P. 141 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sqrtle 14001	Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( sqr ` A ) <_ ( sqr ` B ) ) )
Theorem	sqrtlt 14002	Square root is strictly monotonic. Closed form of sqrtlti 14129. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( sqr ` A ) < ( sqr ` B ) ) )
Theorem	sqrt11 14003	The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = ( sqr ` B ) <-> A = B ) )
Theorem	sqrt00 14004	A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqr ` A ) = 0 <-> A = 0 ) )
Theorem	rpsqrtcl 14005	The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> ( sqr ` A ) e. RR+ )
Theorem	sqrtdiv 14006	Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
Theorem	sqrtneglem 14007	The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( _i x. ( sqr ` A ) ) ^ 2 ) = -u A /\ 0 <_ ( Re ` ( _i x. ( sqr ` A ) ) ) /\ ( _i x. ( _i x. ( sqr ` A ) ) ) e/ RR+ ) )
Theorem	sqrtneg 14008	The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` -u A ) = ( _i x. ( sqr ` A ) ) )
Theorem	sqrtsq2 14009	Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) = B <-> A = ( B ^ 2 ) ) )
Theorem	sqrtsq 14010	Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A ^ 2 ) ) = A )
Theorem	sqrtmsq 14011	Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqr ` ( A x. A ) ) = A )
Theorem	sqrt1 14012	The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
|- ( sqr ` 1 ) = 1
Theorem	sqrt4 14013	The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
|- ( sqr ` 4 ) = 2
Theorem	sqrt9 14014	The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
|- ( sqr ` 9 ) = 3
Theorem	sqrt2gt1lt2 14015	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.)
|- ( 1 < ( sqr ` 2 ) /\ ( sqr ` 2 ) < 2 )
Theorem	sqrtm1 14016	The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of _i, but the definition of sqr df-sqrt 13975 has already been crafted with _i being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 10004 or i2 12965 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)
|- _i = ( sqr ` -u 1 )
Theorem	absneg 14017	Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
|- ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
Theorem	abscl 14018	Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
|- ( A e. CC -> ( abs ` A ) e. RR )
Theorem	abscj 14019	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.)
|- ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) )
Theorem	absvalsq 14020	Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
Theorem	absvalsq2 14021	Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	sqabsadd 14022	Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	sqabssub 14023	Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )
Theorem	absval2 14024	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)
|- ( A e. CC -> ( abs ` A ) = ( sqr ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )
Theorem	abs0 14025	The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( abs ` 0 ) = 0
Theorem	absi 14026	The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( abs ` _i ) = 1
Theorem	absge0 14027	Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> 0 <_ ( abs ` A ) )
Theorem	absrpcl 14028	The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )
Theorem	abs00 14029	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, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00ad 14030	A complex number is zero iff its absolute value is zero. Deduction form of abs00 14029. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) )
Theorem	abs00bd 14031	If a complex number is zero, its absolute value is zero. Converse of abs00d 14185. One-way deduction form of abs00 14029. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = 0 )   =>    |- ( ph -> ( abs ` A ) = 0 )
Theorem	absreimsq 14032	Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` ( A + ( _i x. B ) ) ) ^ 2 ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
Theorem	absreim 14033	Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
|- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A + ( _i x. B ) ) ) = ( sqr ` ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
Theorem	absmul 14034	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
Theorem	absdiv 14035	Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
Theorem	absid 14036	A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
Theorem	abs1 14037	The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- ( abs ` 1 ) = 1
Theorem	absnid 14038	A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
|- ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A )
Theorem	leabs 14039	A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
|- ( A e. RR -> A <_ ( abs ` A ) )
Theorem	absor 14040	The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)
|- ( A e. RR -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
Theorem	absre 14041	Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
|- ( A e. RR -> ( abs ` A ) = ( sqr ` ( A ^ 2 ) ) )
Theorem	absresq 14042	Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
|- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
Theorem	absmod0 14043	A is divisible by B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A mod B ) = 0 <-> ( ( abs ` A ) mod B ) = 0 ) )
Theorem	absexp 14044	Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
Theorem	absexpz 14045	Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
Theorem	abssq 14046	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.)
|- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( abs ` ( A ^ 2 ) ) )
Theorem	sqabs 14047	The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( abs ` A ) = ( abs ` B ) ) )
Theorem	absrele 14048	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.)
