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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsq3 10401 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)

Theoremsq4e2t8 10402 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)

Theoremcu2 10403 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 10404 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 10405 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 10406 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 10407 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 10408 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 10409 Taking to the -th power is the same as using the -th power instead, by i4 10407. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 10410 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsubsq 10411 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 10412 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 10413 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theoremsubsqi 10414 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theorembinom2 10415 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 10416 Special case of binom2 10415 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 10417 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2sub1 10418 Special case of binom2sub 10417 where . (Contributed by AV, 2-Aug-2021.)

Theorembinom2subi 10419 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremmulbinom2 10420 The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)

Theorembinom3 10421 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

Theoremzesq 10422 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremnnesq 10423 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theorembernneq 10424 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)

Theorembernneq2 10425 Variation of Bernoulli's inequality bernneq 10424. (Contributed by NM, 18-Oct-2007.)

Theorembernneq3 10426 A corollary of bernneq 10424. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremexpnbnd 10427* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Theoremexpnlbnd 10428* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 10429* 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 10430 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 10431 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 10432 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 10433 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 10434 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 10435 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 10436 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 10437 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 10438 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 10439 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 10440 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpclzapd 10441 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpap0d 10442 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
#               #

Theoremexpnegapd 10443 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexprecapd 10444 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpp1zapd 10445 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpm1apd 10446 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpsubapd 10447 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremsqmuld 10448 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivapd 10449 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremexpdivapd 10450 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremmulexpd 10451 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 10452 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 10453 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 10454 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 10455 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqoddm1div8 10456 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 10457 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 10458 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 10459 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 10460 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzapd 10461 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremresqcld 10462 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqge0d 10463 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqgt0apd 10464 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremleexp2ad 10465 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2rd 10466 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2sqd 10467 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremle2sqd 10468 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11d 10469 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11ap 10470 Analogue to sq11 10377 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
# #

Theoremsq10 10471 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
; ;;

Theoremsq10e99m1 10472 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
; ;

Theorem3dec 10473 A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
;; ; ;

Theoremexpcanlem 10474 Lemma for expcan 10475. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.)

Theoremexpcan 10475 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpcand 10476 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

4.6.7  Ordered pair theorem for nonnegative integers

Theoremnn0le2msqd 10477 The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthlem1d 10478 A rather pretty lemma for nn0opth2 10482. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthlem2d 10479 Lemma for nn0opth2 10482. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthd 10480 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 3536 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opth2d 10481 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10480. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opth2 10482 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10480. (Contributed by NM, 22-Jul-2004.)

4.6.8  Factorial function

Syntaxcfa 10483 Extend class notation to include the factorial of nonnegative integers.

Definitiondf-fac 10484 Define the factorial function on nonnegative integers. For example, because (ex-fac 13045). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)

Theoremfacnn 10485 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 10486 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 10487 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 10488 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 10489 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 10490 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

Theoremfac4 10491 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
;

Theoremfacnn2 10492 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

Theoremfaccl 10493 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)

Theoremfaccld 10494 Closure of the factorial function, deduction version of faccl 10493. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremfacne0 10495 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)

Theoremfacdiv 10496 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)

Theoremfacndiv 10497 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)

Theoremfacwordi 10498 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)

Theoremfaclbnd 10499 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd2 10500 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

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