:: The First Mean Value Theorem for Integrals
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received October 30, 2007
:: Copyright (c) 2007-2021 Association of Mizar Users


theorem Th1: :: MESFUNC7:1
for X being non empty set
for f, g being PartFunc of X,ExtREAL st ( for x being Element of X st x in dom f holds
f . x <= g . x ) holds
g - f is nonnegative
proof end;

theorem Th2: :: MESFUNC7:2
for X being non empty set
for Y being set
for f being PartFunc of X,ExtREAL
for r being Real holds (r (#) f) | Y = r (#) (f | Y)
proof end;

reconsider jj = 1 as Real ;

theorem Th3: :: MESFUNC7:3
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M & g - f is nonnegative holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral (M,(f | E)) <= Integral (M,(g | E)) )
proof end;

registration
let X be non empty set ;
cluster Relation-like X -defined ExtREAL -valued Function-like extreal-yielding nonnegative for Element of K16(K17(X,ExtREAL));
existence
ex b1 being PartFunc of X,ExtREAL st b1 is nonnegative
proof end;
end;

registration
let X be non empty set ;
let f be PartFunc of X,ExtREAL;
cluster |.f.| -> nonnegative for PartFunc of X,ExtREAL;
correctness
coherence
for b1 being PartFunc of X,ExtREAL st b1 = |.f.| holds
b1 is nonnegative ;
proof end;
end;

theorem :: MESFUNC7:4
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
ex F being sequence of S st
( ( for n being Element of NAT holds F . n = (dom f) /\ (great_eq_dom (|.f.|,(1 / (n + 1)))) ) & (dom f) \ (eq_dom (f,0.)) = union (rng F) & ( for n being Element of NAT holds
( F . n in S & M . (F . n) < +infty ) ) )
proof end;

notation
let F be Relation;
synonym extreal-yielding F for ext-real-valued ;
end;

registration
cluster Relation-like omega -defined Function-like V36() FinSequence-like FinSubsequence-like extreal-yielding for set ;
existence
ex b1 being FinSequence st b1 is extreal-yielding
proof end;
end;

definition
func multextreal -> BinOp of ExtREAL means :Def1: :: MESFUNC7:def 1
 for x, y being Element of ExtREAL holds it . (x,y) = x * y;
existence
ex b1 being BinOp of ExtREAL st
for x, y being Element of ExtREAL holds b1 . (x,y) = x * y from BINOP_1:sch 4();
uniqueness
for b1, b2 being BinOp of ExtREAL st ( for x, y being Element of ExtREAL holds b1 . (x,y) = x * y ) & ( for x, y being Element of ExtREAL holds b2 . (x,y) = x * y ) holds
b1 = b2 from BINOP_2:sch 2();
end;

:: deftheorem Def1 defines multextreal MESFUNC7:def 1 :
for b1 being BinOp of ExtREAL holds
( b1 = multextreal iff for x, y being Element of ExtREAL holds b1 . (x,y) = x * y );

registration
cluster multextreal -> commutative associative ;
coherence
( multextreal is commutative & multextreal is associative )
proof end;
end;

Lm1: 1. is_a_unity_wrt multextreal
proof end;

theorem Th5: :: MESFUNC7:5
the_unity_wrt multextreal = 1 by Lm1, BINOP_1:def 8;

registration
cluster multextreal -> having_a_unity ;
coherence
multextreal is having_a_unity by Lm1, Th5, SETWISEO:14;
end;

definition
let F be extreal-yielding FinSequence;
func Product F -> Element of ExtREAL means :Def2: :: MESFUNC7:def 2
 ex f being FinSequence of ExtREAL st
( f = F & it = multextreal $$ f );
existence
ex b1 being Element of ExtREAL ex f being FinSequence of ExtREAL st
( f = F & b1 = multextreal $$ f )
proof end;
uniqueness
for b1, b2 being Element of ExtREAL st ex f being FinSequence of ExtREAL st
( f = F & b1 = multextreal $$ f ) & ex f being FinSequence of ExtREAL st
( f = F & b2 = multextreal $$ f ) holds
b1 = b2 ;
end;

:: deftheorem Def2 defines Product MESFUNC7:def 2 :
for F being extreal-yielding FinSequence
for b2 being Element of ExtREAL holds
( b2 = Product F iff ex f being FinSequence of ExtREAL st
( f = F & b2 = multextreal $$ f ) );

registration
let x be Element of ExtREAL ;
let n be Nat;
cluster n |-> x -> extreal-yielding ;
coherence
n |-> x is extreal-yielding ;
end;

definition
let x be Element of ExtREAL ;
let k be Nat;
func x |^ k -> number equals :: MESFUNC7:def 3
 Product (k |-> x);
coherence
Product (k |-> x) is number ;
end;

