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Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqeq0 11401 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)

Theoremsqdiv 11402 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)

Theoremsqne0 11403 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)

Theoremresqcl 11404 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)

Theoremsqgt0 11405 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)

Theoremnnsqcl 11406 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremzsqcl 11407 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremqsqcl 11408 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremsq11 11409 The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)

Theoremlt2sq 11410 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)

Theoremle2sq 11411 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)

Theoremle2sq2 11412 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)

Theoremsqge0 11413 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)

Theoremzsqcl2 11414 The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremsumsqeq0 11415 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 11416 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)

Theoremsqcli 11417 Closure of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqeq0i 11418 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)

Theoremsqrecii 11419 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)

Theoremsqmuli 11420 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)

Theoremsqdivi 11421 Distribution of square over division. (Contributed by NM, 20-Aug-2001.)

Theoremresqcli 11422 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)

Theoremsqgt0i 11423 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)

Theoremsqge0i 11424 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)

Theoremlt2sqi 11425 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremle2sqi 11426 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremsq11i 11427 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)

Theoremsq0 11428 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)

Theoremsq0i 11429 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)

Theoremsq0id 11430 If a number is zero, its square is zero. Deduction form of sq0i 11429. Converse of sqeq0d 11477. (Contributed by David Moews, 28-Feb-2017.)

Theoremsq1 11431 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)

Theoremsq2 11432 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)

Theoremsq3 11433 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)

Theoremcu2 11434 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 11435 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 11436 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 11437 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 11438 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 11439 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 11440 Taking to the -th power is the same as using the -th power instead, by i4 11438. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 11441 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsqlecan 11442 Cancel one factor of a square in a comparison. Unlike lemul1 9818, the common factor may be zero. (Contributed by NM, 17-Jan-2008.)

Theoremsubsq 11443 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 11444 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 11445 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theorembinom2aiOLD 11446 Product of sum and difference. (Contributed by NM, 7-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsubsqi 11447 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theoremsqeqori 11448 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 11449 The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)

Theoremsqeqor 11450 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 11451 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 11452 Special case of binom2 11451 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 11453 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2subi 11454 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theorembinom3 11455 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

Theoremsq01 11456 If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)

Theoremzesq 11457 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremnnesq 11458 A natural number is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremcrreczi 11459 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 11460 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)

Theorembernneq2 11461 Variation of Bernoulli's inequality bernneq 11460. (Contributed by NM, 18-Oct-2007.)

Theorembernneq3 11462 A corollary of bernneq 11460. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremexpnbnd 11463* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Theoremexpnlbnd 11464* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 11465* 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 11466* Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremdigit2 11467 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 11468 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 11469 Exponentiation property of the modulo operation. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremdiscr1 11470* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremdiscr 11471* 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 11472 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 11473 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 11474 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 11475 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 11476 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 11477 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 11478 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 11479 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 11480 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 11481 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 11482 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpclzd 11483 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpne0d 11484 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpnegd 11485 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexprecd 11486 Nonnegative integer exponentiation of a reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1zd 11487 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpm1d 11488 Value of a complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpsubd 11489 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqmuld 11490 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivd 11491 Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpdivd 11492 Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulexpd 11493 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 11494 Value of zero raised to a natural number power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 11495 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 11496 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 11497 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnsqcld 11498 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 11499 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 11500 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

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