P. 227 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mbfres2 22601	Measurability of a piecewise function: if F is measurable on subsets B and C of its domain, and these pieces make up all of A, then F is measurable on the whole domain. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( ph -> F : A --> RR )   &    |- ( ph -> ( F |` B ) e. MblFn )   &    |- ( ph -> ( F |` C ) e. MblFn )   &    |- ( ph -> ( B u. C ) = A )   =>    |- ( ph -> F e. MblFn )
Theorem	mbfss 22602*	Change the domain of a measurability predicate. (Contributed by Mario Carneiro, 17-Aug-2014.)
|- ( ph -> A C_ B )   &    |- ( ph -> B e. dom vol )   &    |- ( ( ph /\ x e. A ) -> C e. V )   &    |- ( ( ph /\ x e. ( B \ A ) ) -> C = 0 )   &    |- ( ph -> ( x e. A |-> C ) e. MblFn )   =>    |- ( ph -> ( x e. B |-> C ) e. MblFn )
Theorem	mbfmulc2lem 22603	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> B e. RR )   &    |- ( ph -> F : A --> RR )   =>    |- ( ph -> ( ( A X. { B } ) oF x. F ) e. MblFn )
Theorem	mbfmulc2re 22604	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> B e. RR )   &    |- ( ph -> F : A --> CC )   =>    |- ( ph -> ( ( A X. { B } ) oF x. F ) e. MblFn )
Theorem	mbfmax 22605*	The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( ph -> F : A --> RR )   &    |- ( ph -> F e. MblFn )   &    |- ( ph -> G : A --> RR )   &    |- ( ph -> G e. MblFn )   &    |- H = ( x e. A |-> if ( ( F ` x ) <_ ( G ` x ) , ( G ` x ) , ( F ` x ) ) )   =>    |- ( ph -> H e. MblFn )
Theorem	mbfneg 22606*	The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> ( x e. A |-> B ) e. MblFn )   =>    |- ( ph -> ( x e. A |-> -u B ) e. MblFn )
Theorem	mbfpos 22607*	The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> ( x e. A |-> B ) e. MblFn )   =>    |- ( ph -> ( x e. A |-> if ( 0 <_ B , B , 0 ) ) e. MblFn )
Theorem	mbfposr 22608*	Converse to mbfpos 22607. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> ( x e. A |-> if ( 0 <_ B , B , 0 ) ) e. MblFn )   &    |- ( ph -> ( x e. A |-> if ( 0 <_ -u B , -u B , 0 ) ) e. MblFn )   =>    |- ( ph -> ( x e. A |-> B ) e. MblFn )
Theorem	mbfposb 22609*	A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ( ph /\ x e. A ) -> B e. RR )   =>    |- ( ph -> ( ( x e. A |-> B ) e. MblFn <-> ( ( x e. A |-> if ( 0 <_ B , B , 0 ) ) e. MblFn /\ ( x e. A |-> if ( 0 <_ -u B , -u B , 0 ) ) e. MblFn ) ) )
Theorem	ismbf3d 22610*	Simplified form of ismbfd 22596. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( ph -> F : A --> RR )   &    |- ( ( ph /\ x e. RR ) -> ( `' F " ( x (,) +oo ) ) e. dom vol )   =>    |- ( ph -> F e. MblFn )
Theorem	mbfimaopnlem 22611*	Lemma for mbfimaopn 22612. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- J = ( TopOpen ` ℂfld )   &    |- G = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )   &    |- B = ( (,) " ( QQ X. QQ ) )   &    |- K = ran ( x e. B , y e. B |-> ( x X. y ) )   =>    |- ( ( F e. MblFn /\ A e. J ) -> ( `' F " A ) e. dom vol )
Theorem	mbfimaopn 22612	The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 22614, which explains why A e. dom vol is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( ( F e. MblFn /\ A e. J ) -> ( `' F " A ) e. dom vol )
Theorem	mbfimaopn2 22613	The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- J = ( TopOpen ` ℂfld )   &    |- K = ( J ↾t B )   =>    |- ( ( ( F e. MblFn /\ F : A --> B /\ B C_ CC ) /\ C e. K ) -> ( `' F " C ) e. dom vol )
Theorem	cncombf 22614	The composition of a continuous function with a measurable function is measurable. (More generally, G can be a Borel-measurable function, but notably the condition that G be only measurable is too weak, the usual counterexample taking G to be the Cantor function and F the indicator function of the G-image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( ( F e. MblFn /\ F : A --> B /\ G e. ( B -cn-> CC ) ) -> ( G o. F ) e. MblFn )
