Home Metamath Proof ExplorerTheorem List (p. 199 of 324) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22341) Hilbert Space Explorer (22342-23864) Users' Mathboxes (23865-32387)

Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvexp3 19801* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 19802 Derivative of the exponential function at 0. The key step in the proof is eftlub 12651, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 19803 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 19804 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 19805 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 19806 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvferm1lem 19807* Lemma for dvferm 19811. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 19808* One-sided version of dvferm 19811. A point which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremdvferm2lem 19809* Lemma for dvferm 19811. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 19810* One-sided version of dvferm 19811. A point which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremdvferm 19811* Fermat's theorem on stationary points. A point which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrollelem 19812* Lemma for rolle 19813. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 19813* Rolle's theorem. If is a real continuous function on which is differentiable on , and , then there is some such that . (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremcmvth 19814* Cauchy's Mean Value Theorem. If are real continuous functions on differentiable on , then there is some such that ' ' . (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremmvth 19815* The Mean Value Theorem. If is a real continuous function on which is differentiable on , then there is some such that is equal to the average slope over . (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)

Theoremdvlip 19816* A function with derivative bounded by is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 19817* A complex function with derivative bounded by on an open ball is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremdvlip2 19818* Combine the results of dvlip 19816 and dvlipcn 19817 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 19819* Lemma for c1lip1 19820. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 19820* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 19821* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremc1lip3 19822* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 19823 If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)

Theoremdv11cn 19824 Two functions defined on a ball whose derivatives are the same and which are equal at any given point in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremdvgt0lem1 19825 Lemma for dvgt0 19827 and dvlt0 19828. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 19826* Lemma for dvgt0 19827 and dvlt0 19828. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 19827 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 19828 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 19829 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 19830* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 19831* Lemma for dvivth 19833. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 19832* Lemma for dvivth 19833. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 19833 Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 19294 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 19834 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 19835 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 19836* Lemma for lhop1 19837. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 19837* L'Hôpital's Rule for limits from the right. If and are differentiable real functions on , and and both approach 0 at , and and ' are not zero on , and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim        lim

Theoremlhop2 19838* L'Hôpital's Rule for limits from the right. If and are differentiable real functions on , and and both approach 0 at , and and ' are not zero on , and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim        lim

Theoremlhop 19839* L'Hôpital's Rule. If is an open set of the reals, and are real functions on containing all of except possibly , which are differentiable everywhere on , and both approach 0, and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 30-Dec-2016.)
lim        lim                      lim        lim

Theoremdvcnvrelem1 19840 Lemma for dvcnvre 19842. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 19841 Lemma for dvcnvre 19842. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 19842* The derivative rule for inverse functions. If is a continuous and differentiable bijective function from to which never has derivative , then is also differentiable, and its derivative is the reciprocal of the derivative of . (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcvx 19843 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 19844* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
..^        ..^        ..^

Theoremdvfsumge 19845* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
..^        ..^        ..^

Theoremdvfsumabs 19846* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
..^        ..^        ..^        ..^ ..^

Theoremdvmptrecl 19847* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 19848* Lemma for dvfsumrlim 19854. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 19849* Lemma for dvfsumrlim 19854. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 19850* Lemma for dvfsumrlim 19854. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 19851* Lemma for dvfsumrlim 19854. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem4 19852* Lemma for dvfsumrlim 19854. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimge0 19853* Lemma for dvfsumrlim 19854. Satisfy the assumption of dvfsumlem4 19852. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim 19854* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if is a decreasing function with antiderivative converging to zero, then the difference between and converges to a constant limit value, with the remainder term bounded by . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim2 19855* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if is a decreasing function with antiderivative converging to zero, then the difference between and converges to a constant limit value, with the remainder term bounded by . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim3 19856* Conjoin the statements of dvfsumrlim 19854 and dvfsumrlim2 19855. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex, and we don't want to repeat ourselves.) (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsum2 19857* The reverse of dvfsumrlim 19854, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremftc1lem1 19858* Lemma for ftc1a 19860 and ftc1 19865. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremftc1lem2 19859* Lemma for ftc1 19865. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremftc1a 19860* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral of is continuous if is integrable. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremftc1lem3 19861* Lemma for ftc1 19865. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 19862* Lemma for ftc1 19865. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 19863* Lemma for ftc1 19865. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 19864* Lemma for ftc1 19865. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 19865* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at with derivative if the original function is continuous at . (Contributed by Mario Carneiro, 1-Sep-2014.)
t        t        fld

Theoremftc1cn 19866* Strengthen the assumptions of ftc1 19865 to when the function is continuous on the entire interval ; in this case we can calculate exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremftc2 19867* The Fundamental Theorem of Calculus, part two. If is a function continuous on and continuously differentiable on , then the integral of the derivative of is equal to . (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremftc2ditglem 19868* Lemma for ftc2ditg 19869. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 19869* Directed integral analog of ftc2 19867. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 19870* Integration by parts. If is the derivative of and is the derivative of , and and , then under suitable integrability and differentiability assumptions, the integral of from to is equal to minus the integral of . (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremitgsubstlem 19871* Lemma for itgsubst 19872. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 19872* Integration by -substitution. If is a continuous, differentiable function from to , whose derivative is continuous and integrable, and is a continuous function on , then the integral of from to is equal to the integral of from to . In this part of the proof we discharge the assumptions in itgsubstlem 19871, which use the fact that is open to shrink the interval a little to where - this is possible because is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
_ _

PART 13  BASIC REAL AND COMPLEX FUNCTIONS

13.1  Polynomials

13.1.1  Abstract polynomials, continued

Theoremevlslem6 19873* Lemma for evlseu 19876. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                      g g

Theoremevlslem3 19874* Lemma for evlseu 19876. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                                    g

Theoremevlslem1 19875* Lemma for evlseu 19876, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom               algSc       RingHom

Theoremevlseu 19876* For a given intepretation of the variables and of the scalars , this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly               algSc       mVar                             RingHom               RingHom

Theoremreldmevls 19877 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub

Theoremmpfrcl 19878 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub        SubRing

Theoremevlsval 19879* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsval2 19880* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsrhm 19881 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        s        s               SubRing RingHom

Theoremevlssca 19882 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
evalSub        mPoly        s               algSc                     SubRing

Theoremevlsvar 19883* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mVar        s                             SubRing

Theoremevlval 19884 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
eval               evalSub

Theoremevlrhm 19885 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval               mPoly        s        RingHom

Theoremevl1fval 19886* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval

Theoremevl1val 19887* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval               mPoly

Theoremevl1rhm 19888 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1       s               RingHom

Theoremevl1sca 19889 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1              algSc

Theoremevl1scad 19890 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1              algSc

Theoremevl1var 19891 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       var1

Theoremevl1vard 19892 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       var1              Poly1

Theoremevl1addd 19893 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1subd 19894 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1muld 19895 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1vsd 19896 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1expd 19897 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1                                          .gmulGrp       .gmulGrp

Theoremmpfconst 19898 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub                      SubRing

Theoremmpfproj 19899* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
evalSub                      SubRing

Theoremmpfsubrg 19900 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
evalSub        SubRing SubRing s

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 32301-32387
 Copyright terms: Public domain < Previous  Next >