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Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsvalsqi 12201 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 12202 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 12203 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 12204 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 12205 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 12206 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.)

Theoremabsgt0i 12207 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsnegi 12208 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 12209 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 12210 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 12211 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 12212 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 12213 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 12214 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivzi 12215 Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)

Theoremabstrii 12216 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs3difi 12217 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 12218 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrcld 12219 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrgt0d 12220 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 12221 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 12222 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsord 12223 The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 12224 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrcld 12225 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrmsqd 12226 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsqd 12227 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrge0d 12228 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrnegd 12229 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 12230 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrdivd 12231 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrmuld 12232 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsq2d 12233 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrled 12234 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrltd 12235 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 12236 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 12237 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 12238 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 12239 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 12240 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 12241 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 12242 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscld 12243 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrcld 12244 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrrege0d 12245 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqsqrd 12246 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremmsqsqrd 12247 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr00d 12248 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsqd 12249 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsq2d 12250 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsge0d 12251 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsval2d 12252 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs00d 12253 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 12254 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 12255 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnegd 12256 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscjd 12257 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 12258 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 12259 Absolute value of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubd 12260 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsmuld 12261 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdivd 12262 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabstrid 12263 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difd 12264 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2dif2d 12265 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difabsd 12266 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3difd 12267 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3lemd 12268 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 12269 Extend class notation to include the limsup function.

Definitiondf-limsup 12270* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 12273 for its value. (Contributed by NM, 26-Oct-2005.)

Theoremlimsupgord 12271 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupcl 12272 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupval 12273* 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 12274* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupgval 12275* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupgle 12276* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsuple 12277* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsuplt 12278* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

Theoremlimsupval2 12279* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremlimsupgre 12280* 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 12281* 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 12282* 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 12283 Extend class notation with convergence relation for limits.

Syntaxcrli 12284 Extend class notation with real convergence relation for limits.

Syntaxco1 12285 Extend class notation with the set of all eventually bounded functions.

Syntaxclo1 12286 Extend class notation with the set of all eventually upper bounded functions.

Definitiondf-clim 12287* Define the limit relation for complex number sequences. See clim 12293 for its relational expression. (Contributed by NM, 28-Aug-2005.)

Definitiondf-rlim 12288* Define the limit relation for partial functions on the reals. See rlim 12294 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)

Definitiondf-o1 12289* 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 12290* 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 12291 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimrel 12292 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)

Theoremclim 12293* 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 12294* 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 12295* Rewrite rlim 12294 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremrlim2lt 12296* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremrlim3 12297* Restrict the range of the domain bound to reals greater than some . (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremclimcl 12298 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremrlimpm 12299 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimf 12300 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

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