P. 196 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mbfimaopn 19501	The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 19503, 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 19502	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 19503	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 19504	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 19505	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 o F + G ) e. MblFn )
Theorem	mbfadd 19506	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 o F + G ) e. MblFn )
Theorem	mbfsub 19507	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 o F - G ) e. MblFn )
Theorem	mbfmulc2 19508*	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 19509*	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 19510*	The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed 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 y <_ B )   =>    |- ( ph -> G e. MblFn )
Theorem	mbflimsup 19511*	The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 9-May-2016.)
|- 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 19512*	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 19513*	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 19514	Extend class notation to include the zero polynomial.
class 0 p
Definition	df-0p 19515	Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- 0 p = ( CC X. { 0 } )
Theorem	0pval 19516	The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. CC -> ( 0 p ` A ) = 0 )
Theorem	0plef 19517	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 /\ 0 p o R <_ F ) )
Theorem	0pledm 19518	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 -> ( 0 p o R <_ F <-> ( A X. { 0 } ) o R <_ F ) )
Theorem	isi1f 19519	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 19469); 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 19520	Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( F e. dom S.1 -> F e. MblFn )
Theorem	i1ff 19521	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 19522	A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
|- ( F e. dom S.1 -> ran F e. Fin )
Theorem	i1fima 19523	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 19524	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 19525	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 19526*	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 19527	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 19528*	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 19529*	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 19530	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 19531	Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- ( ( F e. dom S.1 /\ 0 p o R <_ F ) -> 0 <_ ( S.1 ` F ) )
Theorem	i1f0 19532	The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( RR X. { 0 } ) e. dom S.1
Theorem	itg10 19533	The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
|- ( S.1 ` ( RR X. { 0 } ) ) = 0
Theorem	i1f1lem 19534*	Lemma for i1f1 19535 and itg11 19536. (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 19535*	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 19536*	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 19537*	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 19538*	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 o F + G ) " { A } ) = U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )
Theorem	i1fmullem 19539*	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 o F x. G ) " { A } ) = U_ y e. ( ran G \ { 0 } ) ( ( `' F " { ( A / y ) } ) i^i ( `' G " { y } ) ) )
Theorem	i1fadd 19540	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 o F + G ) e. dom S.1 )
Theorem	i1fmul 19541	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 o F x. G ) e. dom S.1 )
Theorem	itg1addlem2 19542*	Lemma for itg1add 19546. 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 19544 and itg1addlem5 19545. (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 19543*	Lemma for itg1add 19546. (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 19544*	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 o F + G ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )
Theorem	itg1addlem5 19545*	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 o F + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )
Theorem	itg1add 19546	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 o F + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )
Theorem	i1fmulclem 19547	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 } ) o F x. F ) " { B } ) = ( `' F " { ( B / A ) } ) )
Theorem	i1fmulc 19548	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 } ) o F x. F ) e. dom S.1 )
Theorem	itg1mulc 19549	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 } ) o F x. F ) ) = ( A x. ( S.1 ` F ) ) )
Theorem	i1fres 19550*	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 19551*	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 19552*	Deduction form of i1fposd 19552. (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 19553	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 o F - G ) e. dom S.1 )
