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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqrt0rlem 9601 Lemma for sqrt0 9602. (Contributed by Jim Kingdon, 26-Aug-2020.)

Theoremsqrt0 9602 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrexlem1arp 9603* Lemma for resqrex 9624. is a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)

Theoremresqrexlemp1rp 9604* Lemma for resqrex 9624. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)

Theoremresqrexlemf 9605* Lemma for resqrex 9624. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrexlemf1 9606* Lemma for resqrex 9624. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrexlemfp1 9607* Lemma for resqrex 9624. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrexlemover 9608* Lemma for resqrex 9624. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrexlemdec 9609* Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)

Theoremresqrexlemdecn 9610* Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)

Theoremresqrexlemlo 9611* Lemma for resqrex 9624. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)

Theoremresqrexlemcalc1 9612* Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)

Theoremresqrexlemcalc2 9613* Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)

Theoremresqrexlemcalc3 9614* Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)

Theoremresqrexlemnmsq 9615* Lemma for resqrex 9624. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)

Theoremresqrexlemnm 9616* Lemma for resqrex 9624. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)

Theoremresqrexlemcvg 9617* Lemma for resqrex 9624. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)

Theoremresqrexlemgt0 9618* Lemma for resqrex 9624. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)

Theoremresqrexlemoverl 9619* Lemma for resqrex 9624. Every term in the sequence is an overestimate compared with the limit . Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)

Theoremresqrexlemglsq 9620* Lemma for resqrex 9624. The sequence formed by squaring each term of converges to . (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)

Theoremresqrexlemga 9621* Lemma for resqrex 9624. The sequence formed by squaring each term of converges to . (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)

Theoremresqrexlemsqa 9622* Lemma for resqrex 9624. The square of a limit is . (Contributed by Jim Kingdon, 7-Aug-2021.)

Theoremresqrexlemex 9623* Lemma for resqrex 9624. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrex 9624* Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremrsqrmo 9625* Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremrersqreu 9626* Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremresqrtcl 9627 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremrersqrtthlem 9628 Lemma for resqrtth 9629. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremresqrtth 9629 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremremsqsqrt 9630 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrtge0 9631 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)

Theoremsqrtgt0 9632 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtmul 9633 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtle 9634 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrtlt 9635 Square root is strictly monotonic. Closed form of sqrtlti 9733. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrt11ap 9636 Analogue to sqrt11 9637 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
# #

Theoremsqrt11 9637 The square root function is one-to-one. Also see sqrt11ap 9636 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremsqrt00 9638 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremrpsqrtcl 9639 The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)

Theoremsqrtdiv 9640 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremsqrtsq2 9641 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq 9642 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsq 9643 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrt1 9644 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

Theoremsqrt4 9645 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

Theoremsqrt9 9646 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

Theoremsqrt2gt1lt2 9647 The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsneg 9648 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

Theoremabscl 9649 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)

Theoremabscj 9650 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)

Theoremabsvalsq 9651 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)

Theoremabsvalsq2 9652 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)

Theoremsqabsadd 9653 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)

Theoremsqabssub 9654 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Theoremabsval2 9655 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

Theoremabs0 9656 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsi 9657 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 9658 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpclap 9659 The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabs00ap 9660 The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
# #

Theoremabsext 9661 Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
# #

Theoremabs00 9662 The absolute value of a number is zero iff the number is zero. Also see abs00ap 9660 which is similar but for apartness. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00ad 9663 A complex number is zero iff its absolute value is zero. Deduction form of abs00 9662. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 9664 If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 9665 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 9666 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 9667 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsdivap 9668 Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabsid 9669 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabs1 9670 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremabsnid 9671 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremleabs 9672 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremabsre 9673 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 9674 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsexp 9675 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpzap 9676 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabssq 9677 Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqabs 9678 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 9679 The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)

Theoremabsimle 9680 The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremnn0abscl 9681 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)

Theoremzabscl 9682 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremltabs 9683 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)

Theoremabslt 9684 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 9685 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubap0 9686 If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
#

Theoremabssubne0 9687 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 9686 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)

Theoremabsdiflt 9688 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 9689 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 9690 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 9691 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 9692 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 1-Apr-2005.)

Theoremrecvalap 9693 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsidm 9694 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0ap 9695 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremnnabscl 9696 The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremabssub 9697 Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabssubge0 9698 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 9699 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabstri 9700 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

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