:: Differentiable Functions into Real Normed Spaces
:: by Hiroyuki Okazaki , Noboru Endou , Keiko Narita and Yasunari Shidama
::
:: Received October 13, 2010
:: Copyright (c) 2010-2011 Association of Mizar Users

begin

reconsider cs = NAT --> 0 as Real_Sequence by FUNCOP_1:57;

theorem Th1: :: NDIFF_3:1

for G being RealNormSpace
for s1 being Real_Sequence
for seq being sequence of G st ( for n being Element of NAT holds ||.(seq . n).|| <= s1 . n ) & s1 is convergent & lim s1 = 0 holds
( seq is convergent & lim seq = 0. G )

proof end;

theorem Th2: :: NDIFF_3:2

for G being RealNormSpace
for k being Element of NAT
for s1 being Real_Sequence
for seq being sequence of G holds (s1 ^\ k) (#) (seq ^\ k) = (s1 (#) seq) ^\ k

proof end;

definition

let F be non trivial RealNormSpace;
let IT be PartFunc of REAL, the carrier of F;
attr IT is REST-like means :Def1: :: NDIFF_3:def 1
( IT is total & ( for h being convergent_to_0 Real_Sequence holds
( (h ") (#) (IT /* h) is convergent & lim ((h ") (#) (IT /* h)) = 0. F ) ) );

end;

:: deftheorem Def1 defines REST-like NDIFF_3:def 1 :
for F being non trivial RealNormSpace
for IT being PartFunc of REAL, the carrier of F holds
( IT is REST-like iff ( IT is total & ( for h being convergent_to_0 Real_Sequence holds
( (h ") (#) (IT /* h) is convergent & lim ((h ") (#) (IT /* h)) = 0. F ) ) ) );

registration

let F be non trivial RealNormSpace;
cluster V13() V16( REAL ) V17( the carrier of F) Function-like REST-like Element of K6(K7(REAL, the carrier of F));
existence
ex b₁ being PartFunc of REAL, the carrier of F st b₁ is REST-like

proof end;

end;

definition

let F be non trivial RealNormSpace;
mode REST of F is REST-like PartFunc of REAL, the carrier of F;

end;

definition

let F be non trivial RealNormSpace;
let IT be Function of REAL, the carrier of F;
attr IT is linear means :Def2: :: NDIFF_3:def 2
ex r being Point of F st
for p being Real holds IT . p = p * r;

end;

:: deftheorem Def2 defines linear NDIFF_3:def 2 :
for F being non trivial RealNormSpace
for IT being Function of REAL, the carrier of F holds
( IT is linear iff ex r being Point of F st
for p being Real holds IT . p = p * r );

registration

let F be non trivial RealNormSpace;
cluster V1() V13() V16( REAL ) V17( the carrier of F) Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of F));
existence
ex b₁ being Function of REAL, the carrier of F st b₁ is linear

proof end;

end;

definition

let F be non trivial RealNormSpace;
mode LINEAR of F is linear Function of REAL, the carrier of F;

end;

theorem Th3: :: NDIFF_3:3

for F being non trivial RealNormSpace
for L1, L2 being LINEAR of F holds
( L1 + L2 is LINEAR of F & L1 - L2 is LINEAR of F )

proof end;

theorem Th4: :: NDIFF_3:4

for F being non trivial RealNormSpace
for r being Real
for L being LINEAR of F holds r (#) L is LINEAR of F

proof end;

theorem Th5: :: NDIFF_3:5

for F being non trivial RealNormSpace
for h1, h2 being PartFunc of REAL, the carrier of F
for seq being Real_Sequence st rng seq c= (dom h1) /\ (dom h2) holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )

proof end;

theorem Th6: :: NDIFF_3:6

for F being non trivial RealNormSpace
for h1, h2 being PartFunc of REAL, the carrier of F
for seq being Real_Sequence st h1 is total & h2 is total holds
( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) )

proof end;

theorem Th7: :: NDIFF_3:7

for F being non trivial RealNormSpace
for R1, R2 being REST of F holds
( R1 + R2 is REST of F & R1 - R2 is REST of F )

proof end;

theorem Th8: :: NDIFF_3:8

for F being non trivial RealNormSpace
for r being Real
for R being REST of F holds r (#) R is REST of F

proof end;

definition

let F be non trivial RealNormSpace;
let f be PartFunc of REAL, the carrier of F;
let x0 be real number ;
pred f is_differentiable_in x0 means :Def3: :: NDIFF_3:def 3
ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) );

end;

