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

Theoremitg2val 19501* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2l 19502* Elementhood in the set of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lr 19503* Sufficient condition for elementhood in the set . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremxrge0f 19504 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 19505 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2ub 19506 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 19507* 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 19508 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2itg1 19509 The integral of a nonnegative simple function using is the same as its value under . (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg20 19510 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2lecl 19511 If an integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2le 19512 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitg2const 19513* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitg2const2 19514* 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 19515* 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 19528, but unlike that theorem this one doesn't require to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitg2uba 19516* Approximate version of itg2ub 19506. If approximately dominates , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2lea 19517* Approximate version of itg2le 19512. If for almost all , then . (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2eqa 19518* Approximate equality of integrals. If for almost all , then . (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitg2mulclem 19519 Lemma for itg2mulc 19520. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremitg2mulc 19520 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 19521* Lemma for itg2split 19522. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2split 19522* The integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19532 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2monolem1 19523* Lemma for itg2mono 19526. We show that for any constant less than one, is less than , and so , which is one half of the equality in itg2mono 19526. 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 19487. 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 19524* Lemma for itg2mono 19526. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2monolem3 19525* Lemma for itg2mono 19526. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2mono 19526* 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 19527* Subject to the conditions coming from mbfi1fseq 19494, the sequence of simple functions are all less than the target function . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq 19528* Subject to the conditions coming from mbfi1fseq 19494, 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 19529* In an extension to the results of itg2i1fseq 19528, 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 19530* Special case of itg2i1fseq2 19529: 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 19532 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 19533* 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 19534* Lemma for itgcn 19615. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem2 19535* Lemma for itgcn 19615. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremitg2cn 19536* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19802 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
MblFn

Theoremibllem 19537 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremisibl 19538* 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 19539* 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 19540 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
MblFn

Theoremiblitg 19541* 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 19542* Evaluate the class substitution in df-itg 19397. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgex 19543 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1f 19544 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1 19545* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg1 19546 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg 19547* 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 19548* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitgv 19549* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq2 19550 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgresr 19551 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitg0 19552 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgz 19553 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgeq2dv 19554* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremitgmpt 19555* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgcl 19556* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgvallem 19557* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgvallem3 19558* Lemma for itgposval 19568 and itgreval 19569. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremibl0 19559 The zero function is integrable on any measurable set. (Unlike iblconst 19590, this does not require to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremiblcnlem1 19560* Lemma for iblcnlem 19561. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblcnlem 19561* Expand out the forall in isibl2 19539. (Contributed by Mario Carneiro, 6-Aug-2014.)
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Theoremitgcnlem 19562* Expand out the sum in dfitg 19542. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblrelem 19563* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblposlem 19564* Lemma for iblpos 19565. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblpos 19565* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblre 19566* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgrevallem1 19567* Lemma for itgposval 19568 and itgreval 19569. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgposval 19568* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgreval 19569* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremitgrecl 19570* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremiblcn 19571* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgcnval 19572* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgre 19573* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitgim 19574* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremiblneg 19575* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgneg 19576* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblss 19577* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremiblss2 19578* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgitg2 19579* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fibl 19580 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgitg1 19581* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgle 19582* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgge0 19583* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgss 19584* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgss2 19585* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgeqa 19586* Approximate equality of integrals. If for almost all , then and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)

Theoremitgss3 19587* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)

Theoremitgioo 19588* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremitgless 19589* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremiblconst 19590 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitgconst 19591* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)

MblFn       MblFn

Theoremibladd 19593* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremiblsub 19594* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgadd 19597* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremitgsub 19598* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgfsum 19599* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)

Theoremiblabslem 19600* Lemma for iblabs 19601. (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn

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