Article FDIFF_1, MML version 4.99.1005
:: FDIFF_1:th 1
theorem
for b1 being Element of bool REAL holds
for b2 being Element of REAL holds
b2 in b1
iff
b2 in REAL
iff
b1 = REAL;
:: FDIFF_1:attrnot 1 => FDIFF_1:attr 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is convergent_to_0 means
a1 is non-empty & a1 is convergent & lim a1 = 0;
end;
:: FDIFF_1:dfs 1
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is convergent_to_0
it is sufficient to prove
thus a1 is non-empty & a1 is convergent & lim a1 = 0;
:: FDIFF_1:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is convergent_to_0
iff
b1 is non-empty & b1 is convergent & lim b1 = 0;
:: FDIFF_1:exreg 1
registration
cluster Relation-like Function-like non empty total complex-valued ext-real-valued real-valued quasi_total convergent_to_0 Relation of NAT,REAL;
end;
:: FDIFF_1:exreg 2
registration
cluster Relation-like Function-like constant non empty total complex-valued ext-real-valued real-valued quasi_total Relation of NAT,REAL;
end;
:: FDIFF_1:condreg 1
registration
let a1 be Function-like constant quasi_total Relation of NAT,REAL;
cluster -> constant (subsequence of a1);
end;
:: FDIFF_1:attrnot 2 => FDIFF_1:attr 2
definition
let a1 be Function-like Relation of REAL,REAL;
attr a1 is REST-like means
a1 is total(REAL, REAL) &
(for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
b1 " (#) (a1 * b1) is convergent &
lim (b1 " (#) (a1 * b1)) = 0);
end;
:: FDIFF_1:dfs 2
definiens
let a1 be Function-like Relation of REAL,REAL;
To prove
a1 is REST-like
it is sufficient to prove
thus a1 is total(REAL, REAL) &
(for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
b1 " (#) (a1 * b1) is convergent &
lim (b1 " (#) (a1 * b1)) = 0);
:: FDIFF_1:def 3
theorem
for b1 being Function-like Relation of REAL,REAL holds
b1 is REST-like
iff
b1 is total(REAL, REAL) &
(for b2 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL holds
b2 " (#) (b1 * b2) is convergent &
lim (b2 " (#) (b1 * b2)) = 0);
:: FDIFF_1:exreg 3
registration
cluster Relation-like Function-like complex-valued ext-real-valued real-valued REST-like Relation of REAL,REAL;
end;
:: FDIFF_1:modenot 1
definition
mode REST is Function-like REST-like Relation of REAL,REAL;
end;
:: FDIFF_1:attrnot 3 => FDIFF_1:attr 3
definition
let a1 be Function-like Relation of REAL,REAL;
attr a1 is linear means
a1 is total(REAL, REAL) &
(ex b1 being Element of REAL st
for b2 being Element of REAL holds
a1 . b2 = b1 * b2);
end;
:: FDIFF_1:dfs 3
definiens
let a1 be Function-like Relation of REAL,REAL;
To prove
a1 is linear
it is sufficient to prove
thus a1 is total(REAL, REAL) &
(ex b1 being Element of REAL st
for b2 being Element of REAL holds
a1 . b2 = b1 * b2);
:: FDIFF_1:def 4
theorem
for b1 being Function-like Relation of REAL,REAL holds
b1 is linear
iff
b1 is total(REAL, REAL) &
(ex b2 being Element of REAL st
for b3 being Element of REAL holds
b1 . b3 = b2 * b3);
:: FDIFF_1:exreg 4
registration
cluster Relation-like Function-like complex-valued ext-real-valued real-valued linear Relation of REAL,REAL;
end;
:: FDIFF_1:modenot 2
definition
mode LINEAR is Function-like linear Relation of REAL,REAL;
end;
:: FDIFF_1:th 6
theorem
for b1, b2 being Function-like linear Relation of REAL,REAL holds
b1 + b2 is Function-like linear Relation of REAL,REAL &
b1 - b2 is Function-like linear Relation of REAL,REAL;
:: FDIFF_1:th 7
theorem
for b1 being Element of REAL
for b2 being Function-like linear Relation of REAL,REAL holds
b1 (#) b2 is Function-like linear Relation of REAL,REAL;
:: FDIFF_1:th 8
theorem
for b1, b2 being Function-like REST-like Relation of REAL,REAL holds
b1 + b2 is Function-like REST-like Relation of REAL,REAL &
b1 - b2 is Function-like REST-like Relation of REAL,REAL &
b1 (#) b2 is Function-like REST-like Relation of REAL,REAL;
:: FDIFF_1:th 9
theorem
for b1 being Element of REAL
for b2 being Function-like REST-like Relation of REAL,REAL holds
b1 (#) b2 is Function-like REST-like Relation of REAL,REAL;
:: FDIFF_1:th 10
theorem
for b1, b2 being Function-like linear Relation of REAL,REAL holds
b1 (#) b2 is REST-like;
:: FDIFF_1:th 11
theorem
for b1 being Function-like REST-like Relation of REAL,REAL
for b2 being Function-like linear Relation of REAL,REAL holds