:: deftheorem Def3 defines is_differentiable_in NDIFF_3:def 3 :
for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for x0 being real number holds
( f is_differentiable_in x0 iff ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) );

definition

let F be non trivial RealNormSpace;
let f be PartFunc of REAL, the carrier of F;
let x0 be real number ;
assume A1: f is_differentiable_in x0 ;
func diff (f,x0) -> Point of F means :Def4: :: NDIFF_3:def 4
ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
( it = L . 1 & ( for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) ) );
existence
ex b₁ being Point of F ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
( b₁ = L . 1 & ( for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) ) )

proof end;

uniqueness
for b₁, b₂ being Point of F st ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
( b₁ = L . 1 & ( for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) ) ) & ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
( b₂ = L . 1 & ( for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines diff NDIFF_3:def 4 :
for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for x0 being real number st f is_differentiable_in x0 holds
for b₄ being Point of F holds
( b₄ = diff (f,x0) iff ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LINEAR of F ex R being REST of F st
( b₄ = L . 1 & ( for x being Real st x in N holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) ) ) ) );

definition

let F be non trivial RealNormSpace;
let f be PartFunc of REAL, the carrier of F;
let X be set ;
pred f is_differentiable_on X means :Def5: :: NDIFF_3:def 5
( X c= dom f & ( for x being Real st x in X holds
f | X is_differentiable_in x ) );

end;

:: deftheorem Def5 defines is_differentiable_on NDIFF_3:def 5 :
for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for X being set holds
( f is_differentiable_on X iff ( X c= dom f & ( for x being Real st x in X holds
f | X is_differentiable_in x ) ) );

theorem Th9: :: NDIFF_3:9

for F being non trivial RealNormSpace
for X being set
for f being PartFunc of REAL, the carrier of F st f is_differentiable_on X holds
X is Subset of REAL

proof end;

theorem Th10: :: NDIFF_3:10

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F holds
( f is_differentiable_on Z iff ( Z c= dom f & ( for x being Real st x in Z holds
f is_differentiable_in x ) ) )

proof end;

theorem :: NDIFF_3:11

for F being non trivial RealNormSpace
for Y being Subset of REAL
for f being PartFunc of REAL, the carrier of F st f is_differentiable_on Y holds
Y is open

proof end;

definition

let F be non trivial RealNormSpace;
let f be PartFunc of REAL, the carrier of F;
let X be set ;
assume A1: f is_differentiable_on X ;
func f `| X -> PartFunc of REAL, the carrier of F means :Def6: :: NDIFF_3:def 6
( dom it = X & ( for x being Real st x in X holds
it . x = diff (f,x) ) );
existence
ex b₁ being PartFunc of REAL, the carrier of F st
( dom b₁ = X & ( for x being Real st x in X holds
b₁ . x = diff (f,x) ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of REAL, the carrier of F st dom b₁ = X & ( for x being Real st x in X holds
b₁ . x = diff (f,x) ) & dom b₂ = X & ( for x being Real st x in X holds
b₂ . x = diff (f,x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines `| NDIFF_3:def 6 :
for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for X being set st f is_differentiable_on X holds
for b<sub>4</sub> being PartFunc of REAL, the carrier of F holds
( b<sub>4</sub> = f `| X iff ( dom b₄ = X & ( for x being Real st x in X holds
b₄ . x = diff (f,x) ) ) );

theorem :: NDIFF_3:12

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F st Z c= dom f & ex r being Point of F st rng f = {r} holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) /. x = 0. F ) )

proof end;

