P. 118 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	absdivapd 11701	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
Theorem	abstrid 11702	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Theorem	abs2difd 11703	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
Theorem	abs2dif2d 11704	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) )
Theorem	abs2difabsd 11705	Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ ( abs ` ( A - B ) ) )
Theorem	abs3difd 11706	Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
Theorem	abs3lemd 11707	Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. RR )   &    |- ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) )   &    |- ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) )   =>    |- ( ph -> ( abs ` ( A - B ) ) < D )
Theorem	qdenre 11708*	The rational numbers are dense in RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 10471. (Contributed by BJ, 15-Oct-2021.)
|- ( ( A e. RR /\ B e. RR+ ) -> E. x e. QQ ( abs ` ( x - A ) ) < B )
4.8.5  The maximum of two real numbers
Theorem	maxcom 11709	The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
|- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < )
Theorem	maxabsle 11710	An upper bound for { A , B }. (Contributed by Jim Kingdon, 20-Dec-2021.)
|- ( ( A e. RR /\ B e. RR ) -> A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
Theorem	maxleim 11711	Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> sup ( { A , B } , RR , < ) = B ) )
Theorem	maxabslemab 11712	Lemma for maxabs 11715. A variation of maxleim 11711- 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.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) = B )
Theorem	maxabslemlub 11713	Lemma for maxabs 11715. A least upper bound for { A , B }. (Contributed by Jim Kingdon, 20-Dec-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> C < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )   =>    |- ( ph -> ( C < A \/ C < B ) )
Theorem	maxabslemval 11714*	Lemma for maxabs 11715. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\ A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) ) )
Theorem	maxabs 11715	Maximum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 20-Dec-2021.)
|- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
Theorem	maxcl 11716	The maximum of two real numbers is a real number. (Contributed by Jim Kingdon, 22-Dec-2021.)
|- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
Theorem	maxle1 11717	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.)
|- ( ( A e. RR /\ B e. RR ) -> A <_ sup ( { A , B } , RR , < ) )
Theorem	maxle2 11718	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.)
|- ( ( A e. RR /\ B e. RR ) -> B <_ sup ( { A , B } , RR , < ) )
Theorem	maxleast 11719	The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ C /\ B <_ C ) ) -> sup ( { A , B } , RR , < ) <_ C )
Theorem	maxleastb 11720	Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) <_ C <-> ( A <_ C /\ B <_ C ) ) )
Theorem	maxleastlt 11721	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ C < sup ( { A , B } , RR , < ) ) ) -> ( C < A \/ C < B ) )
Theorem	maxleb 11722	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> sup ( { A , B } , RR , < ) = B ) )
Theorem	dfabsmax 11723	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.)
|- ( A e. RR -> ( abs ` A ) = sup ( { A , -u A } , RR , < ) )
Theorem	maxltsup 11724	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) )
Theorem	max0addsup 11725	The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
|- ( A e. RR -> ( sup ( { A , 0 } , RR , < ) + sup ( { -u A , 0 } , RR , < ) ) = ( abs ` A ) )
Theorem	rexanre 11726*	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	rexico 11727*	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	maxclpr 11728	The maximum of two real numbers is one of those numbers if and only if dichotomy ( A <_ B \/ B <_ A) holds. For example, this can be combined with zletric 9486 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.)
|- ( ( A e. RR /\ B e. RR ) -> ( sup ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )
Theorem	rpmaxcl 11729	The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { A , B } , RR , < ) e. RR+ )
Theorem	zmaxcl 11730	The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> sup ( { A , B } , RR , < ) e. ZZ )
Theorem	nn0maxcl 11731	The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> sup ( { A , B } , RR , < ) e. NN0 )
Theorem	2zsupmax 11732	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.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> sup ( { A , B } , RR , < ) = if ( A <_ B , B , A ) )
Theorem	fimaxre2 11733*	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.)
|- ( ( A C_ RR /\ A e. Fin ) -> E. x e. RR A. y e. A y <_ x )
Theorem	negfi 11734*	The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
|- ( ( A C_ RR /\ A e. Fin ) -> { n e. RR | -u n e. A } e. Fin )
4.8.6  The minimum of two real numbers
Theorem	mincom 11735	The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
|- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < )
Theorem	minmax 11736	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = -u sup ( { -u A , -u B } , RR , < ) )
Theorem	mincl 11737	The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
Theorem	min1inf 11738	The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) <_ A )
Theorem	min2inf 11739	The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) <_ B )
Theorem	lemininf 11740	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ inf ( { B , C } , RR , < ) <-> ( A <_ B /\ A <_ C ) ) )
Theorem	ltmininf 11741	Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < inf ( { B , C } , RR , < ) <-> ( A < B /\ A < C ) ) )
Theorem	minabs 11742	The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) )
Theorem	minclpr 11743	The minimum of two real numbers is one of those numbers if and only if dichotomy ( A <_ B \/ B <_ A) holds. For example, this can be combined with zletric 9486 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.)
|- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )
Theorem	rpmincl 11744	The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR , < ) e. RR+ )
Theorem	bdtrilem 11745	Lemma for bdtri 11746. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( ( abs ` ( A - C ) ) + ( abs ` ( B - C ) ) ) <_ ( C + ( abs ` ( ( A + B ) - C ) ) ) )
Theorem	bdtri 11746	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> inf ( { ( A + B ) , C } , RR , < ) <_ (inf ( { A , C } , RR , < ) + inf ( { B , C } , RR , < ) ) )
Theorem	mul0inf 11747	Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11568 and mulap0bd 8800 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) = 0 ) )
Theorem	mingeb 11748	Equivalence of <_ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> inf ( { A , B } , RR , < ) = A ) )
Theorem	2zinfmin 11749	Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> inf ( { A , B } , RR , < ) = if ( A <_ B , A , B ) )
4.8.7  The maximum of two extended reals
Theorem	xrmaxleim 11750	Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )
Theorem	xrmaxiflemcl 11751	Lemma for xrmaxif 11757. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) e. RR* )
Theorem	xrmaxifle 11752	An upper bound for { A , B } in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
Theorem	xrmaxiflemab 11753	Lemma for xrmaxif 11757. A variation of xrmaxleim 11750- 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.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )
Theorem	xrmaxiflemlub 11754	Lemma for xrmaxif 11757. A least upper bound for { A , B }. (Contributed by Jim Kingdon, 28-Apr-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> C < if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )   =>    |- ( ph -> ( C < A \/ C < B ) )
Theorem	xrmaxiflemcom 11755	Lemma for xrmaxif 11757. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
Theorem	xrmaxiflemval 11756*	Lemma for xrmaxif 11757. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- M = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )   =>    |- ( ( A e. RR* /\ B e. RR* ) -> ( M e. RR* /\ A. x e. { A , B } -. M < x /\ A. x e. RR* ( x < M -> E. z e. { A , B } x < z ) ) )
Theorem	xrmaxif 11757	Maximum of two extended reals in terms of if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
Theorem	xrmaxcl 11758	The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) e. RR* )
Theorem	xrmax1sup 11759	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.)
|- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
Theorem	xrmax2sup 11760	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.)
|- ( ( A e. RR* /\ B e. RR* ) -> B <_ sup ( { A , B } , RR* , < ) )
Theorem	xrmaxrecl 11761	The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR* , < ) = sup ( { A , B } , RR , < ) )
Theorem	xrmaxleastlt 11762	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ C < sup ( { A , B } , RR* , < ) ) ) -> ( C < A \/ C < B ) )
Theorem	xrltmaxsup 11763	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < sup ( { A , B } , RR* , < ) <-> ( C < A \/ C < B ) ) )
Theorem	xrmaxltsup 11764	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) < C <-> ( A < C /\ B < C ) ) )
Theorem	xrmaxlesup 11765	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.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) <_ C <-> ( A <_ C /\ B <_ C ) ) )
Theorem	xrmaxaddlem 11766	Lemma for xrmaxadd 11767. The case where A is real. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
Theorem	xrmaxadd 11767	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
4.8.8  The minimum of two extended reals
Theorem	xrnegiso 11768	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|- F = ( x e. RR* |-> -e x )   =>    |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )
Theorem	infxrnegsupex 11769*	The infimum of a set of extended reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ph -> E. x e. RR* ( A. y e. A -. y < x /\ A. y e. RR* ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR* )   =>    |- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )
Theorem	xrnegcon1d 11770	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   =>    |- ( ph -> ( -e A = B <-> -e B = A ) )
Theorem	xrminmax 11771	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
Theorem	xrmincl 11772	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) e. RR* )
Theorem	xrmin1inf 11773	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ A )
Theorem	xrmin2inf 11774	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ B )
Theorem	xrmineqinf 11775	The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ B <_ A ) -> inf ( { A , B } , RR* , < ) = B )
Theorem	xrltmininf 11776	Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )
Theorem	xrlemininf 11777	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ inf ( { B , C } , RR* , < ) <-> ( A <_ B /\ A <_ C ) ) )
Theorem	xrminltinf 11778	Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> ( B < A \/ C < A ) ) )
Theorem	xrminrecl 11779	The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR* , < ) = inf ( { A , B } , RR , < ) )
Theorem	xrminrpcl 11780	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Theorem	xrminadd 11781	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +einf ( { B , C } , RR* , < ) ) )
Theorem	xrbdtri 11782	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) /\ ( C e. RR* /\ 0 < C ) ) -> inf ( { ( A +e B ) , C } , RR* , < ) <_ (inf ( { A , C } , RR* , < ) +einf ( { B , C } , RR* , < ) ) )
Theorem	iooinsup 11783	Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) )
4.9  Elementary limits and convergence
4.9.1  Limits
Syntax	cli 11784	Extend class notation with convergence relation for limits.
class ~~>
Definition	df-clim 11785*	Define the limit relation for complex number sequences. See clim 11787 for its relational expression. (Contributed by NM, 28-Aug-2005.)
|- ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
Theorem	climrel 11786	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Rel ~~>
Theorem	clim 11787*	Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ph -> F e. V )   &    |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	climcl 11788	Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( F ~~> A -> A e. CC )
Theorem	clim2 11789*	Express the predicate: The limit of complex number sequence F is A, or F converges to A, with more general quantifier restrictions than clim 11787. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	clim2c 11790*	Express the predicate F converges to A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )
Theorem	clim0 11791*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) )
Theorem	clim0c 11792*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
Theorem	climi 11793*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )
Theorem	climi2 11794*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )
Theorem	climi0 11795*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> 0 )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )
Theorem	climconst 11796*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   =>    |- ( ph -> F ~~> A )
Theorem	climconst2 11797	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ>= ` M ) C_ Z   &    |- Z e. _V   =>    |- ( ( A e. CC /\ M e. ZZ ) -> ( Z X. { A } ) ~~> A )
Theorem	climz 11798	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ X. { 0 } ) ~~> 0
Theorem	climuni 11799	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- ( ( F ~~> A /\ F ~~> B ) -> A = B )
Theorem	fclim 11800	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ~~> : dom ~~> --> CC