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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremseqfeq2 11301* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfveq 11302* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfeq 11303* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqshft2 11304* Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqres 11305 Restricting its characteristic function to does not affect the function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserf 11306* An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserfre 11307* An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremmonoord 11308* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremmonoord2 11309* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)

Theoremsermono 11310* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)

Theoremseqsplit 11311* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseq1p 11312* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqcaopr3 11313* Lemma for seqcaopr2 11314. (Contributed by Mario Carneiro, 25-Apr-2016.)
..^

Theoremseqcaopr2 11314* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)

Theoremseqcaopr 11315* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremseqf1olem2a 11316* Lemma for seqf1o 11319. (Contributed by Mario Carneiro, 24-Apr-2016.)

Theoremseqf1olem1 11317* Lemma for seqf1o 11319. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf1olem2 11318* Lemma for seqf1o 11319. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremseqf1o 11319* Rearrange a sum via an arbitrary bijection on . (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremseradd 11320* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremsersub 11321* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqid3 11322* A sequence that consists entirely of zeroes (or whatever the identity is for operation ) sums to zero. (Contributed by Mario Carneiro, 15-Dec-2014.)

Theoremseqid 11323* Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity is for operation ). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqid2 11324* The last few terms of a sequence that ends with all zeroes (or whatever the identity is for operation ) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqhomo 11325* Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqz 11326* If the operation has an absorbing element (a.k.a. zero element), then any sequence containing a evaluates to . (Contributed by Mario Carneiro, 27-May-2014.)

Theoremseqfeq4 11327* Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)

Theoremseqfeq3 11328* Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)

Theoremseqdistr 11329* The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremser0 11330 The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)

Theoremser0f 11331 A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)

Theoremserge0 11332* A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserle 11333* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremser1const 11334 Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Theoremseqof 11335* Distribute function operation through a sequence. Note that is an implicit function on . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremseqof2 11336* Distribute function operation through a sequence. Maps-to notation version of seqof 11335. (Contributed by Mario Carneiro, 7-Jul-2017.)

5.6.4  Integer powers

Syntaxcexp 11337 Extend class notation to include exponentiation of a complex number to an integer power.

Definitiondf-exp 11338* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 11341 and expp1 11343 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case gives the value , so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)

Theoremexpval 11339 Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpnnval 11340 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexp0 11341 Value of a complex number raised to the 0th power. Note that under our definition, , following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexp1 11342 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)

Theoremexpp1 11343 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 NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)

Theoremexpneg 11344 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpneg2 11345 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpn1 11346 A number to the negative one power is the reciprocal. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpcllem 11347* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)

Theoremexpcl2lem 11348* Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)

Theoremnnexpcl 11349 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)

Theoremnn0expcl 11350 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)

Theoremzexpcl 11351 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)

Theoremqexpcl 11352 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)

Theoremreexpcl 11353 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)

Theoremexpcl 11354 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)

Theoremrpexpcl 11355 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)

Theoremreexpclz 11356 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)

Theoremqexpclz 11357 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theoremm1expcl2 11358 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremm1expcl 11359 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremexpclzlem 11360 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpclz 11361 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremnn0expcli 11362 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremexpm1t 11363 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)

Theorem1exp 11364 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpeq0 11365 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)

Theoremexpne0 11366 Natural number exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.)

Theoremexpne0i 11367 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpgt0 11368 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpnegz 11369 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theorem0exp 11370 Value of zero raised to a natural number power. (Contributed by NM, 19-Aug-2004.)

Theoremexpge0 11371 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpge1 11372 Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpgt1 11373 Natural number exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremmulexp 11374 Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)

Theoremmulexpz 11375 Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexprec 11376 Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpadd 11377 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)

Theoremexpaddz 11379 Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpmul 11380 Product of exponents law for natural number exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)

Theoremexpmulz 11381 Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpsub 11382 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpp1z 11383 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpm1 11384 Value of a complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpdiv 11385 Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremltexp2a 11386 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpcan 11387 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremltexp2 11388 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremleexp2 11389 Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremleexp2a 11390 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremltexp2r 11391 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremleexp2r 11392 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremleexp1a 11393 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)

Theoremexple1 11394 Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremexpubnd 11395 An upper bound on when . (Contributed by NM, 19-Dec-2005.)

Theoremsqval 11396 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsqneg 11397 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)

Theoremsqsubswap 11398 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremsqcl 11399 Closure of square. (Contributed by NM, 10-Aug-1999.)

Theoremsqmul 11400 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)

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