P. 111 of Theorem List - Intuitionistic Logic Explorer

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

Theorem	absi 11001	The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	absge0 11002	Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)


Theorem	absrpclap 11003	The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
$/\$ #

Theorem	abs00ap 11004	The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
# #

Theorem	absext 11005	Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
$/\$ # #

Theorem	abs00 11006	The absolute value of a number is zero iff the number is zero. Also see abs00ap 11004 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.)


Theorem	abs00ad 11007	A complex number is zero iff its absolute value is zero. Deduction form of abs00 11006. (Contributed by David Moews, 28-Feb-2017.)


Theorem	abs00bd 11008	If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)


Theorem	absreimsq 11009	Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
$/\$

Theorem	absreim 11010	Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
$/\$

Theorem	absmul 11011	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.)
$/\$

Theorem	absdivap 11012	Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
$/\$ $/\$ #

Theorem	absid 11013	A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
$/\$

Theorem	abs1 11014	The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)


Theorem	absnid 11015	A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
$/\$

Theorem	leabs 11016	A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)


Theorem	qabsor 11017	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
$\/$

Theorem	qabsord 11018	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
$\/$

Theorem	absre 11019	Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)


Theorem	absresq 11020	Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)


Theorem	absexp 11021	Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
$/\$

Theorem	absexpzap 11022	Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
$/\$ # $/\$

Theorem	abssq 11023	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.)


Theorem	sqabs 11024	The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
$/\$

Theorem	absrele 11025	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.)


Theorem	absimle 11026	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.)


Theorem	nn0abscl 11027	The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)


Theorem	zabscl 11028	The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)


Theorem	ltabs 11029	A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
$/\$

Theorem	abslt 11030	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
$/\$ $/\$

Theorem	absle 11031	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
$/\$ $/\$

Theorem	abssubap0 11032	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.)
$/\$ $/\$ #

Theorem	abssubne0 11033	If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 11032 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)
$/\$ $/\$

Theorem	absdiflt 11034	The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
$/\$ $/\$ $/\$

Theorem	absdifle 11035	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
$/\$ $/\$ $/\$

Theorem	elicc4abs 11036	Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)
$/\$ $/\$

Theorem	lenegsq 11037	Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)
$/\$ $/\$ $/\$

Theorem	releabs 11038	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.)


Theorem	recvalap 11039	Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
$/\$ #

Theorem	absidm 11040	The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)


Theorem	absgt0ap 11041	The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theorem	nnabscl 11042	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.)
$/\$

Theorem	abssub 11043	Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
$/\$

Theorem	abssubge0 11044	Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
$/\$ $/\$

Theorem	abssuble0 11045	Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
$/\$ $/\$

Theorem	abstri 11046	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.)
$/\$

Theorem	abs3dif 11047	Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
$/\$ $/\$

Theorem	abs2dif 11048	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
$/\$

Theorem	abs2dif2 11049	Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$

Theorem	abs2difabs 11050	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
$/\$

Theorem	recan 11051*	Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
$/\$

Theorem	absf 11052	Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	abs3lem 11053	Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
$/\$ $/\$ $/\$ $/\$

Theorem	fzomaxdiflem 11054	Lemma for fzomaxdif 11055. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ $/\$ ..^ $/\$ ..^

Theorem	fzomaxdif 11055	A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ $/\$ ..^ ..^

Theorem	cau3lem 11056*	Lemma for cau3 11057. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cau3 11057*	Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of in the assertion, so it can be used with rexanuz 10930 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
$/\$ $/\$

Theorem	cau4 11058*	Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
$/\$ $/\$

Theorem	caubnd2 11059*	A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
$/\$

Theorem	amgm2 11060	Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	sqrtthi 11061	Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)


Theorem	sqrtcli 11062	The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)


Theorem	sqrtgt0i 11063	The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)


Theorem	sqrtmsqi 11064	Square root of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqrtsqi 11065	Square root of square. (Contributed by NM, 11-Aug-1999.)


Theorem	sqsqrti 11066	Square of square root. (Contributed by NM, 11-Aug-1999.)


Theorem	sqrtge0i 11067	The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)


Theorem	absidi 11068	A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)


Theorem	absnidi 11069	A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)


Theorem	leabsi 11070	A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)


Theorem	absrei 11071	Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)


Theorem	sqrtpclii 11072	The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)


Theorem	sqrtgt0ii 11073	The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)


Theorem	sqrt11i 11074	The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)
$/\$

Theorem	sqrtmuli 11075	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
$/\$

Theorem	sqrtmulii 11076	Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)


Theorem	sqrtmsq2i 11077	Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)
$/\$

Theorem	sqrtlei 11078	Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
$/\$

Theorem	sqrtlti 11079	Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
$/\$

Theorem	abslti 11080	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
$/\$

Theorem	abslei 11081	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
$/\$

Theorem	absvalsqi 11082	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)


Theorem	absvalsq2i 11083	Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)


Theorem	abscli 11084	Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)


Theorem	absge0i 11085	Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)


Theorem	absval2i 11086	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)


Theorem	abs00i 11087	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.)


Theorem	absgt0api 11088	The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
#

Theorem	absnegi 11089	Absolute value of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	abscji 11090	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.)


Theorem	releabsi 11091	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.)


Theorem	abssubi 11092	Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)


Theorem	absmuli 11093	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)


Theorem	sqabsaddi 11094	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)


Theorem	sqabssubi 11095	Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)


Theorem	absdivapzi 11096	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theorem	abstrii 11097	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.)


Theorem	abs3difi 11098	Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)


Theorem	abs3lemi 11099	Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
$/\$

Theorem	rpsqrtcld 11100	The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

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