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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnpw2blenfzo2 40901 A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
#b #b ..^#b

Theoremnnpw2pmod 40902 Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
#b #b

Theoremblen1 40903 The binary length of 1. (Contributed by AV, 21-May-2020.)
#b

Theoremblen2 40904 The binary length of 2. (Contributed by AV, 21-May-2020.)
#b

Theoremnnpw2p 40905* Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
..^

Theoremnnpw2pb 40906* A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
..^

Theoremblen1b 40907 The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
#b

Theoremblennnt2 40908 The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.)
#b #b

Theoremnnolog2flm1 40909 The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.)
logb logb

Theoremblennn0em1 40910 The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.)
#b #b

Theoremblennngt2o2 40911 The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.)
#b #b

Theoremblengt1fldiv2p1 40912 The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.)
#b #b

Theoremblennn0e2 40913 The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.)
#b #b

21.33.16.10  Digits

Generalisation of df-bits 14474. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 40933: digit 2 ) N ) = 1 <-> K e. ( bits .

Syntaxcdig 40914 Extend class notation with the class of the digit extraction operation.
digit

Definitiondf-dig 40915* Definition of an operation to obtain the th digit of a nonnegative real number in the positional system with base . corresponds to the first digit of the fractional part (for the first digit after the decimal point), corresponds to the last digit of the integer part (for the first digit before the decimal point). See also digit1 12444. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.)
digit

Theoremdigfval 40916* Operation to obtain the th digit of a nonnegative real number in the positional system with base . (Contributed by AV, 23-May-2020.)
digit

Theoremdigval 40917 The th digit of a nonnegative real number in the positional system with base . (Contributed by AV, 23-May-2020.)
digit

Theoremdigvalnn0 40918 The th digit of a nonnegative real number in the positional system with base is a nonnegative integer. (Contributed by AV, 28-May-2020.)
digit

Theoremnn0digval 40919 The th digit of a nonnegative real number in the positional system with base . (Contributed by AV, 23-May-2020.)
digit

Theoremdignn0fr 40920 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
digit

Theoremdignn0ldlem 40921 Lemma for dignnld 40922. (Contributed by AV, 25-May-2020.)
logb

Theoremdignnld 40922 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
logb digit

Theoremdig2nn0ld 40923 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
#b digit

Theoremdig2nn1st 40924 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
#b digit

Theoremdig0 40925 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
digit

Theoremdigexp 40926 The th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
digit

Theoremdig1 40927 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
digit

Theorem0dig1 40928 The th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
digit

Theorem0dig2pr01 40929 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
digit

Theoremdig2nn0 40930 A digit of a nonnegative integer in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
digit

Theorem0dig2nn0e 40931 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
digit

Theorem0dig2nn0o 40932 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
digit

Theoremdig2bits 40933 The th digit of a nonnegative integer in a binary system is its th bit. (Contributed by AV, 24-May-2020.)
digit bits

21.33.16.11  Nonnegative integer as sum of its shifted digits

Theoremdignn0flhalflem1 40934 Lemma 1 for dignn0flhalf 40937. (Contributed by AV, 7-Jun-2012.)

Theoremdignn0flhalflem2 40935 Lemma 2 for dignn0flhalf 40937. (Contributed by AV, 7-Jun-2012.)

Theoremdignn0ehalf 40936 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
digit digit

Theoremdignn0flhalf 40937 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
digit digit

Theoremnn0sumshdiglemA 40938* Lemma for nn0sumshdig 40942 (induction step, even multiplicant). (Contributed by AV, 3-Jun-2020.)
#b ..^digit #b ..^ digit

Theoremnn0sumshdiglemB 40939* Lemma for nn0sumshdig 40942 (induction step, odd multiplicant). (Contributed by AV, 7-Jun-2020.)
#b ..^digit #b ..^ digit

Theoremnn0sumshdiglem1 40940* Lemma 1 for nn0sumshdig 40942 (induction step). (Contributed by AV, 7-Jun-2020.)
#b ..^digit #b ..^ digit

Theoremnn0sumshdiglem2 40941* Lemma 2 for nn0sumshdig 40942. (Contributed by AV, 7-Jun-2020.)
#b ..^digit

Theoremnn0sumshdig 40942* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
..^#bdigit

21.33.16.12  Algorithms for the multiplication of nonnegative integers

Theoremnn0mulfsum 40943* Trivial algorithm to calculate the product of two nonnegative integers and by adding up times. (Contributed by AV, 17-May-2020.)

Theoremnn0mullong 40944* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers and by multiplying the multiplicand by each digit of the multiplier and then add up all the properly shifted results. Here, the binary representation of the multiplier is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 14546. (Contributed by AV, 7-Jun-2020.)
..^#bdigit

21.34  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

21.34.1  Natural deduction

Theorem19.8ad 40945 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1955. (Contributed by DAW, 13-Feb-2017.)

