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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreleabsi 11901 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.)
 |-  A  e.  CC   =>    |-  ( Re `  A )  <_  ( abs `  A )
 
Theoremabssubi 11902 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) )
 
Theoremabsmuli 11903 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) )
 
Theoremsqabsaddi 11904 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( abs `  ( A  +  B )
 ) ^ 2 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  +  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) )
 
Theoremsqabssubi 11905 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( abs `  ( A  -  B ) ) ^ 2 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  -  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) )
 
Theoremabsdivzi 11906 Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrii 11907 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  +  B )
 )  <_  ( ( abs `  A )  +  ( abs `  B )
 )
 
Theoremabs3difi 11908 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( abs `  ( A  -  B ) )  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) )
 
Theoremabs3lemi 11909 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 )
 
Theoremrpsqrcld 11910 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( sqr `  A )  e.  RR+ )
 
Theoremsqrgt0d 11911 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  ( sqr `  A ) )
 
Theoremabsnidd 11912 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A 
 <_  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  -u A )
 
Theoremleabsd 11913 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( abs `  A ) )
 
Theoremabsord 11914 The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A ) )
 
Theoremabsred 11915 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrcld 11916 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  A )  e. 
 RR )
 
Theoremsqrmsqd 11917 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrsqd 11918 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A ^
 2 ) )  =  A )
 
Theoremsqrge0d 11919 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( sqr `  A ) )
 
Theoremsqrnegd 11920 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  -u A )  =  ( _i  x.  ( sqr `  A ) ) )
 
Theoremabsidd 11921 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( abs `  A )  =  A )
 
Theoremsqrdivd 11922 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( sqr `  ( A  /  B ) )  =  ( ( sqr `  A )  /  ( sqr `  B ) ) )
 
Theoremsqrmuld 11923 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) ) )
 
Theoremsqrsq2d 11924 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  (
 ( sqr `  A )  =  B  <->  A  =  ( B ^ 2 ) ) )
 
Theoremsqrled 11925 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( sqr `  A )  <_  ( sqr `  B ) ) )
 
Theoremsqrltd 11926 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  ( sqr `  A )  <  ( sqr `  B ) ) )
 
Theoremsqr11d 11927 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  ( sqr `  A )  =  ( sqr `  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremabsltd 11928 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <  B  <->  ( -u B  <  A  /\  A  <  B ) ) )
 
Theoremabsled 11929 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <_  B  <->  ( -u B  <_  A  /\  A  <_  B ) ) )
 
Theoremabssubge0d 11930 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( B  -  A ) )  =  ( B  -  A ) )
 
Theoremabssuble0d 11931 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A ) )
 
Theoremabsdifltd 11932 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <  C 
 <->  ( ( B  -  C )  <  A  /\  A  <  ( B  +  C ) ) ) )
 
Theoremabsdifled 11933 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <_  C 
 <->  ( ( B  -  C )  <_  A  /\  A  <_  ( B  +  C ) ) ) )
 
Theoremabscld 11934 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremsqrcld 11935 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sqr `  A )  e. 
 CC )
 
Theoremsqrrege0d 11936 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( Re `  ( sqr `  A ) ) )
 
Theoremsqsqrd 11937 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( sqr `  A ) ^ 2 )  =  A )
 
Theoremmsqsqrd 11938 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqr00d 11939 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( sqr `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsvalsqd 11940 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2d 11941 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremabsge0d 11942 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 11943 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  (
 ( ( Re `  A ) ^ 2
 )  +  ( ( Im `  A ) ^ 2 ) ) ) )
 
Theoremabs00d 11944 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.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsne0d 11945 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.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  =/=  0 )
 
Theoremabsrpcld 11946 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  e.  RR+ )
 
Theoremabsnegd 11947 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 11948 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.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( * `  A ) )  =  ( abs `  A ) )
 
Theoremreleabsd 11949 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.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  <_  ( abs `  A )
 )
 
Theoremabsexpd 11950 Absolute value of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssubd 11951 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) ) )
 
Theoremabsmuld 11952 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) ) )
 
Theoremabsdivd 11953 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrid 11954 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 ) ) )
 
Theoremabs2difd 11955 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 ) ) )
 
Theoremabs2dif2d 11956 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 ) ) )
 
Theoremabs2difabsd 11957 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 ) ) )
 
Theoremabs3difd 11958 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 ) ) ) )
 
Theoremabs3lemd 11959 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 )
 
5.8  Elementary limits and convergence
 
5.8.1  Superior limit (lim sup)
 
Syntaxclsp 11960 Extend class notation to include the limsup function.
 class  limsup
 
Definitiondf-limsup 11961* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 11964 for its value. (Contributed by NM, 26-Oct-2005.)
 |-  limsup  =  ( x  e. 
 _V  |->  sup ( ran  (
 k  e.  RR  |->  sup ( ( ( x
 " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 ) ,  RR* ,  `'  <  ) )
 
Theoremlimsupgord 11962 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sup ( ( ( F " ( B [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) 
 <_  sup ( ( ( F " ( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupcl 11963 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( F  e.  V  ->  ( limsup `  F )  e.  RR* )
 
Theoremlimsupval 11964* The superior limit of an infinite sequence  F of extended real numbers, which is the infimum (indicated by  `'  <) of the set of suprema of all upper infinite subsequences of  F. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( F  e.  V  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
 
