Theorem List for Intuitionistic Logic Explorer - 9601-9700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | sqrt0rlem 9601 |
Lemma for sqrt0 9602. (Contributed by Jim Kingdon, 26-Aug-2020.)
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Theorem | sqrt0 9602 |
Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
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Theorem | resqrexlem1arp 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.)
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Theorem | resqrexlemp1rp 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.)
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Theorem | resqrexlemf 9605* |
Lemma for resqrex 9624. The sequence is a function. (Contributed
by
Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
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Theorem | resqrexlemf1 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.)
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Theorem | resqrexlemfp1 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.)
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Theorem | resqrexlemover 9608* |
Lemma for resqrex 9624. Each element of the sequence is an
overestimate. (Contributed by Mario Carneiro and Jim Kingdon,
27-Jul-2021.)
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Theorem | resqrexlemdec 9609* |
Lemma for resqrex 9624. The sequence is decreasing. (Contributed
by
Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
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Theorem | resqrexlemdecn 9610* |
Lemma for resqrex 9624. The sequence is decreasing. (Contributed
by
Jim Kingdon, 31-Jul-2021.)
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Theorem | resqrexlemlo 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.)
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Theorem | resqrexlemcalc1 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.)
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Theorem | resqrexlemcalc2 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.)
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Theorem | resqrexlemcalc3 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.)
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Theorem | resqrexlemnmsq 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.)
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Theorem | resqrexlemnm 9616* |
Lemma for resqrex 9624. The difference between two terms of the
sequence. (Contributed by Mario Carneiro and Jim Kingdon,
31-Jul-2021.)
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Theorem | resqrexlemcvg 9617* |
Lemma for resqrex 9624. The sequence has a limit. (Contributed by
Jim
Kingdon, 6-Aug-2021.)
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Theorem | resqrexlemgt0 9618* |
Lemma for resqrex 9624. A limit is nonnegative. (Contributed by
Jim
Kingdon, 7-Aug-2021.)
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Theorem | resqrexlemoverl 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.)
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Theorem | resqrexlemglsq 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.)
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Theorem | resqrexlemga 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.)
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Theorem | resqrexlemsqa 9622* |
Lemma for resqrex 9624. The square of a limit is .
(Contributed by Jim Kingdon, 7-Aug-2021.)
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Theorem | resqrexlemex 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.)
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Theorem | resqrex 9624* |
Existence of a square root for positive reals. (Contributed by Mario
Carneiro, 9-Jul-2013.)
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Theorem | rsqrmo 9625* |
Uniqueness for the square root function. (Contributed by Jim Kingdon,
10-Aug-2021.)
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Theorem | rersqreu 9626* |
Existence and uniqueness for the real square root function.
(Contributed by Jim Kingdon, 10-Aug-2021.)
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Theorem | resqrtcl 9627 |
Closure of the square root function. (Contributed by Mario Carneiro,
9-Jul-2013.)
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Theorem | rersqrtthlem 9628 |
Lemma for resqrtth 9629. (Contributed by Jim Kingdon, 10-Aug-2021.)
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Theorem | resqrtth 9629 |
Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29.
(Contributed by Mario Carneiro, 9-Jul-2013.)
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Theorem | remsqsqrt 9630 |
Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
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Theorem | sqrtge0 9631 |
The square root function is nonnegative for nonnegative input.
(Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro,
9-Jul-2013.)
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Theorem | sqrtgt0 9632 |
The square root function is positive for positive input. (Contributed by
Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
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Theorem | sqrtmul 9633 |
Square root distributes over multiplication. (Contributed by NM,
30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
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Theorem | sqrtle 9634 |
Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof
shortened by Mario Carneiro, 29-May-2016.)
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Theorem | sqrtlt 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.)
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Theorem | sqrt11ap 9636 |
Analogue to sqrt11 9637 but for apartness. (Contributed by Jim
Kingdon,
11-Aug-2021.)
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Theorem | sqrt11 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.)
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Theorem | sqrt00 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.)
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Theorem | rpsqrtcl 9639 |
The square root of a positive real is a positive real. (Contributed by
NM, 22-Feb-2008.)
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Theorem | sqrtdiv 9640 |
Square root distributes over division. (Contributed by Mario Carneiro,
5-May-2016.)
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Theorem | sqrtsq2 9641 |
Relationship between square root and squares. (Contributed by NM,
31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
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Theorem | sqrtsq 9642 |
Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by
Mario Carneiro, 29-May-2016.)
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Theorem | sqrtmsq 9643 |
Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by
Mario Carneiro, 29-May-2016.)
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Theorem | sqrt1 9644 |
The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
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Theorem | sqrt4 9645 |
The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
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Theorem | sqrt9 9646 |
The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
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Theorem | sqrt2gt1lt2 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.)
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Theorem | absneg 9648 |
Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
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Theorem | abscl 9649 |
Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
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Theorem | abscj 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.)
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Theorem | absvalsq 9651 |
Square of value of absolute value function. (Contributed by NM,
16-Jan-2006.)
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Theorem | absvalsq2 9652 |
Square of value of absolute value function. (Contributed by NM,
1-Feb-2007.)