Theorem	cnmbf 22615	A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
|- ( ( A e. dom vol /\ F e. ( A -cn-> CC ) ) -> F e. MblFn )
Theorem	mbfaddlem 22616	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   &    |- ( ph -> F : A --> RR )   &    |- ( ph -> G : A --> RR )   =>    |- ( ph -> ( F oF + G ) e. MblFn )
Theorem	mbfadd 22617	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   =>    |- ( ph -> ( F oF + G ) e. MblFn )
Theorem	mbfsub 22618	The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   =>    |- ( ph -> ( F oF - G ) e. MblFn )
Theorem	mbfmulc2 22619*	A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
|- ( ph -> C e. CC )   &    |- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> ( x e. A |-> B ) e. MblFn )   =>    |- ( ph -> ( x e. A |-> ( C x. B ) ) e. MblFn )
Theorem	mbfsup 22620*	The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, B ( n , x ) is a function of both n and x, since it is an n-indexed sequence of functions on x. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
|- Z = ( ZZ>= ` M )   &    |- G = ( x e. A |-> sup ( ran ( n e. Z |-> B ) , RR , < ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   &    |- ( ( ph /\ x e. A ) -> E. y e. RR A. n e. Z B <_ y )   =>    |- ( ph -> G e. MblFn )
Theorem	mbfinf 22621*	The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)
|- Z = ( ZZ>= ` M )   &    |- G = ( x e. A |-> inf ( ran ( n e. Z |-> B ) , RR , < ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   &    |- ( ( ph /\ x e. A ) -> E. y e. RR A. n e. Z y <_ B )   =>    |- ( ph -> G e. MblFn )
Theorem	mbfinfOLD 22622*	The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) Obsolete version of mbfinf 22621 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- Z = ( ZZ>= ` M )   &    |- G = ( x e. A |-> sup ( ran ( n e. Z |-> B ) , RR , `' < ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   &    |- ( ( ph /\ x e. A ) -> E. y e. RR A. n e. Z y <_ B )   =>    |- ( ph -> G e. MblFn )
Theorem	mbflimsup 22623*	The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
|- Z = ( ZZ>= ` M )   &    |- G = ( x e. A |-> ( limsup ` ( n e. Z |-> B ) ) )   &    |- H = ( m e. RR |-> sup ( ( ( ( n e. Z |-> B ) " ( m [,) +oo ) ) i^i RR* ) , RR* , < ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. A ) -> ( limsup ` ( n e. Z |-> B ) ) e. RR )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   =>    |- ( ph -> G e. MblFn )
Theorem	mbflimsupOLD 22624*	The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) Obsolete version of mbflimsup 22623 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- Z = ( ZZ>= ` M )   &    |- G = ( x e. A |-> ( limsup ` ( n e. Z |-> B ) ) )   &    |- H = ( m e. RR |-> sup ( ( ( ( n e. Z |-> B ) " ( m [,) +oo ) ) i^i RR* ) , RR* , < ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. A ) -> ( limsup ` ( n e. Z |-> B ) ) e. RR )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   =>    |- ( ph -> G e. MblFn )
Theorem	mbflimlem 22625*	The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. A ) -> ( n e. Z |-> B ) ~~> C )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. RR )   =>    |- ( ph -> ( x e. A |-> C ) e. MblFn )
Theorem	mbflim 22626*	The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. A ) -> ( n e. Z |-> B ) ~~> C )   &    |- ( ( ph /\ n e. Z ) -> ( x e. A |-> B ) e. MblFn )   &    |- ( ( ph /\ ( n e. Z /\ x e. A ) ) -> B e. V )   =>    |- ( ph -> ( x e. A |-> C ) e. MblFn )
Syntax	c0p 22627	Extend class notation to include the zero polynomial.
