Home Metamath Proof ExplorerTheorem List (p. 116 of 327) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22413) Hilbert Space Explorer (22414-23936) Users' Mathboxes (23937-32689)

Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexpnlbnd 11501* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 11502* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)

Theoremexpmulnbnd 11503* Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremdigit2 11504 Two ways to express the th digit in the decimal (when base ) expansion of a number . corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)

Theoremdigit1 11505 Two ways to express the th digit in the decimal expansion of a number (when base ). corresponds to the first digit after the decimal point. (Contributed by NM, 3-Jan-2009.)

Theoremmodexp 11506 Exponentiation property of the modulo operation. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremdiscr1 11507* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremdiscr 11508* If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is non-positive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexp0d 11509 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 11510 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 11511 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 11512 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 11513 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 11514 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 11515 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 11516 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpaddd 11517 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpmuld 11518 Product of exponents law for natural number exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqrecd 11519 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpclzd 11520 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpne0d 11521 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpnegd 11522 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexprecd 11523 Nonnegative integer exponentiation of a reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1zd 11524 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpm1d 11525 Value of a complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpsubd 11526 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqmuld 11527 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivd 11528 Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpdivd 11529 Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulexpd 11530 Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)

Theorem0expd 11531 Value of zero raised to a natural number power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 11532 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 11533 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 11534 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnsqcld 11535 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 11536 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 11537 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 11538 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltexp2rd 11539 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzd 11540 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremresqcld 11541 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqge0d 11542 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqgt0d 11543 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltexp2d 11544 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2d 11545 Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcand 11546 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2ad 11547 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2rd 11548 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2sqd 11549 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremle2sqd 11550 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11d 11551 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)

5.6.5  Ordered pair theorem for nonnegative integers

Theoremnn0le2msqi 11552 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem1 11553 A rather pretty lemma for nn0opthi 11555. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem2 11554 Lemma for nn0opthi 11555. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)

Theoremnn0opthi 11555 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers and by . If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3815 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)

Theoremnn0opth2i 11556 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 11555. (Contributed by NM, 22-Jul-2004.)

Theoremnn0opth2 11557 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 11555. (Contributed by NM, 22-Jul-2004.)

5.6.6  Factorial function

Syntaxcfa 11558 Extend class notation to include the factorial of nonnegative integers.

Definitiondf-fac 11559 Define the factorial function on nonnegative integers. For example, ; because ; (fac4 11566). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)

Theoremfacnn 11560 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 11561 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 11562 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 11563 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 11564 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 11565 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

Theoremfac4 11566 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
;

Theoremfacnn2 11567 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

Theoremfaccl 11568 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)

Theoremfacne0 11569 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)

Theoremfacdiv 11570 A natural number divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)

Theoremfacndiv 11571 No natural number (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)

Theoremfacwordi 11572 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)

Theoremfaclbnd 11573 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd2 11574 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd3 11575 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)

Theoremfaclbnd4lem1 11576 Lemma for faclbnd4 11580. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)

Theoremfaclbnd4lem2 11577 Lemma for faclbnd4 11580. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 11576 to antecedents. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd4lem3 11578 Lemma for faclbnd4 11580. The case. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd4lem4 11579 Lemma for faclbnd4 11580. Prove the case by induction on . (Contributed by NM, 19-Dec-2005.)

Theoremfaclbnd4 11580 Variant of faclbnd5 11581 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd5 11581 The factorial function grows faster than powers and exponentiations. If we consider and to be constants, the right-hand side of the inequality is a constant times -factorial. (Contributed by NM, 24-Dec-2005.)

Theoremfaclbnd6 11582 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)

Theoremfacubnd 11583 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremfacavg 11584 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)

5.6.7  The binomial coefficient operation

Syntaxcbc 11585 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).

Definitiondf-bc 11586* Define the binomial coefficient operation. In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". is read " choose ." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when does not hold. (Contributed by NM, 10-Jul-2005.)

Theorembcval 11587 Value of the binomial coefficient, choose . Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when does not hold. See bcval2 11588 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval2 11588 Value of the binomial coefficient, choose , in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval3 11589 Value of the binomial coefficient, choose , outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcval4 11590 Value of the binomial coefficient, choose , outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcrpcl 11591 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 11606.) (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembccmpl 11592 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)

Theorembcn0 11593 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembc0k 11594 The binomial coefficient " 0 choose " is 0 for a positive integer K. Note that (see bcn0 11593). (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theorembcnn 11595 choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcn1 11596 Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnp1n 11597 Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcm1k 11598 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1n 11599 The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1nk 11600 The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

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 32401-32500 326 32501-32600 327 32601-32689
 Copyright terms: Public domain < Previous  Next >