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Theorem | sqabsadd 9653 |
Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason]
p. 133. (Contributed by NM, 21-Jan-2007.)
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Theorem | sqabssub 9654 |
Square of absolute value of difference. (Contributed by NM,
21-Jan-2007.)
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Theorem | absval2 9655 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by NM, 17-Mar-2005.)
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Theorem | abs0 9656 |
The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by
Mario Carneiro, 29-May-2016.)
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Theorem | absi 9657 |
The absolute value of the imaginary unit. (Contributed by NM,
26-Mar-2005.)
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Theorem | absge0 9658 |
Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.)
(Revised by Mario Carneiro, 29-May-2016.)
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Theorem | absrpclap 9659 |
The absolute value of a number apart from zero is a positive real.
(Contributed by Jim Kingdon, 11-Aug-2021.)
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Theorem | abs00ap 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.)
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Theorem | absext 9661 |
Strong extensionality for absolute value. (Contributed by Jim Kingdon,
12-Aug-2021.)
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Theorem | abs00 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.)
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Theorem | abs00ad 9663 |
A complex number is zero iff its absolute value is zero. Deduction form
of abs00 9662. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | abs00bd 9664 |
If a complex number is zero, its absolute value is zero. (Contributed
by David Moews, 28-Feb-2017.)
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Theorem | absreimsq 9665 |
Square of the absolute value of a number that has been decomposed into
real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
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Theorem | absreim 9666 |
Absolute value of a number that has been decomposed into real and
imaginary parts. (Contributed by NM, 14-Jan-2006.)
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Theorem | absmul 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.)
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Theorem | absdivap 9668 |
Absolute value distributes over division. (Contributed by Jim Kingdon,
11-Aug-2021.)
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Theorem | absid 9669 |
A nonnegative number is its own absolute value. (Contributed by NM,
11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
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Theorem | abs1 9670 |
The absolute value of 1. Common special case. (Contributed by David A.
Wheeler, 16-Jul-2016.)
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Theorem | absnid 9671 |
A negative number is the negative of its own absolute value. (Contributed
by NM, 27-Feb-2005.)
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Theorem | leabs 9672 |
A real number is less than or equal to its absolute value. (Contributed
by NM, 27-Feb-2005.)
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Theorem | absre 9673 |
Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
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Theorem | absresq 9674 |
Square of the absolute value of a real number. (Contributed by NM,
16-Jan-2006.)
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Theorem | absexp 9675 |
Absolute value of positive integer exponentiation. (Contributed by NM,
5-Jan-2006.)
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Theorem | absexpzap 9676 |
Absolute value of integer exponentiation. (Contributed by Jim Kingdon,
11-Aug-2021.)
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Theorem | abssq 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.)
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Theorem | sqabs 9678 |
The squares of two reals are equal iff their absolute values are equal.
(Contributed by NM, 6-Mar-2009.)
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Theorem | absrele 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.)
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Theorem | absimle 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.)
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Theorem | nn0abscl 9681 |
The absolute value of an integer is a nonnegative integer. (Contributed
by NM, 27-Feb-2005.)
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Theorem | zabscl 9682 |
The absolute value of an integer is an integer. (Contributed by Stefan
O'Rear, 24-Sep-2014.)
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Theorem | ltabs 9683 |
A number which is less than its absolute value is negative. (Contributed
by Jim Kingdon, 12-Aug-2021.)
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Theorem | abslt 9684 |
Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
(Revised by Mario Carneiro, 29-May-2016.)
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Theorem | absle 9685 |
Absolute value and 'less than or equal to' relation. (Contributed by NM,
6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
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Theorem | abssubap0 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.)
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Theorem | abssubne0 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.)
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Theorem | absdiflt 9688 |
The absolute value of a difference and 'less than' relation. (Contributed
by Paul Chapman, 18-Sep-2007.)
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Theorem | absdifle 9689 |
The absolute value of a difference and 'less than or equal to' relation.
(Contributed by Paul Chapman, 18-Sep-2007.)
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Theorem | elicc4abs 9690 |
Membership in a symmetric closed real interval. (Contributed by Stefan
O'Rear, 16-Nov-2014.)
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Theorem | lenegsq 9691 |
Comparison to a nonnegative number based on comparison to squares.
(Contributed by NM, 16-Jan-2006.)
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Theorem | releabs 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.)
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Theorem | recvalap 9693 |
Reciprocal expressed with a real denominator. (Contributed by Jim
Kingdon, 13-Aug-2021.)
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Theorem | absidm 9694 |
The absolute value function is idempotent. (Contributed by NM,
20-Nov-2004.)
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Theorem | absgt0ap 9695 |
The absolute value of a number apart from zero is positive. (Contributed
by Jim Kingdon, 13-Aug-2021.)
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Theorem | nnabscl 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.)
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Theorem | abssub 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.)
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Theorem | abssubge0 9698 |
Absolute value of a nonnegative difference. (Contributed by NM,
14-Feb-2008.)
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Theorem | abssuble0 9699 |
Absolute value of a nonpositive difference. (Contributed by FL,
3-Jan-2008.)
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Theorem | abstri 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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