class 0p
Definition	df-0p 22628	Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- 0p = ( CC X. { 0 } )
Theorem	0pval 22629	The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. CC -> ( 0p ` A ) = 0 )
Theorem	0plef 22630	Two ways to say that the function F on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
|- ( F : RR --> ( 0 [,) +oo ) <-> ( F : RR --> RR /\ 0p oR <_ F ) )
Theorem	0pledm 22631	Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A C_ CC )   &    |- ( ph -> F Fn A )   =>    |- ( ph -> ( 0p oR <_ F <-> ( A X. { 0 } ) oR <_ F ) )
Theorem	isi1f 22632	The predicate " F is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom F e. dom S.1 to represent this concept because S.1 is the first preparation function for our final definition S. (see df-itg 22581); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( F e. dom S.1 <-> ( F e. MblFn /\ ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )
Theorem	i1fmbf 22633	Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( F e. dom S.1 -> F e. MblFn )
Theorem	i1ff 22634	A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( F e. dom S.1 -> F : RR --> RR )
Theorem	i1frn 22635	A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( F e. dom S.1 -> ran F e. Fin )
Theorem	i1fima 22636	Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( F e. dom S.1 -> ( `' F " A ) e. dom vol )
Theorem	i1fima2 22637	Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ( F e. dom S.1 /\ -. 0 e. A ) -> ( vol ` ( `' F " A ) ) e. RR )
Theorem	i1fima2sn 22638	Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ( F e. dom S.1 /\ A e. ( B \ { 0 } ) ) -> ( vol ` ( `' F " { A } ) ) e. RR )
Theorem	i1fd 22639*	A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ph -> F : RR --> RR )   &    |- ( ph -> ran F e. Fin )   &    |- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( `' F " { x } ) e. dom vol )   &    |- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { x } ) ) e. RR )   =>    |- ( ph -> F e. dom S.1 )
Theorem	i1f0rn 22640	Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( F e. dom S.1 -> 0 e. ran F )
Theorem	itg1val 22641*	The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ x e. ( ran F \ { 0 } ) ( x x. ( vol ` ( `' F " { x } ) ) ) )
Theorem	itg1val2 22642*	The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ( F e. dom S.1 /\ ( A e. Fin /\ ( ran F \ { 0 } ) C_ A /\ A C_ ( RR \ { 0 } ) ) ) -> ( S.1 ` F ) = sum_ x e. A ( x x. ( vol ` ( `' F " { x } ) ) ) )
Theorem	itg1cl 22643	Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( F e. dom S.1 -> ( S.1 ` F ) e. RR )
Theorem	itg1ge0 22644	Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( F e. dom S.1 /\ 0p oR <_ F ) -> 0 <_ ( S.1 ` F ) )
Theorem	i1f0 22645	The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( RR X. { 0 } ) e. dom S.1
Theorem	itg10 22646	The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( S.1 ` ( RR X. { 0 } ) ) = 0
Theorem	i1f1lem 22647*	Lemma for i1f1 22648 and itg11 22649. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- F = ( x e. RR |-> if ( x e. A , 1 , 0 ) )   =>    |- ( F : RR --> { 0 , 1 } /\ ( A e. dom vol -> ( `' F " { 1 } ) = A ) )
Theorem	i1f1 22648*	Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- F = ( x e. RR |-> if ( x e. A , 1 , 0 ) )   =>    |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> F e. dom S.1 )
Theorem	itg11 22649*	The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- F = ( x e. RR |-> if ( x e. A , 1 , 0 ) )   =>    |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( S.1 ` F ) = ( vol ` A ) )
Theorem	itg1addlem1 22650*	Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
|- ( ph -> F : X --> Y )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B C_ ( `' F " { k } ) )   &    |- ( ( ph /\ k e. A ) -> B e. dom vol )   &    |- ( ( ph /\ k e. A ) -> ( vol ` B ) e. RR )   =>    |- ( ph -> ( vol ` U_ k e. A B ) = sum_ k e. A ( vol ` B ) )
Theorem	i1faddlem 22651*	Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   =>    |- ( ( ph /\ A e. CC ) -> ( `' ( F oF + G ) " { A } ) = U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )
