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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremi1fposd 19601* Deduction form of i1fposd 19601. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fsub 19602 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitg1sub 19603 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitg10a 19604* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg1ge0a 19605* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg1lea 19606* Approximate version of itg1le 19607. If for almost all , then . (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)

Theoremitg1le 19607 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.)

Theoremitg1climres 19608* Restricting the simple function to the increasing sequence of measurable sets whose union is yields a sequence of simple functions whose integrals approach the integral of . (Contributed by Mario Carneiro, 15-Aug-2014.)

Theoremmbfi1fseqlem1 19609* Lemma for mbfi1fseq 19615. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem2 19610* Lemma for mbfi1fseq 19615. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1fseqlem3 19611* Lemma for mbfi1fseq 19615. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem4 19612* Lemma for mbfi1fseq 19615. This lemma is not as interesting as it is long - it is simply checking that is in fact a sequence of simple functions, by verifying that its range is in (which is to say, the numbers from to in increments of ), and also that the preimage of each point is measurable, because it is equal to for and for . (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem5 19613* Lemma for mbfi1fseq 19615. Verify that describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremmbfi1fseqlem6 19614* Lemma for mbfi1fseq 19615. Verify that converges pointwise to , and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1fseq 19615* 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 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremmbfi1flimlem 19616* Lemma for mbfi1flim 19617. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfi1flim 19617* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
MblFn

Theoremmbfmullem2 19618* Lemma for mbfmul 19620. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                                                 MblFn

Theoremmbfmullem 19619 Lemma for mbfmul 19620. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn                     MblFn

Theoremmbfmul 19620 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
MblFn       MblFn       MblFn

Theoremitg2lcl 19621* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2val 19622* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2l 19623* Elementhood in the set of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lr 19624* Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremxrge0f 19625 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.)

Theoremitg2cl 19626 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2ub 19627 The integral of a nonnegative real function is an upper bound on the integrals of all simple functions dominated by . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2leub 19628* Any upper bound on the integrals of all simple functions dominated by is greater than , the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2ge0 19629 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2itg1 19630 The integral of a nonnegative simple function using is the same as its value under . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg20 19631 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lecl 19632 If an integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2le 19633 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2const 19634* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitg2const2 19635* 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.)

Theoremitg2seq 19636* Definitional property of the integral: for any function there is a countable sequence of simple functions less than whose integrals converge to the integral of . (This theorem is for the most part unnecessary in lieu of itg2i1fseq 19649, but unlike that theorem this one doesn't require to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitg2uba 19637* Approximate version of itg2ub 19627. If approximately dominates , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2lea 19638* Approximate version of itg2le 19633. If for almost all , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2eqa 19639* Approximate equality of integrals. If for almost all , then . (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitg2mulclem 19640 Lemma for itg2mulc 19641. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremitg2mulc 19641 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.)

Theoremitg2splitlem 19642* Lemma for itg2split 19643. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2split 19643* The integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19653 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2monolem1 19644* Lemma for itg2mono 19647. We show that for any constant less than one, is less than , and so , which is one half of the equality in itg2mono 19647. Consider the sequence . This is an increasing sequence of measurable sets whose union is , and so has an integral which equals in the limit, by itg1climres 19608. Then by taking the limit in , we get as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremitg2monolem2 19645* Lemma for itg2mono 19647. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2monolem3 19646* Lemma for itg2mono 19647. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2mono 19647* The Monotone Convergence Theorem for nonnegative functions. If is a monotone increasing sequence of positive, measurable, real-valued functions, and is the pointwise limit of the sequence, then is the limit of the sequence . (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2i1fseqle 19648* Subject to the conditions coming from mbfi1fseq 19615, the sequence of simple functions are all less than the target function . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq 19649* Subject to the conditions coming from mbfi1fseq 19615, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq2 19650* In an extension to the results of itg2i1fseq 19649, if there is an upper bound on the integrals of the simple functions approaching , then is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq3 19651* Special case of itg2i1fseq2 19650: if the integral of is a real number, then the standard limit relation holds on the integrals of simple functions approaching . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

MblFn                     MblFn

Theoremitg2add 19653 The integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn                     MblFn

Theoremitg2gt0 19654* If the function is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem1 19655* Lemma for itgcn 19736. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem2 19656* Lemma for itgcn 19736. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremitg2cn 19657* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19923 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
MblFn

Theoremibllem 19658 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremisibl 19659* The predicate " is integrable". The "integrable" predicate corresponds roughly to the range of validity of , which is to say that the expression doesn't make sense unless . (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremisibl2 19660* The predicate " is integrable" when is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblmbf 19661 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
MblFn

Theoremiblitg 19662* If a function is integrable, then the integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremdfitg 19663* Evaluate the class substitution in df-itg 19518. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgex 19664 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1f 19665 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1 19666* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg1 19667 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg 19668* Bound-variable hypothesis builder for an integral: if is (effectively) not free in and , it is not free in . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitg 19669* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitgv 19670* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq2 19671 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgresr 19672 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitg0 19673 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgz 19674 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgeq2dv 19675* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremitgmpt 19676* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgcl 19677* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgvallem 19678* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgvallem3 19679* Lemma for itgposval 19689 and itgreval 19690. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremibl0 19680 The zero function is integrable on any measurable set. (Unlike iblconst 19711, this does not require to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremiblcnlem1 19681* Lemma for iblcnlem 19682. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblcnlem 19682* Expand out the forall in isibl2 19660. (Contributed by Mario Carneiro, 6-Aug-2014.)
MblFn

Theoremitgcnlem 19683* Expand out the sum in dfitg 19663. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblrelem 19684* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblposlem 19685* Lemma for iblpos 19686. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblpos 19686* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblre 19687* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgrevallem1 19688* Lemma for itgposval 19689 and itgreval 19690. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgposval 19689* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgreval 19690* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremitgrecl 19691* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremiblcn 19692* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgcnval 19693* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgre 19694* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitgim 19695* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremiblneg 19696* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgneg 19697* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblss 19698* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremiblss2 19699* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgitg2 19700* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

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