:: NDIFF_1 semantic presentation
theorem Th1: :: NDIFF_1:1
theorem Th2: :: NDIFF_1:2
theorem Th3: :: NDIFF_1:3
theorem Th4: :: NDIFF_1:4
theorem Th5: :: NDIFF_1:5
:: deftheorem Def1 defines being_not_0 NDIFF_1:def 1 :
theorem Th6: :: NDIFF_1:6
theorem Th7: :: NDIFF_1:7
:: deftheorem Def2 defines (#) NDIFF_1:def 2 :
:: deftheorem Def3 defines * NDIFF_1:def 3 :
theorem Th8: :: NDIFF_1:8
theorem Th9: :: NDIFF_1:9
theorem Th10: :: NDIFF_1:10
theorem Th11: :: NDIFF_1:11
theorem Th12: :: NDIFF_1:12
theorem Th13: :: NDIFF_1:13
theorem Th14: :: NDIFF_1:14
theorem Th15: :: NDIFF_1:15
theorem Th16: :: NDIFF_1:16
theorem Th17: :: NDIFF_1:17
theorem Th18: :: NDIFF_1:18
theorem Th19: :: NDIFF_1:19
theorem Th20: :: NDIFF_1:20
:: deftheorem Def4 defines convergent_to_0 NDIFF_1:def 4 :
theorem Th21: :: NDIFF_1:21
theorem Th22: :: NDIFF_1:22
theorem Th23: :: NDIFF_1:23
theorem Th24: :: NDIFF_1:24
theorem Th25: :: NDIFF_1:25
:: deftheorem Def5 defines REST-like NDIFF_1:def 5 :
theorem Th26: :: NDIFF_1:26
theorem Th27: :: NDIFF_1:27
theorem Th28: :: NDIFF_1:28
theorem Th29: :: NDIFF_1:29
theorem Th30: :: NDIFF_1:30
theorem Th31: :: NDIFF_1:31
theorem Th32: :: NDIFF_1:32
theorem Th33: :: NDIFF_1:33
theorem Th34: :: NDIFF_1:34
:: deftheorem Def6 defines is_differentiable_in NDIFF_1:def 6 :
definition
let S be non
trivial RealNormSpace;
let T be non
trivial RealNormSpace;
let f be
PartFunc of
S,
T;
let x0 be
Point of
S;
assume E43:
f is_differentiable_in x0
;
func diff c3,
c4 -> Point of
(R_NormSpace_of_BoundedLinearOperators a1,a2) means :
Def7:
:: NDIFF_1:def 7
ex
N being
Neighbourhood of
x0 st
(
N c= dom f & ex
R being
REST of
S,
T st
for
x being
Point of
S st
x in N holds
(f /. x) - (f /. x0) = (it . (x - x0)) + (R /. (x - x0)) );
existence
ex b1 being Point of (R_NormSpace_of_BoundedLinearOperators S,T) ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b1 . (x - x0)) + (R /. (x - x0)) )
uniqueness
for b1, b2 being Point of (R_NormSpace_of_BoundedLinearOperators S,T) st ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b1 . (x - x0)) + (R /. (x - x0)) ) & ex N being Neighbourhood of x0 st
( N c= dom f & ex R being REST of S,T st
for x being Point of S st x in N holds
(f /. x) - (f /. x0) = (b2 . (x - x0)) + (R /. (x - x0)) ) holds
b1 = b2
end;
:: deftheorem Def7 defines diff NDIFF_1:def 7 :
:: deftheorem Def8 defines is_differentiable_on NDIFF_1:def 8 :
theorem Th35: :: NDIFF_1:35
theorem Th36: :: NDIFF_1:36
theorem Th37: :: NDIFF_1:37
definition
let S be non
trivial RealNormSpace;
let T be non
trivial RealNormSpace;
let f be
PartFunc of
S,
T;
let X be
set ;
assume E43:
f is_differentiable_on X
;
func c3 `| c4 -> PartFunc of
a1,
(R_NormSpace_of_BoundedLinearOperators a1,a2) means :
Def9:
:: NDIFF_1:def 9
(
dom it = X & ( for
x being
Point of
S st
x in X holds
it /. x = diff f,
x ) );
existence
ex b1 being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators S,T) st
( dom b1 = X & ( for x being Point of S st x in X holds
b1 /. x = diff f,x ) )
uniqueness
for b1, b2 being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators S,T) st dom b1 = X & ( for x being Point of S st x in X holds
b1 /. x = diff f,x ) & dom b2 = X & ( for x being Point of S st x in X holds
b2 /. x = diff f,x ) holds
b1 = b2
end;
:: deftheorem Def9 defines `| NDIFF_1:def 9 :
theorem Th38: :: NDIFF_1:38
theorem Th39: :: NDIFF_1:39
theorem Th40: :: NDIFF_1:40
theorem Th41: :: NDIFF_1:41
theorem Th42: :: NDIFF_1:42
theorem Th43: :: NDIFF_1:43
theorem Th44: :: NDIFF_1:44
theorem Th45: :: NDIFF_1:45
theorem Th46: :: NDIFF_1:46
theorem Th47: :: NDIFF_1:47
theorem Th48: :: NDIFF_1:48
theorem Th49: :: NDIFF_1:49
theorem Th50: :: NDIFF_1:50
theorem Th51: :: NDIFF_1:51
theorem Th52: :: NDIFF_1:52