Home Metamath Proof ExplorerTheorem List (p. 197 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

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

Theoremitg2i1fseq2 19601* In an extension to the results of itg2i1fseq 19600, 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 19602* Special case of itg2i1fseq2 19601: 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 19604 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 19605* 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 19606* Lemma for itgcn 19687. (Contributed by Mario Carneiro, 30-Aug-2014.)
MblFn

Theoremitg2cnlem2 19607* Lemma for itgcn 19687. (Contributed by Mario Carneiro, 31-Aug-2014.)
MblFn

Theoremitg2cn 19608* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19874 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
MblFn

Theoremibllem 19609 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremisibl 19610* 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 19611* 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 19612 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
MblFn

Theoremiblitg 19613* 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 19614* Evaluate the class substitution in df-itg 19469. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgex 19615 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1f 19616 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq1 19617* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg1 19618 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremnfitg 19619* 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 19620* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremcbvitgv 19621* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgeq2 19622 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremitgresr 19623 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitg0 19624 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgz 19625 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgeq2dv 19626* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremitgmpt 19627* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgcl 19628* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)

Theoremitgvallem 19629* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgvallem3 19630* Lemma for itgposval 19640 and itgreval 19641. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremibl0 19631 The zero function is integrable on any measurable set. (Unlike iblconst 19662, this does not require to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)

Theoremiblcnlem1 19632* Lemma for iblcnlem 19633. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblcnlem 19633* Expand out the forall in isibl2 19611. (Contributed by Mario Carneiro, 6-Aug-2014.)
MblFn

Theoremitgcnlem 19634* Expand out the sum in dfitg 19614. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblrelem 19635* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblposlem 19636* Lemma for iblpos 19637. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremiblpos 19637* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
MblFn

Theoremiblre 19638* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgrevallem1 19639* Lemma for itgposval 19640 and itgreval 19641. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgposval 19640* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgreval 19641* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)

Theoremitgrecl 19642* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremiblcn 19643* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgcnval 19644* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgre 19645* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremitgim 19646* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)

Theoremiblneg 19647* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgneg 19648* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblss 19649* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremiblss2 19650* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremitgitg2 19651* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremi1fibl 19652 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgitg1 19653* Transfer an integral using to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremitgle 19654* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgge0 19655* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgss 19656* 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 19657* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgeqa 19658* 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 19659* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)

Theoremitgioo 19660* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremitgless 19661* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremiblconst 19662 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitgconst 19663* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)

MblFn       MblFn

Theoremibladd 19665* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremiblsub 19666* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgadd 19669* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

Theoremitgsub 19670* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgfsum 19671* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)

Theoremiblabslem 19672* Lemma for iblabs 19673. (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn

Theoremiblabs 19673* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremiblabsr 19674* A measurable function is integrable iff its absolute value is integrable. (See iblabs 19673 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
MblFn

Theoremiblmulc2 19675* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2lem1 19676* Lemma for itgmulc2 19678: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2lem2 19677* Lemma for itgmulc2 19678: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgmulc2 19678* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgabs 19679* The triangle inequality for integrals. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremitgsplit 19680* The integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)

Theoremitgspliticc 19681* The integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.)

Theoremitgsplitioo 19682* The integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theorembddmulibl 19683* A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
MblFn

Theorembddibl 19684* A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
MblFn

Theoremcniccibl 19685 A continuous function on a closed interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremitggt0 19686* The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremitgcn 19687* Transfer itg2cn 19608 to the full Lebesgue integral. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremditgeq1 19688* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgeq2 19689* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgeq3 19690* 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 19696 first and use the equality theorems for df-itg 19469.) (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgeq3dv 19691* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgex 19692 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
_

Theoremditg0 19693* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
_

Theoremcbvditg 19694* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
_ _

Theoremcbvditgv 19695* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
_ _

Theoremditgpos 19696* Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
_

Theoremditgneg 19697* Value of the directed integral in the backward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
_

Theoremditgcl 19698* Closure of a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_

Theoremditgswap 19699* Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _

Theoremditgsplitlem 19700* Lemma for ditgsplit 19701. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _ _

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
 Copyright terms: Public domain < Previous  Next >