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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcu2 11201 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 11202 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 11203 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 11204 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 11205 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 11206 A natural number is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremiexpcyc 11207 Taking to the -th power is the same as using the -th power instead, by i4 11205. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 11208 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsqlecan 11209 Cancel one factor of a square in a comparison. Unlike lemul1 9608, the common factor may be zero. (Contributed by NM, 17-Jan-2008.)

Theoremsubsq 11210 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 11211 Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)

Theorembinom2i 11212 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theorembinom2aiOLD 11213 Product of sum and difference. (Contributed by NM, 7-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsubsqi 11214 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theoremsqeqori 11215 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.)

Theoremsubsq0i 11216 The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)

Theoremsqeqor 11217 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008.)

Theorembinom2 11218 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 11219 Special case of binom2 11218 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 11220 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2subi 11221 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theorembinom3 11222 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

Theoremsq01 11223 If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)

Theoremzesq 11224 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremnnesq 11225 A natural number is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremcrreczi 11226 Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Jeff Hankins, 16-Dec-2013.)

Theorembernneq 11227 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)

Theorembernneq2 11228 Variation of Bernoulli's inequality bernneq 11227. (Contributed by NM, 18-Oct-2007.)

Theorembernneq3 11229 A corollary of bernneq 11227. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremexpnbnd 11230* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Theoremexpnlbnd 11231* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 11232* 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 11233* Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremdigit2 11234 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 11235 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 11236 Exponentiation property of the modulo operation. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremdiscr1 11237* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremdiscr 11238* 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 11239 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 11240 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 11241 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 11242 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 11243 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 11244 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 11245 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 11246 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 11247 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 11248 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 11249 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpclzd 11250 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpne0d 11251 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpnegd 11252 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexprecd 11253 Nonnegative integer exponentiation of a reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1zd 11254 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpm1d 11255 Value of a complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpsubd 11256 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqmuld 11257 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivd 11258 Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpdivd 11259 Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulexpd 11260 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 11261 Value of zero raised to a natural number power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 11262 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 11263 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 11264 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnsqcld 11265 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 11266 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 11267 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 11268 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltexp2rd 11269 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzd 11270 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremresqcld 11271 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqge0d 11272 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqgt0d 11273 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltexp2d 11274 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2d 11275 Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcand 11276 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2ad 11277 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2rd 11278 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2sqd 11279 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremle2sqd 11280 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11d 11281 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 11282 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem1 11283 A rather pretty lemma for nn0opthi 11285. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem2 11284 Lemma for nn0opthi 11285. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)

Theoremnn0opthi 11285 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 3649 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)

Theoremnn0opth2i 11286 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 11285. (Contributed by NM, 22-Jul-2004.)

Theoremnn0opth2 11287 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 11285. (Contributed by NM, 22-Jul-2004.)

5.6.6  Factorial function

Syntaxcfa 11288 Extend class notation to include the factorial of nonnegative integers.

Definitiondf-fac 11289 Define the factorial function on nonnegative integers. For example, ; because ; (fac4 11296). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)

Theoremfacnn 11290 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 11291 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 11292 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 11293 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 11294 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 11295 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

Theoremfac4 11296 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
;

Theoremfacnn2 11297 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

Theoremfaccl 11298 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)

Theoremfacne0 11299 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)

Theoremfacdiv 11300 A natural number divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)

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