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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqrmsqd 11901 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsqd 11902 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrge0d 11903 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrnegd 11904 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 11905 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrdivd 11906 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrmuld 11907 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsq2d 11908 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrled 11909 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrltd 11910 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 11911 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 11912 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 11913 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 11914 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 11915 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 11916 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 11917 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscld 11918 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrcld 11919 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrrege0d 11920 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqsqrd 11921 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremmsqsqrd 11922 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr00d 11923 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsqd 11924 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsq2d 11925 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsge0d 11926 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsval2d 11927 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs00d 11928 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 11929 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.)

Theoremabsrpcld 11930 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnegd 11931 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscjd 11932 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 11933 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 11934 Absolute value of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubd 11935 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsmuld 11936 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdivd 11937 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabstrid 11938 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difd 11939 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2dif2d 11940 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difabsd 11941 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3difd 11942 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3lemd 11943 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)

5.8  Elementary limits and convergence

5.8.1  Superior limit (lim sup)

Syntaxclsp 11944 Extend class notation to include the limsup function.

Definitiondf-limsup 11945* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 11948 for its value. (Contributed by NM, 26-Oct-2005.)

Theoremlimsupgord 11946 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupcl 11947 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupval 11948* The superior limit of an infinite sequence of extended real numbers, which is the infimum (indicated by ) of the set of suprema of all upper infinite subsequences of . Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)

Theoremlimsupgf 11949* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupgval 11950* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupgle 11951* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsuple 11952* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsuplt 11953* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupval2 11954* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremlimsupgre 11955* If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.)

Theoremlimsupbnd1 11956* If a sequence is eventually at most , then the limsup is also at most . (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupbnd2 11957* If a sequence is eventually greater than , then the limsup is also greater than . (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

5.8.2  Limits

Syntaxcli 11958 Extend class notation with convergence relation for limits.

Syntaxcrli 11959 Extend class notation with real convergence relation for limits.

Syntaxco1 11960 Extend class notation with the set of all eventually bounded functions.

Syntaxclo1 11961 Extend class notation with the set of all eventually upper bounded functions.

Definitiondf-clim 11962* Define the limit relation for complex number sequences. See clim 11968 for its relational expression. (Contributed by NM, 28-Aug-2005.)

Definitiondf-rlim 11963* Define the limit relation for partial functions on the reals. See rlim 11969 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)

Definitiondf-o1 11964* Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)

Definitiondf-lo1 11965* Define the set of eventually upper bounded real functions. This fills a gap in coverage, to express statements like via . (Contributed by Mario Carneiro, 25-May-2016.)

Theoremclimrel 11966 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimrel 11967 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)

Theoremclim 11968* Express the predicate: The limit of complex number sequence is , or converges to . This means that for any real , no matter how small, there always exists an integer such that the absolute difference of any later complex number in the sequence and the limit is less than . (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremrlim 11969* Express the predicate: The limit of complex number function is , or converges to , in the real sense. This means that for any real , no matter how small, there always exists a number such that the absolute difference of any number in the function beyond and the limit is less than . (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremrlim2 11970* Rewrite rlim 11969 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremrlim2lt 11971* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremrlim3 11972* Restrict the range of the domain bound to reals greater than some . (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremclimcl 11973 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremrlimpm 11974 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimf 11975 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimss 11976 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimcl 11977 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremclim2 11978* Express the predicate: The limit of complex number sequence is , or converges to , with more general quantifier restrictions than clim 11968. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim2c 11979* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim0 11980* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim0c 11981* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlim0 11982* Express the predicate converges to . (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremrlim0lt 11983* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremclimi 11984* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimi2 11985* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimi0 11986* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimi 11987* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)

Theoremrlimi2 11988* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremello1 11989* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello12 11990* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello12r 11991* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1f 11992 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1dm 11993 An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1bdd 11994* The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1mpt 11995* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1mpt2 11996* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1d 11997* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1bdd2 11998* If an eventually bounded function is bounded on every interval by a function , then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremlo1bddrp 11999* Refine o1bdd2 12015 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)

Theoremelo1 12000* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

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