_Lipschitzian_on X ; ::_thesis: ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def17;
- f = (- 1r) (#) f by VFUNCT_2:23;
hence - f is_Lipschitzian_on X by A1, Th106; ::_thesis: ||.f.|| is_Lipschitzian_on X
X c= dom f by A1, Def17;
hence A4: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NCFCONT1:def_21 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= r * ||.(x1 - x2).|| ) )
take s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; ::_thesis: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X & x2 in X implies abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| )
assume that
A5: x1 in X and
A6: x2 in X ; ::_thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A4, A5, PARTFUN1:def_6
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A4, A6, PARTFUN1:def_6
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A4, A5, NORMSP_0:def_3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A4, A6, NORMSP_0:def_3 ;
then A7: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:110;
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A3, A5, A6;
hence abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| by A7, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:110
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_Lipschitzian_on X implies ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X ) )
assume A1: f is_Lipschitzian_on X ; ::_thesis: ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def18;
- f = (- 1) (#) f by VFUNCT_1:23;
hence - f is_Lipschitzian_on X by A1, Th107; ::_thesis: ||.f.|| is_Lipschitzian_on X
X c= dom f by A1, Def18;
hence A4: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NCFCONT1:def_21 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= r * ||.(x1 - x2).|| ) )
take s ; ::_thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A2; ::_thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X & x2 in X implies abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| )
assume that
A5: x1 in X and
A6: x2 in X ; ::_thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A4, A5, PARTFUN1:def_6
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A4, A6, PARTFUN1:def_6
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A4, A5, NORMSP_0:def_3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A4, A6, NORMSP_0:def_3 ;
then A7: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A3, A5, A6;
hence abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| by A7, XXREAL_0:2; ::_thesis: verum
end;
theorem :: NCFCONT1:111
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_Lipschitzian_on X implies ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X ) )
assume A1: f is_Lipschitzian_on X ; ::_thesis: ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
then consider s being Real such that
A2: 0 < s and
A3: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by Def19;
- f = (- 1r) (#) f by VFUNCT_2:23;
hence - f is_Lipschitzian_on X by A1, Th108; ::_thesis: ||.f.|| is_Lipschitzian_on X
X c= dom f by A1, Def19;
hence A4: X c= dom ||.f.|| by NORMSP_0:def_3; :: according to NFCONT_1:def_10 ::_thesis: ex b1 being Element of REAL st
( not b1 <= 0 & ( for b2, b3 being Element of the carrier of RNS holds
( not b2 in X or not b3 in X or abs ((||.f.|| /. b2) - (||.f.|| /. b3)) <= b1 * ||.(b2 - b3).|| ) ) )
take s ; ::_thesis: ( not s <= 0 & ( for b1, b2 being Element of the carrier of RNS holds
( not b1 in X or not b2 in X or abs ((||.f.|| /. b1) - (||.f.|| /. b2)) <= s * ||.(b1 - b2).|| ) ) )
thus 0 < s by A2; ::_thesis: for b1, b2 being Element of the carrier of RNS holds
( not b1 in X or not b2 in X or abs ((||.f.|| /. b1) - (||.f.|| /. b2)) <= s * ||.(b1 - b2).|| )
let x1, x2 be Point of RNS; ::_thesis: ( not x1 in X or not x2 in X or abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| )
assume that
A5: x1 in X and
A6: x2 in X ; ::_thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A4, A5, PARTFUN1:def_6
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A4, A6, PARTFUN1:def_6