Theoremsbidd 40946 An identity theorem for substitution. See sbid 2101. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

Theoremsbidd-misc 40947 An identity theorem for substitution. See sbid 2101. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

21.34.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

Syntaxcge-real 40948 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 40950.

Syntaxcgt 40949 Extend wff notation to include the 'greater than' relation, see df-gt 40951.

Definitiondf-gte 40950 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 9699.

We do not write this as , and similarly we do not write ` > ` as , because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: and but these are very complicated. This definition of , and the similar one for (df-gt 40951), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 40952 for a more conventional expression of the relationship between and . As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Definitiondf-gt 40951 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 9570. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 40950 for a discussion on why this approach is used for the definition. See gt-lt 40953 and gt-lth 40955 for more conventional expression of the relationship between and .

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremgte-lte 40952 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Theoremgt-lt 40953 Simple relationship between and . (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremgte-lteh 40954 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Theoremgt-lth 40955 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremex-gt 40956 Simple example of , in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)

Theoremex-gte 40957 Simple example of , in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)

21.34.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as . However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

Syntaxcsinh 40958 Extend class notation to include the hyperbolic sine function, see df-sinh 40961.
sinh

Syntaxccosh 40959 Extend class notation to include the hyperbolic cosine function. see df-cosh 40962.
cosh

Syntaxctanh 40960 Extend class notation to include the hyperbolic tangent function, see df-tanh 40963.
tanh

Definitiondf-sinh 40961 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4454, which requires the form . See sinhval-named 40964 for a simple way to evaluate it. We define this function by dividing by , which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 40967 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
sinh

Definitiondf-cosh 40962 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4454, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.)
cosh

Definitiondf-tanh 40963 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4454, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.)
tanh cosh

Theoremsinhval-named 40964 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 40961. See sinhval 14285 for a theorem to convert this further. See sinh-conventional 40967 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
sinh

Theoremcoshval-named 40965 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 40962. See coshval 14286 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
cosh

Theoremtanhval-named 40966 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 40963. (Contributed by David A. Wheeler, 10-May-2015.)
cosh tanh

Theoremsinh-conventional 40967 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
sinh

Theoremsinhpcosh 40968 Prove that sinh cosh using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
sinh cosh

21.34.4  Reciprocal trigonometric functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

Syntaxcsec 40969 Extend class notation to include the secant function, see df-sec 40972.

Syntaxccsc 40970 Extend class notation to include the cosecant function, see df-csc 40973.

Syntaxccot 40971 Extend class notation to include the cotangent function, see df-cot 40974.

Definitiondf-sec 40972* Define the secant function. We define it this way for cmpt 4454, which requires the form . The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Definitiondf-csc 40973* Define the cosecant function. We define it this way for cmpt 4454, which requires the form . The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Definitiondf-cot 40974* Define the cotangent function. We define it this way for cmpt 4454, which requires the form . The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremsecval 40975 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcscval 40976 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcotval 40977 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremseccl 40978 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcsccl 40979 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcotcl 40980 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremreseccl 40981 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecsccl 40982 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecotcl 40983 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecsec 40984 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremreccsc 40985 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremreccot 40986 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)

Theoremrectan 40987 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)

Theoremsec0 40988 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)

Theoremonetansqsecsq 40989 Prove the tangent squared secant squared identity A ) ^ 2 ) ) = ( ( sec . (Contributed by David A. Wheeler, 25-May-2015.)

Theoremcotsqcscsq 40990 Prove the tangent squared cosecant squared identity A ) ^ 2 ) ) = ( ( csc . (Contributed by David A. Wheeler, 27-May-2015.)

21.34.5  Identities for "if"

Utility theorems for "if".

Theoremifnmfalse 40991 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3881 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)

21.34.6  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 40995 and df-dp2 40994 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 11075.

TODO: Fix non-existent label dfpval.

Syntaxcdp2 40992 Constant used for decimal fraction constructor. See df-dp2 40994.
_

Syntaxcdp 40993 Decimal point operator. See df-dp 40995.

Definitiondf-dp2 40994 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11075. (Contributed by David A. Wheeler, 15-May-2015.)
_

Definitiondf-dp 40995* Define the (decimal point) operator. For example, , and ;__ ;;;; ;;; Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is , not ; this should simplify some proofs. The LHS is , since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

_

Theoremdp2cl 40996 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
_

Theoremdpval 40997 Define the value of the decimal point operator. See df-dp 40995. (Contributed by David A. Wheeler, 15-May-2015.)
_

Theoremdpcl 40998 Prove that the closure of the decimal point is as we have defined it. See df-dp 40995. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremdpfrac1 40999 Prove a simple equivalence involving the decimal point. See df-dp 40995 and dpcl 40998. (Contributed by David A. Wheeler, 15-May-2015.)
;

21.34.7  Logarithms generalized to arbitrary base using ` logb `

Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".

Theoremlogb2aval 41000 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
logb

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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 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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 32601-32700 328 32701-32800 329 32801-32900 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