Theorem	i1fmullem 22652*	Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   =>    |- ( ( ph /\ A e. ( CC \ { 0 } ) ) -> ( `' ( F oF x. G ) " { A } ) = U_ y e. ( ran G \ { 0 } ) ( ( `' F " { ( A / y ) } ) i^i ( `' G " { y } ) ) )
Theorem	i1fadd 22653	The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   =>    |- ( ph -> ( F oF + G ) e. dom S.1 )
Theorem	i1fmul 22654	The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   =>    |- ( ph -> ( F oF x. G ) e. dom S.1 )
Theorem	itg1addlem2 22655*	Lemma for itg1add 22659. The function I represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both i and j are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 22657 and itg1addlem5 22658. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   &    |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )   =>    |- ( ph -> I : ( RR X. RR ) --> RR )
Theorem	itg1addlem3 22656*	Lemma for itg1add 22659. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   &    |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )   =>    |- ( ( ( A e. RR /\ B e. RR ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A I B ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )
Theorem	itg1addlem4 22657*	Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   &    |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )   &    |- P = ( + |` ( ran F X. ran G ) )   =>    |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )
Theorem	itg1addlem5 22658*	Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   &    |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )   &    |- P = ( + |` ( ran F X. ran G ) )   =>    |- ( ph -> ( S.1 ` ( F oF + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )
Theorem	itg1add 22659	The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> G e. dom S.1 )   =>    |- ( ph -> ( S.1 ` ( F oF + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )
Theorem	i1fmulclem 22660	Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A e. RR )   =>    |- ( ( ( ph /\ A =/= 0 ) /\ B e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { B } ) = ( `' F " { ( B / A ) } ) )
Theorem	i1fmulc 22661	A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
Theorem	itg1mulc 22662	The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
Theorem	i1fres 22663*	The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
|- G = ( x e. RR |-> if ( x e. A , ( F ` x ) , 0 ) )   =>    |- ( ( F e. dom S.1 /\ A e. dom vol ) -> G e. dom S.1 )
Theorem	i1fpos 22664*	The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- G = ( x e. RR |-> if ( 0 <_ ( F ` x ) , ( F ` x ) , 0 ) )   =>    |- ( F e. dom S.1 -> G e. dom S.1 )
Theorem	i1fposd 22665*	Deduction form of i1fposd 22665. (Contributed by Mario Carneiro, 6-Aug-2014.)
|- ( ph -> ( x e. RR |-> A ) e. dom S.1 )   =>    |- ( ph -> ( x e. RR |-> if ( 0 <_ A , A , 0 ) ) e. dom S.1 )
Theorem	i1fsub 22666	The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF - G ) e. dom S.1 )
Theorem	itg1sub 22667	The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( S.1 ` ( F oF - G ) ) = ( ( S.1 ` F ) - ( S.1 ` G ) ) )
Theorem	itg10a 22668*	The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol* ` A ) = 0 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> ( F ` x ) = 0 )   =>    |- ( ph -> ( S.1 ` F ) = 0 )
Theorem	itg1ge0a 22669*	The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol* ` A ) = 0 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> 0 <_ ( F ` x ) )   =>    |- ( ph -> 0 <_ ( S.1 ` F ) )
Theorem	itg1lea 22670*	Approximate version of itg1le 22671. If F <_ G for almost all x, then S.1 F <_ S.1 G. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
|- ( ph -> F e. dom S.1 )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol* ` A ) = 0 )   &    |- ( ph -> G e. dom S.1 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> ( F ` x ) <_ ( G ` x ) )   =>    |- ( ph -> ( S.1 ` F ) <_ ( S.1 ` G ) )
Theorem	itg1le 22671	If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
|- ( ( F e. dom S.1 /\ G e. dom S.1 /\ F oR <_ G ) -> ( S.1 ` F ) <_ ( S.1 ` G ) )
Theorem	itg1climres 22672*	Restricting the simple function F to the increasing sequence A ( n ) of measurable sets whose union is RR yields a sequence of simple functions whose integrals approach the integral of F. (Contributed by Mario Carneiro, 15-Aug-2014.)