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A4, A5, NORMSP_0:def_3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A4, A6, NORMSP_0:def_3 ;
then A7: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:110;
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A3, A5, A6;
hence abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| by A7, XXREAL_0:2; ::_thesis: verum
end;
theorem Th112: :: NCFCONT1:112
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let f be PartFunc of CNS1,CNS2; ::_thesis: ( X c= dom f & f | X is V20() implies f is_Lipschitzian_on X )
assume that
A1: X c= dom f and
A2: f | X is V20() ; ::_thesis: f is_Lipschitzian_on X
now__::_thesis:_for_x1,_x2_being_Point_of_CNS1_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in X and
A4: x2 in X ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
A5: ( x1 in X /\ (dom f) & x2 in X /\ (dom f) ) by A1, A3, A4, XBOOLE_0:def_4;
f /. x1 = f . x1 by A1, A3, PARTFUN1:def_6
.= f . x2 by A2, A5, PARTFUN2:58
.= f /. x2 by A1, A4, PARTFUN1:def_6 ;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. CNS2).|| by RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by CLVECT_1:105; ::_thesis: verum
end;
hence f is_Lipschitzian_on X by A1, Def17; ::_thesis: verum
end;
theorem Th113: :: NCFCONT1:113
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let f be PartFunc of CNS,RNS; ::_thesis: ( X c= dom f & f | X is V20() implies f is_Lipschitzian_on X )
assume that
A1: X c= dom f and
A2: f | X is V20() ; ::_thesis: f is_Lipschitzian_on X
now__::_thesis:_for_x1,_x2_being_Point_of_CNS_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in X and
A4: x2 in X ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
A5: ( x1 in X /\ (dom f) & x2 in X /\ (dom f) ) by A1, A3, A4, XBOOLE_0:def_4;
f /. x1 = f . x1 by A1, A3, PARTFUN1:def_6
.= f . x2 by A2, A5, PARTFUN2:58
.= f /. x2 by A1, A4, PARTFUN1:def_6 ;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. RNS).|| by RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by CLVECT_1:105; ::_thesis: verum
end;
hence f is_Lipschitzian_on X by A1, Def18; ::_thesis: verum
end;
theorem Th114: :: NCFCONT1:114
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st X c= dom f & f | X is V20() holds
f is_Lipschitzian_on X
let f be PartFunc of RNS,CNS; ::_thesis: ( X c= dom f & f | X is V20() implies f is_Lipschitzian_on X )
assume that
A1: X c= dom f and
A2: f | X is V20() ; ::_thesis: f is_Lipschitzian_on X
now__::_thesis:_for_x1,_x2_being_Point_of_RNS_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in X and
A4: x2 in X ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
A5: ( x1 in X /\ (dom f) & x2 in X /\ (dom f) ) by A1, A3, A4, XBOOLE_0:def_4;
f /. x1 = f . x1 by A1, A3, PARTFUN1:def_6
.= f . x2 by A2, A5, PARTFUN2:58
.= f /. x2 by A1, A4, PARTFUN1:def_6 ;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. CNS).|| by RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by NORMSP_1:4; ::_thesis: verum
end;
hence f is_Lipschitzian_on X by A1, Def19; ::_thesis: verum
end;
theorem :: NCFCONT1:115
for CNS being ComplexNormSpace
for Y being Subset of CNS holds id Y is_Lipschitzian_on Y
proof
let CNS be ComplexNormSpace; ::_thesis: for Y being Subset of CNS holds id Y is_Lipschitzian_on Y
reconsider r = 1 as Real ;
let Y be Subset of CNS; ::_thesis: id Y is_Lipschitzian_on Y
thus Y c= dom (id Y) by RELAT_1:45; :: according to NCFCONT1:def_17 ::_thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
take r ; ::_thesis: ( 0 < r & ( for x1, x2 being Point of CNS st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ) )
thus r > 0 ; ::_thesis: for x1, x2 being Point of CNS st x1 in Y & x2 in Y holds
||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in Y & x2 in Y implies ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| )
assume that
A1: x1 in Y and
A2: x2 in Y ; ::_thesis: ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).||
||.(((id Y) /. x1) - ((id Y) /. x2)).|| = ||.(x1 - ((id Y) /. x2)).|| by A1, PARTFUN2:6
.= r * ||.(x1 - x2).|| by A2, PARTFUN2:6 ;
hence ||.(((id Y) /. x1) - ((id Y) /. x2)).|| <= r * ||.(x1 - x2).|| ; ::_thesis: verum