theorem Th13: :: NDIFF_3:13

for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for x0 being real number
for N being Neighbourhood of x0 st f is_differentiable_in x0 & N c= dom f holds
for h being convergent_to_0 Real_Sequence
for c being V20() Real_Sequence st rng c = {x0} & rng (h + c) c= N holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) )

proof end;

theorem Th14: :: NDIFF_3:14

for F being non trivial RealNormSpace
for x0 being Real
for f1, f2 being PartFunc of REAL, the carrier of F st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 + f2 is_differentiable_in x0 & diff ((f1 + f2),x0) = (diff (f1,x0)) + (diff (f2,x0)) )

proof end;

theorem Th15: :: NDIFF_3:15

for F being non trivial RealNormSpace
for x0 being Real
for f1, f2 being PartFunc of REAL, the carrier of F st f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 - f2 is_differentiable_in x0 & diff ((f1 - f2),x0) = (diff (f1,x0)) - (diff (f2,x0)) )

proof end;

theorem Th16: :: NDIFF_3:16

for F being non trivial RealNormSpace
for x0 being Real
for f being PartFunc of REAL, the carrier of F
for r being Real st f is_differentiable_in x0 holds
( r (#) f is_differentiable_in x0 & diff ((r (#) f),x0) = r * (diff (f,x0)) )

proof end;

theorem :: NDIFF_3:17

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL, the carrier of F st Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) )

proof end;

theorem :: NDIFF_3:18

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL, the carrier of F st Z c= dom (f1 - f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = (diff (f1,x)) - (diff (f2,x)) ) )

proof end;

theorem :: NDIFF_3:19

for F being non trivial RealNormSpace
for r being Real
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F st Z c= dom (r (#) f) & f is_differentiable_on Z holds
( r (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) f) `| Z) . x = r * (diff (f,x)) ) )

proof end;

theorem :: NDIFF_3:20

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F st Z c= dom f & f | Z is constant holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0. F ) )

proof end;

theorem Th21: :: NDIFF_3:21

for F being non trivial RealNormSpace
for r, p being Point of F
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F st Z c= dom f & ( for x being Real st x in Z holds
f /. x = (x * r) + p ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = r ) )

proof end;

theorem Th22: :: NDIFF_3:22

for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F
for x0 being Real st f is_differentiable_in x0 holds
f is_continuous_in x0

proof end;

theorem :: NDIFF_3:23

for F being non trivial RealNormSpace
for X being set
for f being PartFunc of REAL, the carrier of F st f is_differentiable_on X holds
f | X is continuous

proof end;

theorem Th24: :: NDIFF_3:24

for F being non trivial RealNormSpace
for X being set
for Z being open Subset of REAL
for f being PartFunc of REAL, the carrier of F st f is_differentiable_on X & Z c= X holds
f is_differentiable_on Z

proof end;

Lm1: {} REAL is closed

proof end;

theorem :: NDIFF_3:25

for F being non trivial RealNormSpace ex R being REST of F st
( R /. 0 = 0. F & R is_continuous_in 0 )

proof end;

definition

let F be non trivial RealNormSpace;
let f be PartFunc of REAL, the carrier of F;
attr f is differentiable means :Def7: :: NDIFF_3:def 7
f is_differentiable_on dom f;

end;

:: deftheorem Def7 defines differentiable NDIFF_3:def 7 :
for F being non trivial RealNormSpace
for f being PartFunc of REAL, the carrier of F holds
( f is differentiable iff f is_differentiable_on dom f );

Lm2: [#] REAL is open

proof end;

registration

let F be non trivial RealNormSpace;
cluster V1() V13() V16( REAL ) V17( the carrier of F) Function-like total quasi_total differentiable Element of K6(K7(REAL, the carrier of F));
existence
ex b₁ being Function of REAL, the carrier of F st b₁ is differentiable

proof end;

end;

theorem :: NDIFF_3:26

for F being non trivial RealNormSpace
for Z being open Subset of REAL
for f being differentiable PartFunc of REAL, the carrier of F st Z c= dom f holds
f is_differentiable_on Z

proof end;