Theorem	itg1sub 19554	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 o F - G ) ) = ( ( S.1 ` F ) - ( S.1 ` G ) ) )
Theorem	itg10a 19555*	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 19556*	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 19557*	Approximate version of itg1le 19558. 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 19558	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 o R <_ G ) -> ( S.1 ` F ) <_ ( S.1 ` G ) )
Theorem	itg1climres 19559*	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 19560*	Lemma for mbfi1fseq 19566. (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 19561*	Lemma for mbfi1fseq 19566. (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 19562*	Lemma for mbfi1fseq 19566. (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 19563*	Lemma for mbfi1fseq 19566. 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 19564*	Lemma for mbfi1fseq 19566. 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 ) -> ( 0 p o R <_ ( G ` A ) /\ ( G ` A ) o R <_ ( G ` ( A + 1 ) ) ) )
Theorem	mbfi1fseqlem6 19565*	Lemma for mbfi1fseq 19566. 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 ( 0 p o R <_ ( g ` n ) /\ ( g ` n ) o R <_ ( g ` ( n + 1 ) ) ) /\ A. x e. RR ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfi1fseq 19566*	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 ( 0 p o R <_ ( g ` n ) /\ ( g ` n ) o R <_ ( g ` ( n + 1 ) ) ) /\ A. x e. RR ( n e. NN |-> ( ( g ` n ) ` x ) ) ~~> ( F ` x ) ) )
Theorem	mbfi1flimlem 19567*	Lemma for mbfi1flim 19568. (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 19568*	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 19569*	Lemma for mbfmul 19571. (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 o F x. G ) e. MblFn )
Theorem	mbfmullem 19570	Lemma for mbfmul 19571. (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 o F x. G ) e. MblFn )
Theorem	mbfmul 19571	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 o F x. G ) e. MblFn )
Theorem	itg2lcl 19572*	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 o R <_ F /\ x = ( S.1 ` g ) ) }   =>    |- L C_ RR*
Theorem	itg2val 19573*	Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g o R <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( F : RR --> ( 0 [,] +oo ) -> ( S.2 ` F ) = sup ( L , RR* , < ) )
Theorem	itg2l 19574*	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 o R <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( A e. L <-> E. g e. dom S.1 ( g o R <_ F /\ A = ( S.1 ` g ) ) )
Theorem	itg2lr 19575*	Sufficient condition for elementhood in the set L. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- L = { x | E. g e. dom S.1 ( g o R <_ F /\ x = ( S.1 ` g ) ) }   =>    |- ( ( G e. dom S.1 /\ G o R <_ F ) -> ( S.1 ` G ) e. L )
Theorem	xrge0f 19576	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 /\ 0 p o R <_ F ) -> F : RR --> ( 0 [,] +oo ) )
Theorem	itg2cl 19577	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 19578	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 o R <_ F ) -> ( S.1 ` G ) <_ ( S.2 ` F ) )
Theorem	itg2leub 19579*	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 o R <_ F -> ( S.1 ` g ) <_ A ) ) )
Theorem	itg2ge0 19580	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 19581	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 /\ 0 p o R <_ F ) -> ( S.2 ` F ) = ( S.1 ` F ) )
Theorem	itg20 19582	The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
|- ( S.2 ` ( RR X. { 0 } ) ) = 0
Theorem	itg2lecl 19583	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 19584	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 o R <_ G ) -> ( S.2 ` F ) <_ ( S.2 ` G ) )
Theorem	itg2const 19585*	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 19586*	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 19587*	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 19600, 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 ) o R <_ F /\ ( S.2 ` F ) = sup ( ran ( n e. NN |-> ( S.1 ` ( g ` n ) ) ) , RR* , < ) ) )
Theorem	itg2uba 19588*	Approximate version of itg2ub 19578. If F approximately dominates G, then S.1 G <_ S.2 F. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> F : RR --> ( 0 [,] +oo ) )   &    |- ( ph -> G e. dom S.1 )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol * ` A ) = 0 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> ( G ` x ) <_ ( F ` x ) )   =>    |- ( ph -> ( S.1 ` G ) <_ ( S.2 ` F ) )
Theorem	itg2lea 19589*	Approximate version of itg2le 19584. If F <_ G for almost all x, then S.2 F <_ S.2 G. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> F : RR --> ( 0 [,] +oo ) )   &    |- ( ph -> G : RR --> ( 0 [,] +oo ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol * ` A ) = 0 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> ( F ` x ) <_ ( G ` x ) )   =>    |- ( ph -> ( S.2 ` F ) <_ ( S.2 ` G ) )
Theorem	itg2eqa 19590*	Approximate equality of integrals. If F = G for almost all x, then S.2 F = S.2 G. (Contributed by Mario Carneiro, 12-Aug-2014.)