end;
theorem Th116: :: NCFCONT1:116
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of CNS1,CNS2; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by Def17;
A4: X c= dom f by A1, Def17;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_CNS1_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS1; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_CNS1_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_CNS1_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g
let x1 be Point of CNS1; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
||.((f /. x1) - (f /. x0)).|| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then ||.((f /. x1) - (f /. x0)).|| < g by A11, XXREAL_0:2;
then ||.(((f | X) /. x1) - (f /. x0)).|| < g by A9, PARTFUN2:15;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of CNS1 st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th8; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def11; ::_thesis: verum
end;
theorem Th117: :: NCFCONT1:117
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of CNS,RNS; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by Def18;
A4: X c= dom f by A1, Def18;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
||.((f /. x1) - (f /. x0)).|| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then ||.((f /. x1) - (f /. x0)).|| < g by A11, XXREAL_0:2;
then ||.(((f | X) /. x1) - (f /. x0)).|| < g by A9, PARTFUN2:15;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th9; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def12; ::_thesis: verum
end;
theorem Th118: :: NCFCONT1:118
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of RNS,CNS; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by Def19;
A4: X c= dom f by A1, Def19;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_RNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of RNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_RNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_RNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
||.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).||_<_g
let x1 be Point of RNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
||.((f /. x1) - (f /. x0)).|| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then ||.((f /. x1) - (f /. x0)).|| < g by A11, XXREAL_0:2;
then ||.(((f | X) /. x1) - (f /. x0)).|| < g by A9, PARTFUN2:15;
hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
||.(((f | X) /. x1) - ((f | X) /. x0)).|| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th10; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def13; ::_thesis: verum
end;
theorem :: NCFCONT1:119
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of the carrier of CNS,COMPLEX; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| by Def20;
A4: X c= dom f by A1, Def20;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
|.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).|_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
|.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).|_<_g
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies |.(((f | X) /. x1) - ((f | X) /. x0)).| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: |.(((f | X) /. x1) - ((f | X) /. x0)).| < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
|.((f /. x1) - (f /. x0)).| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then |.((f /. x1) - (f /. x0)).| < g by A11, XXREAL_0:2;
then |.(((f | X) /. x1) - (f /. x0)).| < g by A9, PARTFUN2:15;
hence |.(((f | X) /. x1) - ((f | X) /. x0)).| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th12; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def14; ::_thesis: verum
end;
theorem Th120: :: NCFCONT1:120
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,REAL st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
abs ((f /. x1) - (f /. x2)) <= r * ||.(x1 - x2).|| by Def21;
A4: X c= dom f by A1, Def21;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of CNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
