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

Theoremreexpclz 11401 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)

Theoremqexpclz 11402 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theoremm1expcl2 11403 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremm1expcl 11404 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremexpclzlem 11405 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpclz 11406 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremnn0expcli 11407 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremexpm1t 11408 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)

Theorem1exp 11409 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpeq0 11410 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)

Theoremexpne0 11411 Natural number exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.)

Theoremexpne0i 11412 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpgt0 11413 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpnegz 11414 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theorem0exp 11415 Value of zero raised to a natural number power. (Contributed by NM, 19-Aug-2004.)

Theoremexpge0 11416 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpge1 11417 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 11418 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 11419 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 11420 Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexprec 11421 Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpadd 11422 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)

Theoremexpaddzlem 11423 Lemma for expaddz 11424. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpaddz 11424 Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpmul 11425 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 11426 Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpsub 11427 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpp1z 11428 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpm1 11429 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 11430 Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremltexp2a 11431 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpcan 11432 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremltexp2 11433 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremleexp2 11434 Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremleexp2a 11435 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremltexp2r 11436 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 11437 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremleexp1a 11438 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)

Theoremexple1 11439 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 11440 An upper bound on when . (Contributed by NM, 19-Dec-2005.)

Theoremsqval 11441 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsqneg 11442 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)

Theoremsqsubswap 11443 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)

Theoremsqcl 11444 Closure of square. (Contributed by NM, 10-Aug-1999.)

Theoremsqmul 11445 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)

Theoremsqeq0 11446 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)

Theoremsqdiv 11447 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)

Theoremsqne0 11448 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)

Theoremresqcl 11449 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)

Theoremsqgt0 11450 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)

Theoremnnsqcl 11451 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremzsqcl 11452 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremqsqcl 11453 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremsq11 11454 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 11455 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)

Theoremle2sq 11456 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)

Theoremle2sq2 11457 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)

Theoremsqge0 11458 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)

Theoremzsqcl2 11459 The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremsumsqeq0 11460 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 11461 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)

Theoremsqcli 11462 Closure of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqeq0i 11463 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)

Theoremsqrecii 11464 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)

Theoremsqmuli 11465 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)

Theoremsqdivi 11466 Distribution of square over division. (Contributed by NM, 20-Aug-2001.)

Theoremresqcli 11467 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)

Theoremsqgt0i 11468 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)

Theoremsqge0i 11469 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)

Theoremlt2sqi 11470 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremle2sqi 11471 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremsq11i 11472 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)

Theoremsq0 11473 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)

Theoremsq0i 11474 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)

Theoremsq0id 11475 If a number is zero, its square is zero. Deduction form of sq0i 11474. Converse of sqeq0d 11522. (Contributed by David Moews, 28-Feb-2017.)

Theoremsq1 11476 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)

Theoremsq2 11477 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)

Theoremsq3 11478 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)

Theoremcu2 11479 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 11480 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 11481 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 11482 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 11483 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 11484 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 11485 Taking to the -th power is the same as using the -th power instead, by i4 11483. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 11486 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsqlecan 11487 Cancel one factor of a square in a comparison. Unlike lemul1 9862, the common factor may be zero. (Contributed by NM, 17-Jan-2008.)

Theoremsubsq 11488 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 11489 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 11490 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theorembinom2aiOLD 11491 Product of sum and difference. (Contributed by NM, 7-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsubsqi 11492 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theoremsqeqori 11493 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 11494 The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)

Theoremsqeqor 11495 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 11496 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 11497 Special case of binom2 11496 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 11498 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2subi 11499 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theorembinom3 11500 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

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