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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqeq0i 9501 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)

Theoremsqmuli 9502 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)

Theoremsqdivapi 9503 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremresqcli 9504 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)

Theoremsqgt0api 9505 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremsqge0i 9506 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)

Theoremlt2sqi 9507 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremle2sqi 9508 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)

Theoremsq11i 9509 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)

Theoremsq0 9510 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)

Theoremsq0i 9511 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)

Theoremsq0id 9512 If a number is zero, its square is zero. Deduction form of sq0i 9511. Converse of sqeq0d 9548. (Contributed by David Moews, 28-Feb-2017.)

Theoremsq1 9513 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)

Theoremneg1sqe1 9514 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremsq2 9515 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)

Theoremsq3 9516 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)

Theoremcu2 9517 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)

Theoremirec 9518 The reciprocal of . (Contributed by NM, 11-Oct-1999.)

Theoremi2 9519 squared. (Contributed by NM, 6-May-1999.)

Theoremi3 9520 cubed. (Contributed by NM, 31-Jan-2007.)

Theoremi4 9521 to the fourth power. (Contributed by NM, 31-Jan-2007.)

Theoremnnlesq 9522 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 9523 Taking to the -th power is the same as using the -th power instead, by i4 9521. (Contributed by Mario Carneiro, 7-Jul-2014.)

Theoremexpnass 9524 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)

Theoremsubsq 9525 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Theoremsubsq2 9526 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 9527 The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Theoremsubsqi 9528 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)

Theorembinom2 9529 The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Theorembinom21 9530 Special case of binom2 9529 where . (Contributed by Scott Fenton, 11-May-2014.)

Theorembinom2sub 9531 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Theorembinom2subi 9532 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremmulbinom2 9533 The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)

Theorembinom3 9534 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

Theoremzesq 9535 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremnnesq 9536 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theorembernneq 9537 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)

Theorembernneq2 9538 Variation of Bernoulli's inequality bernneq 9537. (Contributed by NM, 18-Oct-2007.)

Theorembernneq3 9539 A corollary of bernneq 9537. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremexpnbnd 9540* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Theoremexpnlbnd 9541* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)

Theoremexpnlbnd2 9542* 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 9543 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexp1d 9544 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpeq0d 9545 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqvald 9546 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqcld 9547 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqeq0d 9548 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpcld 9549 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpp1d 9550 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 9551 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 9552 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 9553 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpclzapd 9554 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpap0d 9555 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
#               #

Theoremexpnegapd 9556 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexprecapd 9557 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpp1zapd 9558 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpm1apd 9559 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremexpsubapd 9560 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theoremsqmuld 9561 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqdivapd 9562 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremexpdivapd 9563 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremmulexpd 9564 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 9565 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpcld 9566 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge0d 9567 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremexpge1d 9568 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqoddm1div8 9569 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 9570 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 9571 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 9572 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 9573 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzapd 9574 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremresqcld 9575 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqge0d 9576 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqgt0apd 9577 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremleexp2ad 9578 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2rd 9579 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2sqd 9580 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremle2sqd 9581 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11d 9582 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11ap 9583 Analogue to sq11 9492 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
# #

Theoremsq10 9584 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
; ;;

Theoremsq10e99m1 9585 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
; ;

Theorem3dec 9586 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.)
;; ; ;

3.6.6  Ordered pair theorem for nonnegative integers

Theoremnn0le2msqd 9587 The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthlem1d 9588 A rather pretty lemma for nn0opth2 9592. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthlem2d 9589 Lemma for nn0opth2 9592. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opthd 9590 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 3412 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opth2d 9591 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 9590. (Contributed by Jim Kingdon, 31-Oct-2021.)

Theoremnn0opth2 9592 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 9590. (Contributed by NM, 22-Jul-2004.)

3.6.7  Factorial function

Syntaxcfa 9593 Extend class notation to include the factorial of nonnegative integers.

Definitiondf-fac 9594 Define the factorial function on nonnegative integers. For example, because (ex-fac 10281). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)

Theoremfacnn 9595 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 9596 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 9597 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 9598 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 9599 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 9600 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

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