abs_(((f_|_X)_/._x1)_-_((f_|_X)_/._x0))_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_CNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
abs_(((f_|_X)_/._x1)_-_((f_|_X)_/._x0))_<_g
let x1 be Point of CNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies abs (((f | X) /. x1) - ((f | X) /. x0)) < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: abs (((f | X) /. x1) - ((f | X) /. x0)) < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
abs ((f /. x1) - (f /. x0)) <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then abs ((f /. x1) - (f /. x0)) < g by A11, XXREAL_0:2;
then abs (((f | X) /. x1) - (f /. x0)) < g by A9, PARTFUN2:15;
hence abs (((f | X) /. x1) - ((f | X) /. x0)) < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of CNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
abs (((f | X) /. x1) - ((f | X) /. x0)) < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th11; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def15; ::_thesis: verum
end;
theorem :: NCFCONT1:121
for RNS being RealNormSpace
for X being set
for f being PartFunc of the carrier of RNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
proof
let RNS be RealNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of RNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of the carrier of RNS,COMPLEX st f is_Lipschitzian_on X holds
f is_continuous_on X
let f be PartFunc of the carrier of RNS,COMPLEX; ::_thesis: ( f is_Lipschitzian_on X implies f is_continuous_on X )
assume A1: f is_Lipschitzian_on X ; ::_thesis: f is_continuous_on X
then consider r being Real such that
A2: 0 < r and
A3: for x1, x2 being Point of RNS st x1 in X & x2 in X holds
|.((f /. x1) - (f /. x2)).| <= r * ||.(x1 - x2).|| by Def22;
A4: X c= dom f by A1, Def22;
then A5: dom (f | X) = X by RELAT_1:62;
now__::_thesis:_for_x0_being_Point_of_RNS_st_x0_in_X_holds_
f_|_X_is_continuous_in_x0
let x0 be Point of RNS; ::_thesis: ( x0 in X implies f | X is_continuous_in x0 )
assume A6: x0 in X ; ::_thesis: f | X is_continuous_in x0
now__::_thesis:_for_g_being_Real_st_0_<_g_holds_
ex_s9_being_Element_of_REAL_st_
(_0_<_s9_&_(_for_x1_being_Point_of_RNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_s9_holds_
|.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).|_<_g_)_)
let g be Real; ::_thesis: ( 0 < g implies ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) ) )
assume A7: 0 < g ; ::_thesis: ex s9 being Element of REAL st
( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) )
set s = g / r;
take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) )
A8: now__::_thesis:_for_x1_being_Point_of_RNS_st_x1_in_dom_(f_|_X)_&_||.(x1_-_x0).||_<_g_/_r_holds_
|.(((f_|_X)_/._x1)_-_((f_|_X)_/._x0)).|_<_g
let x1 be Point of RNS; ::_thesis: ( x1 in dom (f | X) & ||.(x1 - x0).|| < g / r implies |.(((f | X) /. x1) - ((f | X) /. x0)).| < g )
assume that
A9: x1 in dom (f | X) and
A10: ||.(x1 - x0).|| < g / r ; ::_thesis: |.(((f | X) /. x1) - ((f | X) /. x0)).| < g
r * ||.(x1 - x0).|| < (g / r) * r by A2, A10, XREAL_1:68;
then A11: r * ||.(x1 - x0).|| < g by A2, XCMPLX_1:87;
|.((f /. x1) - (f /. x0)).| <= r * ||.(x1 - x0).|| by A3, A5, A6, A9;
then |.((f /. x1) - (f /. x0)).| < g by A11, XXREAL_0:2;
then |.(((f | X) /. x1) - (f /. x0)).| < g by A9, PARTFUN2:15;
hence |.(((f | X) /. x1) - ((f | X) /. x0)).| < g by A5, A6, PARTFUN2:15; ::_thesis: verum
end;
( 0 < r " & s9 = g * (r ") ) by A2, XCMPLX_0:def_9, XREAL_1:122;
hence ( 0 < s9 & ( for x1 being Point of RNS st x1 in dom (f | X) & ||.(x1 - x0).|| < s9 holds
|.(((f | X) /. x1) - ((f | X) /. x0)).| < g ) ) by A7, A8, XREAL_1:129; ::_thesis: verum
end;
hence f | X is_continuous_in x0 by A5, A6, Th13; ::_thesis: verum
end;
hence f is_continuous_on X by A4, Def16; ::_thesis: verum
end;
theorem :: NCFCONT1:122
for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st ex r being Point of CNS2 st rng f = {r} holds
f is_continuous_on dom f
proof
let CNS1, CNS2 be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS1,CNS2 st ex r being Point of CNS2 st rng f = {r} holds
f is_continuous_on dom f