Theoremlimsupgf 11965* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  G : RR --> RR*
 
Theoremlimsupgval 11966* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( M  e.  RR  ->  ( G `  M )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupgle 11967* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( ( B 
 C_  RR  /\  F : B
 --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A 
 <-> 
 A. j  e.  B  ( C  <_  j  ->  ( F `  j ) 
 <_  A ) ) )
 
Theoremlimsuple 11968* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  ->  ( A  <_  ( limsup `
  F )  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
 
Theoremlimsuplt 11969* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  ->  ( ( limsup `  F )  <  A  <->  E. j  e.  RR  ( G `  j )  <  A ) )
 
Theoremlimsupval2 11970* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   =>    |-  ( ph  ->  ( limsup `
  F )  = 
 sup ( ( G
 " A ) , 
 RR* ,  `'  <  ) )
 
Theoremlimsupgre 11971* 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.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   &    |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F : Z --> RR  /\  ( limsup `  F )  <  +oo )  ->  G : RR --> RR )
 
Theoremlimsupbnd1 11972* If a sequence is eventually at most 
A, then the limsup is also at most  A. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence  1  /  n 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.)
 |-  ( ph  ->  B  C_ 
 RR )   &    |-  ( ph  ->  F : B --> RR* )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A ) )   =>    |-  ( ph  ->  ( limsup `
  F )  <_  A )
 
Theoremlimsupbnd2 11973* If a sequence is eventually greater than  A, then the limsup is also greater than  A. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ph  ->  B  C_ 
 RR )   &    |-  ( ph  ->  F : B --> RR* )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  sup ( B ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  A 
 <_  ( F `  j
 ) ) )   =>    |-  ( ph  ->  A 
 <_  ( limsup `  F )
 )
 
5.8.2  Limits
 
Syntaxcli 11974 Extend class notation with convergence relation for limits.
 class  ~~>
 
Syntaxcrli 11975 Extend class notation with real convergence relation for limits.
 class  ~~> r
 
Syntaxco1 11976 Extend class notation with the set of all eventually bounded functions.
 class  O ( 1 )
 
Syntaxclo1 11977 Extend class notation with the set of all eventually upper bounded functions.
 class  <_ O ( 1 )
 
Definitiondf-clim 11978* Define the limit relation for complex number sequences. See clim 11984 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 ) ) }
 
Definitiondf-rlim 11979* Define the limit relation for partial functions on the reals. See rlim 11985 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ~~> r  =  { <. f ,  x >.  |  (
 ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e.  dom  f ( z  <_  w  ->  ( abs `  ( (
 f `  w )  -  x ) )  < 
 y ) ) }
 
Definitiondf-o1 11980* 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  O ( 1 ) 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.)
 |-  O ( 1 )  =  { f  e.  ( CC  ^pm  RR )  |  E. x  e.  RR  E. m  e. 
 RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y ) )  <_  m }
 
Definitiondf-lo1 11981* Define the set of eventually upper bounded real functions. This fills a gap in  O ( 1 ) coverage, to express statements like  f (
x )  <_  g
( x )  +  O ( x ) via  ( x  e.  RR+  |->  ( f ( x )  -  g
( x ) )  /  x )  e. 
<_ O ( 1 ). (Contributed by Mario Carneiro, 25-May-2016.)
 |- 
 <_ O ( 1 )  =  { f  e.  ( RR  ^pm  RR )  |  E. x  e.  RR  E. m  e. 
 RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( f `  y ) 
 <_  m }
 
Theoremclimrel 11982 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |- 
 Rel 
 ~~>
 
Theoremrlimrel 11983 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
 |- 
 Rel 
 ~~> r
 
Theoremclim 11984* 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 ) ) ) )
 
Theoremrlim 11985* Express the predicate: The limit of complex number function  F is  C, or  F converges to  C, in the real sense. This means that for any real  x, no matter how small, there always exists a number  y such that the absolute difference of any number in the function beyond  y and the limit is less than  x. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ( ph  /\  z  e.  A )  ->  ( F `  z )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~> r  C  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  < 
 x ) ) ) )
 
Theoremrlim2 11986* Rewrite rlim 11985 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <_  z  ->  ( abs `  ( B  -  C ) )  < 
 x ) ) )
 
Theoremrlim2lt 11987* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <  z  ->  ( abs `  ( B  -  C ) )  < 
 x ) ) )
 
Theoremrlim3 11988* Restrict the range of the domain bound to reals greater than some  D  e.  RR. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  ( D [,)  +oo ) A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x ) ) )
 
Theoremclimcl 11989 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 )
 
Theoremrlimpm 11990 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F  e.  ( CC 
 ^pm  RR ) )
 
Theoremrlimf 11991 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
 
Theoremrlimss 11992 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  dom  F  C_  RR )
 
Theoremrlimcl 11993 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  ~~> r  A  ->  A  e.  CC )
 
Theoremclim2 11994* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 11984. (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 ) ) ) )
 
Theoremclim2c 11995* 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 ) )
 
Theoremclim0 11996* 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 ) ) )
 
Theoremclim0c 11997* 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 ) )
 
Theoremrlim0 11998* Express the predicate  B ( z ) converges to  0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <_  z  ->  ( abs `  B )  <  x ) ) )
 
Theoremrlim0lt 11999* 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.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <  z  ->  ( abs `  B )  <  x ) ) )
 
Theoremclimi 12000* 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 ) )
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