|- ( ph -> F : RR --> ( 0 [,] +oo ) )   &    |- ( ph -> G : RR --> ( 0 [,] +oo ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> ( vol * ` A ) = 0 )   &    |- ( ( ph /\ x e. ( RR \ A ) ) -> ( F ` x ) = ( G ` x ) )   =>    |- ( ph -> ( S.2 ` F ) = ( S.2 ` G ) )
Theorem	itg2mulclem 19591	Lemma for itg2mulc 19592. (Contributed by Mario Carneiro, 8-Jul-2014.)
|- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- ( ph -> ( S.2 ` F ) e. RR )   &    |- ( ph -> A e. RR+ )   =>    |- ( ph -> ( S.2 ` ( ( RR X. { A } ) o F x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
Theorem	itg2mulc 19592	The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- ( ph -> ( S.2 ` F ) e. RR )   &    |- ( ph -> A e. ( 0 [,) +oo ) )   =>    |- ( ph -> ( S.2 ` ( ( RR X. { A } ) o F x. F ) ) = ( A x. ( S.2 ` F ) ) )
Theorem	itg2splitlem 19593*	Lemma for itg2split 19594. (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> A e. dom vol )   &    |- ( ph -> B e. dom vol )   &    |- ( ph -> ( vol * ` ( A i^i B ) ) = 0 )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ( ph /\ x e. U ) -> C e. ( 0 [,] +oo ) )   &    |- F = ( x e. RR |-> if ( x e. A , C , 0 ) )   &    |- G = ( x e. RR |-> if ( x e. B , C , 0 ) )   &    |- H = ( x e. RR |-> if ( x e. U , C , 0 ) )   &    |- ( ph -> ( S.2 ` F ) e. RR )   &    |- ( ph -> ( S.2 ` G ) e. RR )   =>    |- ( ph -> ( S.2 ` H ) <_ ( ( S.2 ` F ) + ( S.2 ` G ) ) )
Theorem	itg2split 19594*	The S.2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19604 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)
|- ( ph -> A e. dom vol )   &    |- ( ph -> B e. dom vol )   &    |- ( ph -> ( vol * ` ( A i^i B ) ) = 0 )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ( ph /\ x e. U ) -> C e. ( 0 [,] +oo ) )   &    |- F = ( x e. RR |-> if ( x e. A , C , 0 ) )   &    |- G = ( x e. RR |-> if ( x e. B , C , 0 ) )   &    |- H = ( x e. RR |-> if ( x e. U , C , 0 ) )   &    |- ( ph -> ( S.2 ` F ) e. RR )   &    |- ( ph -> ( S.2 ` G ) e. RR )   =>    |- ( ph -> ( S.2 ` H ) = ( ( S.2 ` F ) + ( S.2 ` G ) ) )
Theorem	itg2monolem1 19595*	Lemma for itg2mono 19598. We show that for any constant t less than one, t x. S.1 H is less than S, and so S.1 H <_ S, which is one half of the equality in itg2mono 19598. Consider the sequence A ( n ) = { x | t x. H <_ F ( n ) }. This is an increasing sequence of measurable sets whose union is RR, and so H |` A ( n ) has an integral which equals S.1 H in the limit, by itg1climres 19559. Then by taking the limit in ( t x. H ) |` A ( n ) <_ F ( n ), we get t x. S.1 H <_ S as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- G = ( x e. RR |-> sup ( ran ( n e. NN |-> ( ( F ` n ) ` x ) ) , RR , < ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. MblFn )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) o R <_ ( F ` ( n + 1 ) ) )   &    |- ( ( ph /\ x e. RR ) -> E. y e. RR A. n e. NN ( ( F ` n ) ` x ) <_ y )   &    |- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )   &    |- ( ph -> T e. ( 0 (,) 1 ) )   &    |- ( ph -> H e. dom S.1 )   &    |- ( ph -> H o R <_ G )   &    |- ( ph -> S e. RR )   &    |- A = ( n e. NN |-> { x e. RR | ( T x. ( H ` x ) ) <_ ( ( F ` n ) ` x ) } )   =>    |- ( ph -> ( T x. ( S.1 ` H ) ) <_ S )