|- ( A e. CC -> ( abs ` ( Re ` A ) ) <_ ( abs ` A ) )
Theorem	absimle 14049	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.)
|- ( A e. CC -> ( abs ` ( Im ` A ) ) <_ ( abs ` A ) )
Theorem	max0add 14050	The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Mario Carneiro, 24-Aug-2014.)
|- ( A e. RR -> ( if ( 0 <_ A , A , 0 ) + if ( 0 <_ -u A , -u A , 0 ) ) = ( abs ` A ) )
Theorem	absz 14051	A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. RR -> ( A e. ZZ <-> ( abs ` A ) e. ZZ ) )
Theorem	nn0abscl 14052	The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. ZZ -> ( abs ` A ) e. NN0 )
Theorem	zabscl 14053	The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
|- ( A e. ZZ -> ( abs ` A ) e. ZZ )
Theorem	abslt 14054	Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) )
Theorem	absle 14055	Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )
Theorem	abssubne0 14056	If the absolute value of a complex number is less than a real, its difference from the real is nonzero. (Contributed by NM, 2-Nov-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. CC /\ B e. RR /\ ( abs ` A ) < B ) -> ( B - A ) =/= 0 )
Theorem	absdiflt 14057	The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) < C <-> ( ( B - C ) < A /\ A < ( B + C ) ) ) )
Theorem	absdifle 14058	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )
Theorem	elicc4abs 14059	Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( ( A - B ) [,] ( A + B ) ) <-> ( abs ` ( C - A ) ) <_ B ) )
Theorem	lenegsq 14060	Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)
|- ( ( A e. RR /\ B e. RR /\ 0 <_ B ) -> ( ( A <_ B /\ -u A <_ B ) <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )
Theorem	releabs 14061	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.)
|- ( A e. CC -> ( Re ` A ) <_ ( abs ` A ) )
Theorem	recval 14062	Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) = ( ( * ` A ) / ( ( abs ` A ) ^ 2 ) ) )
Theorem	absidm 14063	The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
|- ( A e. CC -> ( abs ` ( abs ` A ) ) = ( abs ` A ) )
Theorem	absgt0 14064	The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( A e. CC -> ( A =/= 0 <-> 0 < ( abs ` A ) ) )
Theorem	nnabscl 14065	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.)
|- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
Theorem	abssub 14066	Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
Theorem	abssubge0 14067	Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( B - A ) ) = ( B - A ) )
Theorem	abssuble0 14068	Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
Theorem	absmax 14069	The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)
|- ( ( A e. RR /\ B e. RR ) -> if ( A <_ B , B , A ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
Theorem	abstri 14070	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Theorem	abs3dif 14071	Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
Theorem	abs2dif 14072	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
Theorem	abs2dif2 14073	Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Theorem	abs2difabs 14074	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )
Theorem	abs1m 14075*	For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195. (Contributed by NM, 26-Mar-2005.)
|- ( A e. CC -> E. x e. CC ( ( abs ` x ) = 1 /\ ( abs ` A ) = ( x x. A ) ) )
Theorem	recan 14076*	Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( A. x e. CC ( Re ` ( x x. A ) ) = ( Re ` ( x x. B ) ) <-> A = B ) )
Theorem	absf 14077	Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- abs : CC --> RR
Theorem	abs3lem 14078	Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
Theorem	abslem2 14079	Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( * ` ( A / ( abs ` A ) ) ) x. A ) + ( ( A / ( abs ` A ) ) x. ( * ` A ) ) ) = ( 2 x. ( abs ` A ) ) )
Theorem	rddif 14080	The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
|- ( A e. RR -> ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) <_ ( 1 / 2 ) )
Theorem	absrdbnd 14081	Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
|- ( A e. RR -> ( abs ` ( |_ ` ( A + ( 1 / 2 ) ) ) ) <_ ( ( |_ ` ( abs ` A ) ) + 1 ) )
