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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexpubnd 10301 An upper bound on when . (Contributed by NM, 19-Dec-2005.)

Theoremsqval 10302 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsqneg 10303 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)

Theoremsqsubswap 10304 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremsqcl 10305 Closure of square. (Contributed by NM, 10-Aug-1999.)

Theoremsqmul 10306 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)

Theoremsqeq0 10307 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)

Theoremsqdivap 10308 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
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Theoremsqne0 10309 A number is nonzero iff its square is nonzero. See also sqap0 10310 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)

Theoremsqap0 10310 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
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Theoremresqcl 10311 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)

Theoremsqgt0ap 10312 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
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Theoremnnsqcl 10313 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremzsqcl 10314 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremqsqcl 10315 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremsq11 10316 The square function is one-to-one for nonnegative reals. Also see sq11ap 10409 which would easily follow from this given excluded middle, but which for us is proved another way. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)

Theoremlt2sq 10317 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)

Theoremle2sq 10318 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)

Theoremle2sq2 10319 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)

Theoremsqge0 10320 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)

Theoremzsqcl2 10321 The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremsumsqeq0 10322 Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.)

Theoremsqvali 10323 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)

Theoremsqcli 10324 Closure of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqeq0i 10325 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)

Theoremsqmuli 10326 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)

Theoremsqdivapi 10327 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremresqcli 10328 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)

Theoremsqgt0api 10329 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremsqge0i 10330 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)

Theoremlt2sqi 10331 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremle2sqi 10332 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremsq11i 10333 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)

Theoremsq0 10334 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)

Theoremsq0i 10335 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)

Theoremsq0id 10336 If a number is zero, its square is zero. Deduction form of sq0i 10335. Converse of sqeq0d 10374. (Contributed by David Moews, 28-Feb-2017.)

Theoremsq1 10337 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)

Theoremneg1sqe1 10338 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremsq2 10339 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)

Theoremsq3 10340 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)

Theoremsq4e2t8 10341 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)

Theoremcu2 10342 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 10343 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 10344 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 10345 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 10346 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 10347 A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremiexpcyc 10348 Taking to the -th power is the same as using the -th power instead, by i4 10346. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 10349 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsubsq 10350 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 10351 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 10352 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theoremsubsqi 10353 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theorembinom2 10354 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 10355 Special case of binom2 10354 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 10356 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2sub1 10357 Special case of binom2sub 10356 where . (Contributed by AV, 2-Aug-2021.)

Theorembinom2subi 10358 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremmulbinom2 10359 The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)

Theorembinom3 10360 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

Theoremzesq 10361 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremnnesq 10362 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theorembernneq 10363 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)

Theorembernneq2 10364 Variation of Bernoulli's inequality bernneq 10363. (Contributed by NM, 18-Oct-2007.)

Theorembernneq3 10365 A corollary of bernneq 10363. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremexpnbnd 10366* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Theoremexpnlbnd 10367* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 10368* 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.)

Theoremexp0d 10369 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 10370 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 10371 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 10372 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 10373 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 10374 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 10375 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 10376 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 10377 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 10378 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqrecapd 10379 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpclzapd 10380 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpap0d 10381 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpnegapd 10382 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexprecapd 10383 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpp1zapd 10384 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpm1apd 10385 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremexpsubapd 10386 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
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Theoremsqmuld 10387 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivapd 10388 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
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Theoremexpdivapd 10389 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
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Theoremmulexpd 10390 Positive integer 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 10391 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 10392 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 10393 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 10394 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqoddm1div8 10395 A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)

Theoremnnsqcld 10396 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 10397 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 10398 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 10399 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzapd 10400 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
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