Theorem	itg2monolem2 19596*	Lemma for itg2mono 19598. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- G = ( x e. RR |-> sup ( ran ( n e. NN |-> ( ( F ` n ) ` x ) ) , RR , < ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. MblFn )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) o R <_ ( F ` ( n + 1 ) ) )   &    |- ( ( ph /\ x e. RR ) -> E. y e. RR A. n e. NN ( ( F ` n ) ` x ) <_ y )   &    |- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )   &    |- ( ph -> P e. dom S.1 )   &    |- ( ph -> P o R <_ G )   &    |- ( ph -> -. ( S.1 ` P ) <_ S )   =>    |- ( ph -> S e. RR )
Theorem	itg2monolem3 19597*	Lemma for itg2mono 19598. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- G = ( x e. RR |-> sup ( ran ( n e. NN |-> ( ( F ` n ) ` x ) ) , RR , < ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. MblFn )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) o R <_ ( F ` ( n + 1 ) ) )   &    |- ( ( ph /\ x e. RR ) -> E. y e. RR A. n e. NN ( ( F ` n ) ` x ) <_ y )   &    |- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )   &    |- ( ph -> P e. dom S.1 )   &    |- ( ph -> P o R <_ G )   &    |- ( ph -> -. ( S.1 ` P ) <_ S )   =>    |- ( ph -> ( S.1 ` P ) <_ S )
Theorem	itg2mono 19598*	The Monotone Convergence Theorem for nonnegative functions. If { ( F ` n ) : n e. NN } is a monotone increasing sequence of positive, measurable, real-valued functions, and G is the pointwise limit of the sequence, then ( S.2 ` G ) is the limit of the sequence { ( S.2 ` ( F ` n ) ) : n e. NN }. (Contributed by Mario Carneiro, 16-Aug-2014.)
|- G = ( x e. RR |-> sup ( ran ( n e. NN |-> ( ( F ` n ) ` x ) ) , RR , < ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. MblFn )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) : RR --> ( 0 [,) +oo ) )   &    |- ( ( ph /\ n e. NN ) -> ( F ` n ) o R <_ ( F ` ( n + 1 ) ) )   &    |- ( ( ph /\ x e. RR ) -> E. y e. RR A. n e. NN ( ( F ` n ) ` x ) <_ y )   &    |- S = sup ( ran ( n e. NN |-> ( S.2 ` ( F ` n ) ) ) , RR* , < )   =>    |- ( ph -> ( S.2 ` G ) = S )
Theorem	itg2i1fseqle 19599*	Subject to the conditions coming from mbfi1fseq 19566, the sequence of simple functions are all less than the target function F. (Contributed by Mario Carneiro, 17-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- ( ph -> P : NN --> dom S.1 )   &    |- ( ph -> A. n e. NN ( 0 p o R <_ ( P ` n ) /\ ( P ` n ) o R <_ ( P ` ( n + 1 ) ) ) )   &    |- ( ph -> A. x e. RR ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) )   =>    |- ( ( ph /\ M e. NN ) -> ( P ` M ) o R <_ F )
Theorem	itg2i1fseq 19600*	Subject to the conditions coming from mbfi1fseq 19566, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
|- ( ph -> F e. MblFn )   &    |- ( ph -> F : RR --> ( 0 [,) +oo ) )   &    |- ( ph -> P : NN --> dom S.1 )   &    |- ( ph -> A. n e. NN ( 0 p o R <_ ( P ` n ) /\ ( P ` n ) o R <_ ( P ` ( n + 1 ) ) ) )   &    |- ( ph -> A. x e. RR ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) )   &    |- S = ( m e. NN |-> ( S.1 ` ( P ` m ) ) )   =>    |- ( ph -> ( S.2 ` F ) = sup ( ran S , RR* , < ) )