Theorem List for Intuitionistic Logic Explorer - 10901-11000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | leabsd 10901 |
A real number is less than or equal to its absolute value. (Contributed
by Mario Carneiro, 29-May-2016.)
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Theorem | absred 10902 |
Absolute value of a real number. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | resqrtcld 10903 |
The square root of a nonnegative real is a real. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | sqrtmsqd 10904 |
Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | sqrtsqd 10905 |
Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | sqrtge0d 10906 |
The square root of a nonnegative real is nonnegative. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | absidd 10907 |
A nonnegative number is its own absolute value. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | sqrtdivd 10908 |
Square root distributes over division. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | sqrtmuld 10909 |
Square root distributes over multiplication. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | sqrtsq2d 10910 |
Relationship between square root and squares. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | sqrtled 10911 |
Square root is monotonic. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | sqrtltd 10912 |
Square root is strictly monotonic. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | sqr11d 10913 |
The square root function is one-to-one. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | absltd 10914 |
Absolute value and 'less than' relation. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absled 10915 |
Absolute value and 'less than or equal to' relation. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | abssubge0d 10916 |
Absolute value of a nonnegative difference. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | abssuble0d 10917 |
Absolute value of a nonpositive difference. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absdifltd 10918 |
The absolute value of a difference and 'less than' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | absdifled 10919 |
The absolute value of a difference and 'less than or equal to' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | icodiamlt 10920 |
Two elements in a half-open interval have separation strictly less than
the difference between the endpoints. (Contributed by Stefan O'Rear,
12-Sep-2014.)
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Theorem | abscld 10921 |
Real closure of absolute value. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | absvalsqd 10922 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absvalsq2d 10923 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absge0d 10924 |
Absolute value is nonnegative. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | absval2d 10925 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | abs00d 10926 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absne0d 10927 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | absrpclapd 10928 |
The absolute value of a complex number apart from zero is a positive
real. (Contributed by Jim Kingdon, 13-Aug-2021.)
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# |
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Theorem | absnegd 10929 |
Absolute value of negative. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | abscjd 10930 |
The absolute value of a number and its conjugate are the same.
Proposition 10-3.7(b) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | releabsd 10931 |
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 Mario
Carneiro, 29-May-2016.)
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Theorem | absexpd 10932 |
Absolute value of positive integer exponentiation. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | abssubd 10933 |
Swapping order of subtraction doesn't change the absolute value.
Example of [Apostol] p. 363.
(Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | absmuld 10934 |
Absolute value distributes over multiplication. Proposition 10-3.7(f)
of [Gleason] p. 133. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | absdivapd 10935 |
Absolute value distributes over division. (Contributed by Jim
Kingdon, 13-Aug-2021.)
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#
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Theorem | abstrid 10936 |
Triangle inequality for absolute value. Proposition 10-3.7(h) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | abs2difd 10937 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | abs2dif2d 10938 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | abs2difabsd 10939 |
Absolute value of difference of absolute values. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | abs3difd 10940 |
Absolute value of differences around common element. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | abs3lemd 10941 |
Lemma involving absolute value of differences. (Contributed by Mario
Carneiro, 29-May-2016.)
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Theorem | qdenre 10942* |
The rational numbers are dense in : any real number can be
approximated with arbitrary precision by a rational number. For order
theoretic density, see qbtwnre 10002. (Contributed by BJ, 15-Oct-2021.)
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4.7.5 The maximum of two real
numbers
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Theorem | maxcom 10943 |
The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
(Contributed by Jim Kingdon, 21-Dec-2021.)
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Theorem | maxabsle 10944 |
An upper bound for . (Contributed by Jim Kingdon,
20-Dec-2021.)
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Theorem | maxleim 10945 |
Value of maximum when we know which number is larger. (Contributed by
Jim Kingdon, 21-Dec-2021.)
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Theorem | maxabslemab 10946 |
Lemma for maxabs 10949. A variation of maxleim 10945- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 21-Dec-2021.)
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Theorem | maxabslemlub 10947 |
Lemma for maxabs 10949. A least upper bound for .
(Contributed by Jim Kingdon, 20-Dec-2021.)
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Theorem | maxabslemval 10948* |
Lemma for maxabs 10949. Value of the supremum. (Contributed by
Jim
Kingdon, 22-Dec-2021.)
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Theorem | maxabs 10949 |
Maximum of two real numbers in terms of absolute value. (Contributed by
Jim Kingdon, 20-Dec-2021.)
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Theorem | maxcl 10950 |
The maximum of two real numbers is a real number. (Contributed by Jim
Kingdon, 22-Dec-2021.)