b1 (#) b2 is Function-like REST-like Relation of REAL,REAL &
b2 (#) b1 is Function-like REST-like Relation of REAL,REAL;
:: FDIFF_1:prednot 1 => FDIFF_1:pred 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
pred A1 is_differentiable_in A2 means
ex b1 being Neighbourhood of a2 st
b1 c= dom a1 &
(ex b2 being Function-like linear Relation of REAL,REAL st
ex b3 being Function-like REST-like Relation of REAL,REAL st
for b4 being Element of REAL
st b4 in b1
holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2)));
end;
:: FDIFF_1:dfs 4
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
To prove
a1 is_differentiable_in a2
it is sufficient to prove
thus ex b1 being Neighbourhood of a2 st
b1 c= dom a1 &
(ex b2 being Function-like linear Relation of REAL,REAL st
ex b3 being Function-like REST-like Relation of REAL,REAL st
for b4 being Element of REAL
st b4 in b1
holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2)));
:: FDIFF_1:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_differentiable_in b2
iff
ex b3 being Neighbourhood of b2 st
b3 c= dom b1 &
(ex b4 being Function-like linear Relation of REAL,REAL st
ex b5 being Function-like REST-like Relation of REAL,REAL st
for b6 being Element of REAL
st b6 in b3
holds (b1 . b6) - (b1 . b2) = (b4 . (b6 - b2)) + (b5 . (b6 - b2)));
:: FDIFF_1:funcnot 1 => FDIFF_1:func 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
assume a1 is_differentiable_in a2;
func diff(A1,A2) -> Element of REAL means
ex b1 being Neighbourhood of a2 st
b1 c= dom a1 &
(ex b2 being Function-like linear Relation of REAL,REAL st
ex b3 being Function-like REST-like Relation of REAL,REAL st
it = b2 . 1 &
(for b4 being Element of REAL
st b4 in b1
holds (a1 . b4) - (a1 . a2) = (b2 . (b4 - a2)) + (b3 . (b4 - a2))));
end;
:: FDIFF_1:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
st b1 is_differentiable_in b2
for b3 being Element of REAL holds
b3 = diff(b1,b2)
iff
ex b4 being Neighbourhood of b2 st
b4 c= dom b1 &
(ex b5 being Function-like linear Relation of REAL,REAL st
ex b6 being Function-like REST-like Relation of REAL,REAL st
b3 = b5 . 1 &
(for b7 being Element of REAL
st b7 in b4
holds (b1 . b7) - (b1 . b2) = (b5 . (b7 - b2)) + (b6 . (b7 - b2))));
:: FDIFF_1:prednot 2 => FDIFF_1:pred 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_differentiable_on A2 means
a2 c= dom a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_differentiable_in b1);
end;
:: FDIFF_1:dfs 6
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_differentiable_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_differentiable_in b1);
:: FDIFF_1:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_differentiable_on b2
iff
b2 c= dom b1 &
(for b3 being Element of REAL
st b3 in b2
holds b1 | b2 is_differentiable_in b3);
:: FDIFF_1:th 15
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_on b1
holds b1 is Element of bool REAL;
:: FDIFF_1:th 16
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_differentiable_on b1
iff
b1 c= dom b2 &
(for b3 being Element of REAL
st b3 in b1
holds b2 is_differentiable_in b3);
:: FDIFF_1:th 17
theorem
for b1 being Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_on b1
holds b1 is open;
:: FDIFF_1:funcnot 2 => FDIFF_1:func 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
assume a1 is_differentiable_on a2;
func A1 `| A2 -> Function-like Relation of REAL,REAL means
dom it = a2 &
(for b1 being Element of REAL
st b1 in a2
holds it . b1 = diff(a1,b1));
end;
:: FDIFF_1:def 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
st b1 is_differentiable_on b2
for b3 being Function-like Relation of REAL,REAL holds
b3 = b1 `| b2
iff
dom b3 = b2 &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = diff(b1,b4));
:: FDIFF_1:th 19
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 &
(ex b3 being Element of REAL st
rng b2 = {b3})
holds b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (b2 `| b1) . b3 = 0);
:: FDIFF_1:funcreg 1
registration
let a1 be Function-like quasi_total convergent_to_0 Relation of NAT,REAL;
let a2 be Element of NAT;
cluster a1 ^\ a2 -> Function-like quasi_total convergent_to_0;
end;
:: FDIFF_1:funcreg 2
registration
let a1 be Function-like constant quasi_total Relation of NAT,REAL;
let a2 be Element of NAT;
