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Theorem List for Intuitionistic Logic Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqsqrti 11001 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrtge0i 11002 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 11003 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 11004 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 11005 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsrei 11006 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtpclii 11007 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0ii 11008 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrt11i 11009 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtmuli 11010 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmulii 11011 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmsq2i 11012 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)

Theoremsqrtlei 11013 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtlti 11014 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremabslti 11015 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabslei 11016 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabsvalsqi 11017 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 11018 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 11019 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 11020 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 11021 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 11022 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremabsgt0api 11023 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
#

Theoremabsnegi 11024 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 11025 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremreleabsi 11026 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, 2-Oct-1999.)

Theoremabssubi 11027 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 11028 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 11029 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 11030 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivapzi 11031 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrii 11032 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999.)

Theoremabs3difi 11033 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 11034 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrtcld 11035 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtgt0d 11036 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 11037 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 11038 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 11039 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrtcld 11040 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsqd 11041 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsqd 11042 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtge0d 11043 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 11044 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtdivd 11045 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmuld 11046 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq2d 11047 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtled 11048 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtltd 11049 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 11050 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 11051 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 11052 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 11053 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 11054 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 11055 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 11056 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremicodiamlt 11057 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.)

Theoremabscld 11058 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsqd 11059 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsq2d 11060 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsge0d 11061 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsval2d 11062 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs00d 11063 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.)

Theoremabsne0d 11064 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.)

Theoremabsrpclapd 11065 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsnegd 11066 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscjd 11067 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.)

Theoremreleabsd 11068 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.)

Theoremabsexpd 11069 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubd 11070 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsmuld 11071 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdivapd 11072 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrid 11073 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difd 11074 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2dif2d 11075 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difabsd 11076 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3difd 11077 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3lemd 11078 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremqdenre 11079* 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 10134. (Contributed by BJ, 15-Oct-2021.)

4.7.5  The maximum of two real numbers

Theoremmaxcom 11080 The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)

Theoremmaxabsle 11081 An upper bound for . (Contributed by Jim Kingdon, 20-Dec-2021.)

Theoremmaxleim 11082 Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)

Theoremmaxabslemab 11083 Lemma for maxabs 11086. A variation of maxleim 11082- 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.)

Theoremmaxabslemlub 11084 Lemma for maxabs 11086. A least upper bound for . (Contributed by Jim Kingdon, 20-Dec-2021.)

Theoremmaxabslemval 11085* Lemma for maxabs 11086. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)

Theoremmaxabs 11086 Maximum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 20-Dec-2021.)

Theoremmaxcl 11087 The maximum of two real numbers is a real number. (Contributed by Jim Kingdon, 22-Dec-2021.)

Theoremmaxle1 11088 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.)

Theoremmaxle2 11089 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.)

Theoremmaxleast 11090 The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)

Theoremmaxleastb 11091 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.)

Theoremmaxleastlt 11092 The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)

Theoremmaxleb 11093 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.)

Theoremdfabsmax 11094 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.)

Theoremmaxltsup 11095 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)

Theoremmax0addsup 11096 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)

Theoremrexanre 11097* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)

Theoremrexico 11098* Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremmaxclpr 11099 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 9190 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.)

Theoremrpmaxcl 11100 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)

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