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Theorem | maxle1 10951 |
The maximum of two reals is no smaller than the first real. Lemma 3.10 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
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Theorem | maxle2 10952 |
The maximum of two reals is no smaller than the second real. Lemma 3.10
of [Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
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Theorem | maxleast 10953 |
The maximum of two reals is a least upper bound. Lemma 3.11 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 22-Dec-2021.)
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Theorem | maxleastb 10954 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Jim Kingdon, 31-Jan-2022.)
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Theorem | maxleastlt 10955 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
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Theorem | maxleb 10956 |
Equivalence of
and being equal to the maximum of two reals. Lemma
3.12 of [Geuvers], p. 10. (Contributed by
Jim Kingdon, 21-Dec-2021.)
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Theorem | dfabsmax 10957 |
Absolute value of a real number in terms of maximum. Definition 3.13 of
[Geuvers], p. 11. (Contributed by BJ and
Jim Kingdon, 21-Dec-2021.)
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Theorem | maxltsup 10958 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 10-Feb-2022.)
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Theorem | max0addsup 10959 |
The sum of the positive and negative part functions is the absolute value
function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
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Theorem | rexanre 10960* |
Combine two different upper real properties into one. (Contributed by
Mario Carneiro, 8-May-2016.)
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Theorem | rexico 10961* |
Restrict the base of an upper real quantifier to an upper real set.
(Contributed by Mario Carneiro, 12-May-2016.)
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Theorem | maxclpr 10962 |
The maximum of two real numbers is one of those numbers if and only if
dichotomy (
) holds. For example, this
can be
combined with zletric 9066 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 1-Feb-2022.)
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Theorem | rpmaxcl 10963 |
The maximum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 10-Nov-2023.)
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Theorem | zmaxcl 10964 |
The maximum of two integers is an integer. (Contributed by Jim Kingdon,
27-Sep-2022.)
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Theorem | 2zsupmax 10965 |
Two ways to express the maximum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 22-Jan-2023.)
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Theorem | fimaxre2 10966* |
A nonempty finite set of real numbers has an upper bound. (Contributed
by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro,
13-Feb-2014.)
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Theorem | negfi 10967* |
The negation of a finite set of real numbers is finite. (Contributed by
AV, 9-Aug-2020.)
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4.7.6 The minimum of two real
numbers
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Theorem | mincom 10968 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
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inf inf
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Theorem | minmax 10969 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
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inf |
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Theorem | mincl 10970 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
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inf |
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Theorem | min1inf 10971 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
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inf |
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Theorem | min2inf 10972 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
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inf |
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Theorem | lemininf 10973 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by NM, 3-Aug-2007.)
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inf
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Theorem | ltmininf 10974 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
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inf |
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Theorem | minabs 10975 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
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inf
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Theorem | minclpr 10976 |
The minimum of two real numbers is one of those numbers if and only if
dichotomy (
) holds. For example, this
can be
combined with zletric 9066 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 23-May-2023.)
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inf
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Theorem | rpmincl 10977 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
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inf |
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Theorem | bdtrilem 10978 |
Lemma for bdtri 10979. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
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Theorem | bdtri 10979 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
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inf
inf inf |
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Theorem | mul0inf 10980 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 10802 and mulap0bd 8386 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
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inf |
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4.7.7 The maximum of two extended
reals
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Theorem | xrmaxleim 10981 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
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Theorem | xrmaxiflemcl 10982 |
Lemma for xrmaxif 10988. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
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Theorem | xrmaxifle 10983 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxiflemab 10984 |
Lemma for xrmaxif 10988. A variation of xrmaxleim 10981- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxiflemlub 10985 |
Lemma for xrmaxif 10988. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
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Theorem | xrmaxiflemcom 10986 |
Lemma for xrmaxif 10988. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
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Theorem | xrmaxiflemval 10987* |
Lemma for xrmaxif 10988. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
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Theorem | xrmaxif 10988 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxcl 10989 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
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Theorem | xrmax1sup 10990 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
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Theorem | xrmax2sup 10991 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
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Theorem | xrmaxrecl 10992 |
The maximum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
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Theorem | xrmaxleastlt 10993 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
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Theorem | xrltmaxsup 10994 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
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Theorem | xrmaxltsup 10995 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
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Theorem | xrmaxlesup 10996 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 10-May-2023.)
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Theorem | xrmaxaddlem 10997 |
Lemma for xrmaxadd 10998. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
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Theorem | xrmaxadd 10998 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
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4.7.8 The minimum of two extended
reals
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Theorem | xrnegiso 10999 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
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Theorem | infxrnegsupex 11000* |
The infimum of a set of extended reals is the negative of the
supremum of the negatives of its elements. (Contributed by Jim Kingdon,
2-May-2023.)
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inf
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