let f be PartFunc of CNS1,CNS2; ::_thesis: ( ex r being Point of CNS2 st rng f = {r} implies f is_continuous_on dom f )
given r being Point of CNS2 such that A1: rng f = {r} ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_CNS1_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS1; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
f . x2 in rng f by A3, FUNCT_1:def_3;
then f /. x2 in rng f by A3, PARTFUN1:def_6;
then A4: f /. x2 = r by A1, TARSKI:def_1;
f . x1 in rng f by A2, FUNCT_1:def_3;
then f /. x1 in rng f by A2, PARTFUN1:def_6;
then f /. x1 = r by A1, TARSKI:def_1;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. CNS2).|| by A4, RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by CLVECT_1:105; ::_thesis: verum
end;
then f is_Lipschitzian_on dom f by Def17;
hence f is_continuous_on dom f by Th116; ::_thesis: verum
end;
theorem :: NCFCONT1:123
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st ex r being Point of RNS st rng f = {r} holds
f is_continuous_on dom f
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of CNS,RNS st ex r being Point of RNS st rng f = {r} holds
f is_continuous_on dom f
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of CNS,RNS st ex r being Point of RNS st rng f = {r} holds
f is_continuous_on dom f
let f be PartFunc of CNS,RNS; ::_thesis: ( ex r being Point of RNS st rng f = {r} implies f is_continuous_on dom f )
given r being Point of RNS such that A1: rng f = {r} ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_CNS_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
f . x2 in rng f by A3, FUNCT_1:def_3;
then f /. x2 in rng f by A3, PARTFUN1:def_6;
then A4: f /. x2 = r by A1, TARSKI:def_1;
f . x1 in rng f by A2, FUNCT_1:def_3;
then f /. x1 in rng f by A2, PARTFUN1:def_6;
then f /. x1 = r by A1, TARSKI:def_1;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. RNS).|| by A4, RLVECT_1:15
.= 0 by NORMSP_1:1 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by CLVECT_1:105; ::_thesis: verum
end;
then f is_Lipschitzian_on dom f by Def18;
hence f is_continuous_on dom f by Th117; ::_thesis: verum
end;
theorem :: NCFCONT1:124
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st ex r being Point of CNS st rng f = {r} holds
f is_continuous_on dom f
proof
let CNS be ComplexNormSpace; ::_thesis: for RNS being RealNormSpace
for f being PartFunc of RNS,CNS st ex r being Point of CNS st rng f = {r} holds
f is_continuous_on dom f
let RNS be RealNormSpace; ::_thesis: for f being PartFunc of RNS,CNS st ex r being Point of CNS st rng f = {r} holds
f is_continuous_on dom f
let f be PartFunc of RNS,CNS; ::_thesis: ( ex r being Point of CNS st rng f = {r} implies f is_continuous_on dom f )
given r being Point of CNS such that A1: rng f = {r} ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_RNS_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of RNS; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
f . x2 in rng f by A3, FUNCT_1:def_3;
then f /. x2 in rng f by A3, PARTFUN1:def_6;
then A4: f /. x2 = r by A1, TARSKI:def_1;
f . x1 in rng f by A2, FUNCT_1:def_3;
then f /. x1 in rng f by A2, PARTFUN1:def_6;
then f /. x1 = r by A1, TARSKI:def_1;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. CNS).|| by A4, RLVECT_1:15
.= 0 by CLVECT_1:102 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by NORMSP_1:4; ::_thesis: verum
end;
then f is_Lipschitzian_on dom f by Def19;
hence f is_continuous_on dom f by Th118; ::_thesis: verum
end;
theorem :: NCFCONT1:125
for CNS1, CNS2 being ComplexNormSpace
for X being set
for f being PartFunc of CNS1,CNS2 st X c= dom f & f | X is V20() holds
f is_continuous_on X by Th112, Th116;
theorem :: NCFCONT1:126
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of CNS,RNS st X c= dom f & f | X is V20() holds
f is_continuous_on X by Th113, Th117;
theorem :: NCFCONT1:127
for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is V20() holds
f is_continuous_on X by Th114, Th118;
theorem Th128: :: NCFCONT1:128
for CNS being ComplexNormSpace
for f being PartFunc of CNS,CNS st ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = x0 ) holds
