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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremitg2mulclem 19101 Lemma for itg2mulc 19102. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremitg2mulc 19102 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 19103* Lemma for itg2split 19104. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2split 19104* The integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19114 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitg2monolem1 19105* Lemma for itg2mono 19108. We show that for any constant less than one, is less than , and so , which is one half of the equality in itg2mono 19108. 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 19069. 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 19106* Lemma for itg2mono 19108. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2monolem3 19107* Lemma for itg2mono 19108. (Contributed by Mario Carneiro, 16-Aug-2014.)
MblFn

Theoremitg2mono 19108* 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 19109* Subject to the conditions coming from mbfi1fseq 19076, the sequence of simple functions are all less than the target function . (Contributed by Mario Carneiro, 17-Aug-2014.)
MblFn

Theoremitg2i1fseq 19110* Subject to the conditions coming from mbfi1fseq 19076, 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 19111* In an extension to the results of itg2i1fseq 19110, 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 19112* Special case of itg2i1fseq2 19111: 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 19114 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 19115* 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 19116* Lemma for itgcn 19197. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem2 19117* Lemma for itgcn 19197. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremitg2cn 19118* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19384 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
MblFn

Theoremibllem 19119 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremisibl 19120* 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 19121* 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 19122 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
MblFn

Theoremiblitg 19123* 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 19124* Evaluate the class substitution in df-itg 18979. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgex 19125 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1f 19126 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1 19127* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg1 19128 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg 19129* 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 19130* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitgv 19131* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq2 19132 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgresr 19133 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitg0 19134 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgz 19135 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgeq2dv 19136* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremitgmpt 19137* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgcl 19138* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgvallem 19139* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgvallem3 19140* Lemma for itgposval 19150 and itgreval 19151. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremibl0 19141 The zero function is integrable on any measurable set. (Unlike iblconst 19172, this does not require to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremiblcnlem1 19142* Lemma for iblcnlem 19143. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblcnlem 19143* Expand out the forall in isibl2 19121. (Contributed by Mario Carneiro, 6-Aug-2014.)
MblFn

Theoremitgcnlem 19144* Expand out the sum in dfitg 19124. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblrelem 19145* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblposlem 19146* Lemma for iblpos 19147. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblpos 19147* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblre 19148* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgrevallem1 19149* Lemma for itgposval 19150 and itgreval 19151. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgposval 19150* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgreval 19151* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremitgrecl 19152* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremiblcn 19153* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgcnval 19154* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgre 19155* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitgim 19156* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremiblneg 19157* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgneg 19158* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblss 19159* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremiblss2 19160* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgitg2 19161* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fibl 19162 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgitg1 19163* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgle 19164* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgge0 19165* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgss 19166* 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 19167* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgeqa 19168* 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 19169* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)

Theoremitgioo 19170* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremitgless 19171* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremiblconst 19172 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitgconst 19173* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)

MblFn       MblFn

Theoremibladd 19175* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremiblsub 19176* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgadd 19179* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremitgsub 19180* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgfsum 19181* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)

Theoremiblabslem 19182* Lemma for iblabs 19183. (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn

Theoremiblabs 19183* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblabsr 19184* A measurable function is integrable iff its absolute value is integrable. (See iblabs 19183 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn

Theoremiblmulc2 19185* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2lem1 19186* Lemma for itgmulc2 19188: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2lem2 19187* Lemma for itgmulc2 19188: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2 19188* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgabs 19189* The triangle inequality for integrals. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgsplit 19190* The integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgspliticc 19191* The integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgsplitioo 19192* The integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theorembddmulibl 19193* A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
MblFn

Theorembddibl 19194* A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
MblFn

Theoremcniccibl 19195 A continuous function on a closed interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitggt0 19196* The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremitgcn 19197* Transfer itg2cn 19118 to the full Lebesgue integral. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremditgeq1 19198* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgeq2 19199* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
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Theoremditgeq3 19200* Equality theorem for the directed integral. (The domain of the equality here is very rough; for more precise bounds one should decompose it with ditgpos 19206 first and use the equality theorems for df-itg 18979.) (Contributed by Mario Carneiro, 13-Aug-2014.)
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