:: deftheorem defines |^ MESFUNC7:def 3 :
for x being Element of ExtREAL
for k being Nat holds x |^ k = Product (k |-> x);

definition
let x be Element of ExtREAL ;
let k be Nat;
:: original: |^
redefine func x |^ k -> R_eal;
coherence
x |^ k is R_eal ;
end;

registration
cluster <*> ExtREAL -> extreal-yielding ;
coherence
<*> ExtREAL is extreal-yielding ;
end;

registration
let r be Element of ExtREAL ;
cluster <*r*> -> extreal-yielding ;
coherence
<*r*> is extreal-yielding ;
end;

theorem Th6: :: MESFUNC7:6
Product (<*> ExtREAL) = 1
proof end;

theorem Th7: :: MESFUNC7:7
for r being Element of ExtREAL holds Product <*r*> = r
proof end;

registration
let f, g be extreal-yielding FinSequence;
cluster f ^ g -> extreal-yielding ;
coherence
f ^ g is extreal-yielding
proof end;
end;

theorem Th8: :: MESFUNC7:8
for F being extreal-yielding FinSequence
for r being Element of ExtREAL holds Product (F ^ <*r*>) = (Product F) * r
proof end;

theorem Th9: :: MESFUNC7:9
for x being Element of ExtREAL holds x |^ 1 = x
proof end;

theorem Th10: :: MESFUNC7:10
for x being Element of ExtREAL
for k being Nat holds x |^ (k + 1) = (x |^ k) * x
proof end;

definition
let k be Nat;
let X be non empty set ;
let f be PartFunc of X,ExtREAL;
func f |^ k -> PartFunc of X,ExtREAL means :Def4: :: MESFUNC7:def 4
( dom it = dom f & ( for x being Element of X st x in dom it holds
it . x = (f . x) |^ k ) );
existence
ex b1 being PartFunc of X,ExtREAL st
( dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 . x = (f . x) |^ k ) )
proof end;
uniqueness
for b1, b2 being PartFunc of X,ExtREAL st dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 . x = (f . x) |^ k ) & dom b2 = dom f & ( for x being Element of X st x in dom b2 holds
b2 . x = (f . x) |^ k ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines |^ MESFUNC7:def 4 :
for k being Nat
for X being non empty set
for f, b4 being PartFunc of X,ExtREAL holds
( b4 = f |^ k iff ( dom b4 = dom f & ( for x being Element of X st x in dom b4 holds
b4 . x = (f . x) |^ k ) ) );

theorem Th11: :: MESFUNC7:11
for x being Element of ExtREAL
for y being Real
for k being Nat st x = y holds
x |^ k = y |^ k
proof end;

theorem Th12: :: MESFUNC7:12
for x being Element of ExtREAL
for k being Nat st 0 <= x holds
0 <= x |^ k
proof end;

theorem Th13: :: MESFUNC7:13
for k being Nat st 1 <= k holds
+infty |^ k = +infty
proof end;

theorem Th14: :: MESFUNC7:14
for k being Nat
for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for E being Element of S st E c= dom f & f is E -measurable holds
|.f.| |^ k is E -measurable
proof end;

theorem Th15: :: MESFUNC7:15
for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
for E being Element of S st (dom f) /\ (dom g) = E & f is V71() & g is V71() & f is E -measurable & g is E -measurable holds
f (#) g is E -measurable
proof end;

theorem Th16: :: MESFUNC7:16
for X being non empty set
for f being PartFunc of X,ExtREAL st rng f is real-bounded holds
f is V71()
proof end;

:: WP: Mean value theorem for integrals (first)
theorem :: MESFUNC7:17
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL
for E being Element of S
for F being non empty Subset of ExtREAL st (dom f) /\ (dom g) = E & rng f = F & g is V71() & f is E -measurable & rng f is real-bounded & g is_integrable_on M holds
( (f (#) g) | E is_integrable_on M & ex c being Element of REAL st
( c >= inf F & c <= sup F & Integral (M,((f (#) |.g.|) | E)) = c * (Integral (M,(|.g.| | E))) ) )
proof end;

theorem Th18: :: MESFUNC7:18
for X being non empty set
for f being PartFunc of X,ExtREAL
for A being set holds |.f.| | A = |.(f | A).|
proof end;

theorem Th19: :: MESFUNC7:19
for X being non empty set
for f, g being PartFunc of X,ExtREAL holds
( dom (|.f.| + |.g.|) = (dom f) /\ (dom g) & dom |.(f + g).| c= dom |.f.| )
proof end;

theorem Th20: :: MESFUNC7:20
for X being non empty set
for f, g being PartFunc of X,ExtREAL holds (|.f.| | (dom |.(f + g).|)) + (|.g.| | (dom |.(f + g).|)) = (|.f.| + |.g.|) | (dom |.(f + g).|)
proof end;

theorem Th21: :: MESFUNC7:21
for X being non empty set
for f, g being PartFunc of X,ExtREAL
for x being set st x in dom |.(f + g).| holds
|.(f + g).| . x <= (|.f.| + |.g.|) . x
proof end;

theorem :: MESFUNC7:22
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = dom (f + g) & Integral (M,(|.(f + g).| | E)) <= (Integral (M,(|.f.| | E))) + (Integral (M,(|.g.| | E))) )
proof end;

theorem Th23: :: MESFUNC7:23
for X being non empty set
for A being set holds max+ (chi (A,X)) = chi (A,X)
proof end;

theorem Th24: :: MESFUNC7:24
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S st M . E < +infty holds
( chi (E,X) is_integrable_on M & Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E )
proof end;

theorem Th25: :: MESFUNC7:25
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E1, E2 being Element of S st M . (E1 /\ E2) < +infty holds
Integral (M,((chi (E1,X)) | E2)) = M . (E1 /\ E2)
proof end;

theorem :: MESFUNC7:26
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for E being Element of S
for a, b being Real st f is_integrable_on M & E c= dom f & M . E < +infty & ( for x being Element of X st x in E holds
( a <= f . x & f . x <= b ) ) holds
( a * (M . E) <= Integral (M,(f | E)) & Integral (M,(f | E)) <= b * (M . E) )
proof end;