f is_continuous_on dom f
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS,CNS st ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = x0 ) holds
f is_continuous_on dom f
let f be PartFunc of CNS,CNS; ::_thesis: ( ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = x0 ) implies f is_continuous_on dom f )
assume A1: for x0 being Point of CNS st x0 in dom f holds
f /. x0 = x0 ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_CNS_st_x1_in_dom_f_&_x2_in_dom_f_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A2: x1 in dom f and
A3: x2 in dom f ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
f /. x1 = x1 by A1, A2;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by A1, A3; ::_thesis: verum
end;
then f is_Lipschitzian_on dom f by Def17;
hence f is_continuous_on dom f by Th116; ::_thesis: verum
end;
theorem :: NCFCONT1:129
for CNS being ComplexNormSpace
for f being PartFunc of CNS,CNS st f = id (dom f) holds
f is_continuous_on dom f
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS,CNS st f = id (dom f) holds
f is_continuous_on dom f
let f be PartFunc of CNS,CNS; ::_thesis: ( f = id (dom f) implies f is_continuous_on dom f )
assume A1: f = id (dom f) ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_dom_f_holds_
f_/._x0_=_x0
let x0 be Point of CNS; ::_thesis: ( x0 in dom f implies f /. x0 = x0 )
assume A2: x0 in dom f ; ::_thesis: f /. x0 = x0
thus f /. x0 = f . x0 by A2, PARTFUN1:def_6
.= x0 by A1, A2, FUNCT_1:17 ; ::_thesis: verum
end;
hence f is_continuous_on dom f by Th128; ::_thesis: verum
end;
theorem :: NCFCONT1:130
for CNS being ComplexNormSpace
for f being PartFunc of CNS,CNS
for Y being Subset of CNS st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of CNS,CNS
for Y being Subset of CNS st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
let f be PartFunc of CNS,CNS; ::_thesis: for Y being Subset of CNS st Y c= dom f & f | Y = id Y holds
f is_continuous_on Y
let Y be Subset of CNS; ::_thesis: ( Y c= dom f & f | Y = id Y implies f is_continuous_on Y )
assume that
A1: Y c= dom f and
A2: f | Y = id Y ; ::_thesis: f is_continuous_on Y
now__::_thesis:_for_x1,_x2_being_Point_of_CNS_st_x1_in_Y_&_x2_in_Y_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in Y & x2 in Y implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume that
A3: x1 in Y and
A4: x2 in Y ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
x1 in (dom f) /\ Y by A1, A3, XBOOLE_0:def_4;
then A5: x1 in dom (f | Y) by RELAT_1:61;
(f | Y) . x1 = x1 by A2, A3, FUNCT_1:17;
then f . x1 = x1 by A5, FUNCT_1:47;
then A6: f /. x1 = x1 by A1, A3, PARTFUN1:def_6;
x2 in (dom f) /\ Y by A1, A4, XBOOLE_0:def_4;
then A7: x2 in dom (f | Y) by RELAT_1:61;
(f | Y) . x2 = x2 by A2, A4, FUNCT_1:17;
then f . x2 = x2 by A7, FUNCT_1:47;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by A1, A4, A6, PARTFUN1:def_6; ::_thesis: verum
end;
then f is_Lipschitzian_on Y by A1, Def17;
hence f is_continuous_on Y by Th116; ::_thesis: verum
end;
theorem :: NCFCONT1:131
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of CNS,CNS
for z being Complex
for p being Point of CNS st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of CNS,CNS
for z being Complex
for p being Point of CNS st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of CNS,CNS
for z being Complex
for p being Point of CNS st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) holds
f is_continuous_on X
let f be PartFunc of CNS,CNS; ::_thesis: for z being Complex
for p being Point of CNS st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) holds
f is_continuous_on X
let z be Complex; ::_thesis: for p being Point of CNS st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) holds
f is_continuous_on X
let p be Point of CNS; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ) implies f is_continuous_on X )
assume that
A1: X c= dom f and
A2: for x0 being Point of CNS st x0 in X holds
f /. x0 = (z * x0) + p ; ::_thesis: f is_continuous_on X
now__::_thesis:_(_0_<_|.z.|_+_1_&_(_for_x1,_x2_being_Point_of_CNS_st_x1_in_X_&_x2_in_X_holds_