cluster a1 ^\ a2 -> Function-like constant quasi_total;
end;
:: FDIFF_1:th 20
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
for b3 being Neighbourhood of b2
st b1 is_differentiable_in b2 & b3 c= dom b1
for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b5 being Function-like constant quasi_total Relation of NAT,REAL
st rng b5 = {b2} & rng (b4 + b5) c= b3
holds b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)) is convergent &
diff(b1,b2) = lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)));
:: FDIFF_1:th 21
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
holds b2 + b3 is_differentiable_in b1 &
diff(b2 + b3,b1) = (diff(b2,b1)) + diff(b3,b1);
:: FDIFF_1:th 22
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
holds b2 - b3 is_differentiable_in b1 &
diff(b2 - b3,b1) = (diff(b2,b1)) - diff(b3,b1);
:: FDIFF_1:th 23
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_differentiable_in b1
holds b2 (#) b3 is_differentiable_in b1 &
diff(b2 (#) b3,b1) = b2 * diff(b3,b1);
:: FDIFF_1:th 24
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1 & b3 is_differentiable_in b1
holds b2 (#) b3 is_differentiable_in b1 &
diff(b2 (#) b3,b1) = ((b3 . b1) * diff(b2,b1)) + ((b2 . b1) * diff(b3,b1));
:: FDIFF_1:th 25
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 & b2 | b1 = id b1
holds b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (b2 `| b1) . b3 = 1);
:: FDIFF_1:th 26
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= dom (b2 + b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
holds b2 + b3 is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds ((b2 + b3) `| b1) . b4 = (diff(b2,b4)) + diff(b3,b4));
:: FDIFF_1:th 27
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= dom (b2 - b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
holds b2 - b3 is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds ((b2 - b3) `| b1) . b4 = (diff(b2,b4)) - diff(b3,b4));
:: FDIFF_1:th 28
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= dom (b1 (#) b3) & b3 is_differentiable_on b2
holds b1 (#) b3 is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds ((b1 (#) b3) `| b2) . b4 = b1 * diff(b3,b4));
:: FDIFF_1:th 29
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= dom (b2 (#) b3) & b2 is_differentiable_on b1 & b3 is_differentiable_on b1
holds b2 (#) b3 is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds ((b2 (#) b3) `| b1) . b4 = ((b3 . b4) * diff(b2,b4)) + ((b2 . b4) * diff(b3,b4)));
:: FDIFF_1:th 30
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2 & b2 is_constant_on b1
holds b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (b2 `| b1) . b3 = 0);
:: FDIFF_1:th 31
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
st b3 c= dom b4 &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds b4 is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds (b4 `| b3) . b5 = b1);
:: FDIFF_1:th 32
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set
st b1 is_differentiable_in b2
holds b1 is_continuous_in b2;
:: FDIFF_1:th 33
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_on b1
holds b2 is_continuous_on b1;
:: FDIFF_1:th 34
theorem
for b1 being set
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_differentiable_on b1 & b2 c= b1
holds b3 is_differentiable_on b2;
:: FDIFF_1:th 35
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1
holds ex b3 being Function-like REST-like Relation of REAL,REAL st
b3 . 0 = 0 & b3 is_continuous_in 0;
:: FDIFF_1:attrnot 4 => FDIFF_1:attr 4
definition
let a1 be Function-like Relation of REAL,REAL;
attr a1 is differentiable means
a1 is_differentiable_on dom a1;
end;
:: FDIFF_1:dfs 8
definiens
let a1 be Function-like Relation of REAL,REAL;
To prove
a1 is differentiable
it is sufficient to prove
thus a1 is_differentiable_on dom a1;
:: FDIFF_1:def 9
theorem
for b1 being Function-like Relation of REAL,REAL holds
b1 is differentiable
iff
b1 is_differentiable_on dom b1;
:: FDIFF_1:exreg 5
registration
cluster Relation-like Function-like non empty total complex-valued ext-real-valued real-valued quasi_total differentiable Relation of REAL,REAL;
end;
:: FDIFF_1:th 36
theorem
for b1 being open Element of bool REAL
for b2 being Function-like differentiable Relation of REAL,REAL
st b1 c= dom b2
holds b2 is_differentiable_on b1;