|- ( ph -> A : NN --> dom vol )   &    |- ( ( ph /\ n e. NN ) -> ( A ` n ) C_ ( A ` ( n + 1 ) ) )   &    |- ( ph -> U. ran A = RR )   &    |- ( ph -> F e. dom S.1 )   &    |- G = ( x e. RR |-> if ( x e. ( A ` n ) , ( F ` x ) , 0 ) )   =>    |- ( ph -> ( n e. NN |-> ( S.1 ` G ) ) ~~> ( S.1 ` F ) )
Theorem	mbfi1fseqlem1 22673*	Lemma for mbfi1fseq 22679. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   =>    |- ( ph -> J : ( NN X. RR ) --> ( 0 [,) +oo ) )
Theorem	mbfi1fseqlem2 22674*	Lemma for mbfi1fseq 22679. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   &    |- G = ( m e. NN |-> ( x e. RR |-> if ( x e. ( -u m [,] m ) , if ( ( m J x ) <_ m , ( m J x ) , m ) , 0 ) ) )   =>    |- ( A e. NN -> ( G ` A ) = ( x e. RR |-> if ( x e. ( -u A [,] A ) , if ( ( A J x ) <_ A , ( A J x ) , A ) , 0 ) ) )
Theorem	mbfi1fseqlem3 22675*	Lemma for mbfi1fseq 22679. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   &    |- G = ( m e. NN |-> ( x e. RR |-> if ( x e. ( -u m [,] m ) , if ( ( m J x ) <_ m , ( m J x ) , m ) , 0 ) ) )   =>    |- ( ( ph /\ A e. NN ) -> ( G ` A ) : RR --> ran ( m e. ( 0 ... ( A x. ( 2 ^ A ) ) ) |-> ( m / ( 2 ^ A ) ) ) )
Theorem	mbfi1fseqlem4 22676*	Lemma for mbfi1fseq 22679. This lemma is not as interesting as it is long - it is simply checking that G is in fact a sequence of simple functions, by verifying that its range is in ( 0 ... n 2 ^ n ) / ( 2 ^ n ) (which is to say, the numbers from 0 to n in increments of 1 / ( 2 ^ n )), and also that the preimage of each point k is measurable, because it is equal to ( -u n [,] n ) i^i ( `' F " ( k [,) k + 1 / ( 2 ^ n ) ) ) for k < n and ( -u n [,] n ) i^i ( `' F " ( k [,) +oo ) ) for k = n. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   &    |- G = ( m e. NN |-> ( x e. RR |-> if ( x e. ( -u m [,] m ) , if ( ( m J x ) <_ m , ( m J x ) , m ) , 0 ) ) )   =>    |- ( ph -> G : NN --> dom S.1 )
Theorem	mbfi1fseqlem5 22677*	Lemma for mbfi1fseq 22679. Verify that G describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   &    |- G = ( m e. NN |-> ( x e. RR |-> if ( x e. ( -u m [,] m ) , if ( ( m J x ) <_ m , ( m J x ) , m ) , 0 ) ) )   =>    |- ( ( ph /\ A e. NN ) -> ( 0p oR <_ ( G ` A ) /\ ( G ` A ) oR <_ ( G ` ( A + 1 ) ) ) )
Theorem	mbfi1fseqlem6 22678*	Lemma for mbfi1fseq 22679. Verify that G converges pointwise to F, and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )   &    |- G = ( m e. NN |-> ( x e. RR |-> if ( x e. ( -u m [,] m ) , if ( ( m J x ) <_ m , ( m J x ) , m ) , 0 ) ) )   =>    |- ( ph -> E. g ( g : NN --> dom S.1 /\ A. n e. NN ( 0p oR <_ ( g ` n ) /\ ( g ` n ) oR <_ ( g ` ( n + 1 ) ) ) /\ A. x e. RR ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfi1fseq 22679*	A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   =>    |- ( ph -> E. g ( g : NN --> dom S.1 /\ A. n e. NN ( 0p oR <_ ( g ` n ) /\ ( g ` n ) oR <_ ( g ` ( n + 1 ) ) ) /\ A. x e. RR ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfi1flimlem 22680*	Lemma for mbfi1flim 22681. (Contributed by Mario Carneiro, 5-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> RR )   =>    |- ( ph -> E. g ( g : NN --> dom S.1 /\ A. x e. RR ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfi1flim 22681*	Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : A --> RR )   =>    |- ( ph -> E. g ( g : NN --> dom S.1 /\ A. x e. A ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfmullem2 22682*	Lemma for mbfmul 22684. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   &    |- ( ph -> F : A --> RR )   &    |- ( ph -> G : A --> RR )   &    |- ( ph -> P : NN --> dom S.1 )   &    |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) )   &    |- ( ph -> Q : NN --> dom S.1 )   &    |- ( ( ph /\ x e. A ) -> ( n e. NN |-> ( ( Q ` n ) ` x ) ) ~~> ( G ` x ) )   =>    |- ( ph -> ( F oF x. G ) e. MblFn )