||.((f_/._x1)_-_(f_/._x2)).||_<=_(|.z.|_+_1)_*_||.(x1_-_x2).||_)_)
0 + 0 < |.z.| + 1 by COMPLEX1:46, XREAL_1:8;
hence 0 < |.z.| + 1 ; ::_thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= (|.z.| + 1) * ||.(x1 - x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= (|.z.| + 1) * ||.(x1 - x2).|| )
assume ( x1 in X & x2 in X ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= (|.z.| + 1) * ||.(x1 - x2).||
then ( f /. x1 = (z * x1) + p & f /. x2 = (z * x2) + p ) by A2;
then A3: ||.((f /. x1) - (f /. x2)).|| = ||.((z * x1) + (p - (p + (z * x2)))).|| by RLVECT_1:28
.= ||.((z * x1) + ((p - p) - (z * x2))).|| by RLVECT_1:27
.= ||.((z * x1) + ((0. CNS) - (z * x2))).|| by RLVECT_1:15
.= ||.((z * x1) + (- (z * x2))).|| by RLVECT_1:14
.= ||.((z * x1) - (z * x2)).|| by RLVECT_1:def_11
.= ||.(z * (x1 - x2)).|| by CLVECT_1:9
.= |.z.| * ||.(x1 - x2).|| by CLVECT_1:def_13 ;
0 <= ||.(x1 - x2).|| by CLVECT_1:105;
then ||.((f /. x1) - (f /. x2)).|| + 0 <= (|.z.| * ||.(x1 - x2).||) + (1 * ||.(x1 - x2).||) by A3, XREAL_1:7;
hence ||.((f /. x1) - (f /. x2)).|| <= (|.z.| + 1) * ||.(x1 - x2).|| ; ::_thesis: verum
end;
then f is_Lipschitzian_on X by A1, Def17;
hence f is_continuous_on X by Th116; ::_thesis: verum
end;
theorem Th132: :: NCFCONT1:132
for CNS being ComplexNormSpace
for f being PartFunc of the carrier of CNS,REAL st ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on dom f
proof
let CNS be ComplexNormSpace; ::_thesis: for f being PartFunc of the carrier of CNS,REAL st ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on dom f
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( ( for x0 being Point of CNS st x0 in dom f holds
f /. x0 = ||.x0.|| ) implies f is_continuous_on dom f )
assume A1: for x0 being Point of CNS st x0 in dom f holds
f /. x0 = ||.x0.|| ; ::_thesis: f is_continuous_on dom f
now__::_thesis:_for_x1,_x2_being_Point_of_CNS_st_x1_in_dom_f_&_x2_in_dom_f_holds_
abs_((f_/._x1)_-_(f_/._x2))_<=_1_*_||.(x1_-_x2).||
let x1, x2 be Point of CNS; ::_thesis: ( x1 in dom f & x2 in dom f implies abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).|| )
||.x2.|| - ||.x1.|| <= ||.(x2 - x1).|| by CLVECT_1:109;
then ||.x2.|| - ||.x1.|| <= ||.(x1 - x2).|| by CLVECT_1:108;
then A2: ( ||.x1.|| - ||.x2.|| <= ||.(x1 - x2).|| & - (- (||.x1.|| - ||.x2.||)) >= - ||.(x1 - x2).|| ) by CLVECT_1:109, XREAL_1:24;
assume ( x1 in dom f & x2 in dom f ) ; ::_thesis: abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).||
then ( f /. x1 = ||.x1.|| & f /. x2 = ||.x2.|| ) by A1;
hence abs ((f /. x1) - (f /. x2)) <= 1 * ||.(x1 - x2).|| by A2, ABSVALUE:5; ::_thesis: verum
end;
then f is_Lipschitzian_on dom f by Def21;
hence f is_continuous_on dom f by Th120; ::_thesis: verum
end;
theorem :: NCFCONT1:133
for CNS being ComplexNormSpace
for X being set
for f being PartFunc of the carrier of CNS,REAL st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
proof
let CNS be ComplexNormSpace; ::_thesis: for X being set
for f being PartFunc of the carrier of CNS,REAL st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
let X be set ; ::_thesis: for f being PartFunc of the carrier of CNS,REAL st X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = ||.x0.|| ) holds
f is_continuous_on X
let f be PartFunc of the carrier of CNS,REAL; ::_thesis: ( X c= dom f & ( for x0 being Point of CNS st x0 in X holds
f /. x0 = ||.x0.|| ) implies f is_continuous_on X )
assume that
A1: X c= dom f and
A2: for x0 being Point of CNS st x0 in X holds
f /. x0 = ||.x0.|| ; ::_thesis: f is_continuous_on X
X = (dom f) /\ X by A1, XBOOLE_1:28;
then A3: X = dom (f | X) by RELAT_1:61;
now__::_thesis:_for_x0_being_Point_of_CNS_st_x0_in_dom_(f_|_X)_holds_
(f_|_X)_/._x0_=_||.x0.||
let x0 be Point of CNS; ::_thesis: ( x0 in dom (f | X) implies (f | X) /. x0 = ||.x0.|| )
assume A4: x0 in dom (f | X) ; ::_thesis: (f | X) /. x0 = ||.x0.||
hence (f | X) /. x0 = f /. x0 by PARTFUN2:15
.= ||.x0.|| by A2, A3, A4 ;
::_thesis: verum
end;
then f | X is_continuous_on X by A3, Th132;
hence f is_continuous_on X by Th54; ::_thesis: verum
end;