Theorem	fzomaxdiflem 14082	Lemma for fzomaxdif 14083. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( ( A e. ( C..^ D ) /\ B e. ( C..^ D ) ) /\ A <_ B ) -> ( abs ` ( B - A ) ) e. ( 0..^ ( D - C ) ) )
Theorem	fzomaxdif 14083	A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ( C..^ D ) /\ B e. ( C..^ D ) ) -> ( abs ` ( A - B ) ) e. ( 0..^ ( D - C ) ) )
Theorem	uzin2 14084	The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
|- ( ( A e. ran ZZ>= /\ B e. ran ZZ>= ) -> ( A i^i B ) e. ran ZZ>= )
Theorem	rexanuz 14085*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
|- ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <-> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ph /\ E. j e. ZZ A. k e. ( ZZ>= ` j ) ps ) )
Theorem	rexanre 14086*	Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
|- ( A C_ RR -> ( E. j e. RR A. k e. A ( j <_ k -> ( ph /\ ps ) ) <-> ( E. j e. RR A. k e. A ( j <_ k -> ph ) /\ E. j e. RR A. k e. A ( j <_ k -> ps ) ) ) )
Theorem	rexfiuz 14087*	Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
|- ( A e. Fin -> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) A. n e. A ph <-> A. n e. A E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) )
Theorem	rexuz3 14088*	Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ph ) )
Theorem	rexanuz2 14089*	Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) <-> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) )
Theorem	r19.29uz 14090*	A version of 19.29 1801 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( A. k e. Z ph /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ph /\ ps ) )
Theorem	r19.2uz 14091*	A version of r19.2z 4060 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ph -> E. k e. Z ph )
Theorem	rexuzre 14092*	Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ph <-> E. j e. RR A. k e. Z ( j <_ k -> ph ) ) )
Theorem	rexico 14093*	Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
|- ( ( A C_ RR /\ B e. RR ) -> ( E. j e. ( B [,) +oo ) A. k e. A ( j <_ k -> ph ) <-> E. j e. RR A. k e. A ( j <_ k -> ph ) ) )
Theorem	cau3lem 14094*	Lemma for cau3 14095. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
|- Z C_ ZZ   &    |- ( ta -> ps )   &    |- ( ( F ` k ) = ( F ` j ) -> ( ps <-> ch ) )   &    |- ( ( F ` k ) = ( F ` m ) -> ( ps <-> th ) )   &    |- ( ( ph /\ ch /\ ps ) -> ( G ` ( ( F ` j ) D ( F ` k ) ) ) = ( G ` ( ( F ` k ) D ( F ` j ) ) ) )   &    |- ( ( ph /\ th /\ ch ) -> ( G ` ( ( F ` m ) D ( F ` j ) ) ) = ( G ` ( ( F ` j ) D ( F ` m ) ) ) )   &    |- ( ( ph /\ ( ps /\ th ) /\ ( ch /\ x e. RR ) ) -> ( ( ( G ` ( ( F ` k ) D ( F ` j ) ) ) < ( x / 2 ) /\ ( G ` ( ( F ` j ) D ( F ` m ) ) ) < ( x / 2 ) ) -> ( G ` ( ( F ` k ) D ( F ` m ) ) ) < x ) )   =>    |- ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ta /\ ( G ` ( ( F ` k ) D ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ta /\ A. m e. ( ZZ>= ` k ) ( G ` ( ( F ` k ) D ( F ` m ) ) ) < x ) ) )
Theorem	cau3 14095*	Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of j in the assertion, so it can be used with rexanuz 14085 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ A. m e. ( ZZ>= ` k ) ( abs ` ( ( F ` k ) - ( F ` m ) ) ) < x ) )
Theorem	cau4 14096*	Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   =>    |- ( N e. Z -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) <-> A. x e. RR+ E. j e. W A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) ) )
Theorem	caubnd2 14097*	A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) -> E. y e. RR E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < y )
Theorem	caubnd 14098*	A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( A. k e. Z ( F ` k ) e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) -> E. y e. RR A. k e. Z ( abs ` ( F ` k ) ) < y )
Theorem	sqreulem 14099	Lemma for sqreu 14100: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- B = ( ( sqr ` ( abs ` A ) ) x. ( ( ( abs ` A ) + A ) / ( abs ` ( ( abs ` A ) + A ) ) ) )   =>    |- ( ( A e. CC /\ ( ( abs ` A ) + A ) =/= 0 ) -> ( ( B ^ 2 ) = A /\ 0 <_ ( Re ` B ) /\ ( _i x. B ) e/ RR+ ) )
Theorem	sqreu 14100*	Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)
|- ( A e. CC -> E! x e. CC ( ( x ^ 2 ) = A /\ 0 <_ ( Re ` x ) /\ ( _i x. x ) e/ RR+ ) )