Theorem	mbfmullem 22683	Lemma for mbfmul 22684. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   &    |- ( ph -> F : A --> RR )   &    |- ( ph -> G : A --> RR )   =>    |- ( ph -> ( F oF x. G ) e. MblFn )
Theorem	mbfmul 22684	The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> G e. MblFn )   =>    |- ( ph -> ( F oF x. G ) e. MblFn )
Theorem	itg2lcl 22685*	The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }   =>    |- L C_ RR*
Theorem	itg2val 22686*	Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) = sup ( L , RR* , < ) )
Theorem	itg2l 22687*	Elementhood in the set L of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( A e. L <-> E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) )
Theorem	itg2lr 22688*	Sufficient condition for elementhood in the set L. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( ( G e. dom S.1 /\ G oR <_ F ) -> ( S.1 ` G ) e. L )
Theorem	xrge0f 22689	A real function is a nonnegative extended real function if all its values are greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
|- ( ( F : RR --> RR /\ 0p oR <_ F ) -> F : RR --> ( 0 [,] +oo ) )
Theorem	itg2cl 22690	The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) e. RR* )
Theorem	itg2ub 22691	The integral of a nonnegative real function F is an upper bound on the integrals of all simple functions G dominated by F. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ( F : RR --> ( 0 [,] +oo ) /\ G e. dom S.1 /\ G oR <_ F ) -> ( S.1 ` G ) <_ ( S.2 ` F ) )
Theorem	itg2leub 22692*	Any upper bound on the integrals of all simple functions G dominated by F is greater than ( S.2 ` F ), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ( F : RR --> ( 0 [,] +oo ) /\ A e. RR* ) -> ( ( S.2 ` F ) <_ A <-> A. g e. dom S.1 ( g oR <_ F -> ( S.1 ` g ) <_ A ) ) )
Theorem	itg2ge0 22693	The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( F : RR --> ( 0 [,] +oo ) -> 0 <_ ( S.2 ` F ) )
Theorem	itg2itg1 22694	The integral of a nonnegative simple function using S.2 is the same as its value under S.1. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ( F e. dom S.1 /\ 0p oR <_ F ) -> ( S.2 ` F ) = ( S.1 ` F ) )
Theorem	itg20 22695	The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( S.2 ` ( RR X. { 0 } ) ) = 0
Theorem	itg2lecl 22696	If an S.2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ( F : RR --> ( 0 [,] +oo ) /\ A e. RR /\ ( S.2 ` F ) <_ A ) -> ( S.2 ` F ) e. RR )
Theorem	itg2le 22697	If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( ( F : RR --> ( 0 [,] +oo ) /\ G : RR --> ( 0 [,] +oo ) /\ F oR <_ G ) -> ( S.2 ` F ) <_ ( S.2 ` G ) )
Theorem	itg2const 22698*	Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( B x. ( vol ` A ) ) )
Theorem	itg2const2 22699*	When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
|- ( ( A e. dom vol /\ B e. RR+ ) -> ( ( vol ` A ) e. RR <-> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) e. RR ) )
Theorem	itg2seq 22700*	Definitional property of the S.2 integral: for any function F there is a countable sequence g of simple functions less than F whose integrals converge to the integral of F. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 22713, but unlike that theorem this one doesn't require F to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
|- ( F : RR --> ( 0 [,] +oo ) -> E. g ( g : NN --> dom S.1 /\ A. n e. NN ( g ` n ) oR <_ F /\ ( S.2 ` F ) = sup ( ran ( n e. NN |-> ( S.1 ` ( g ` n ) ) ) , RR* , < ) ) )