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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiccleub 8901 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  <_  B )
 
Theoremiccgelb 8902 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremelioo5 8903 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C 
 /\  C  <  B ) ) )
 
Theoremelioo4g 8904 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( C  e.  ( A (,) B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  ( A  <  C  /\  C  <  B ) ) )
 
Theoremioossre 8905 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
 |-  ( A (,) B )  C_  RR
 
Theoremelioc2 8906 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR  /\  A  <  C  /\  C  <_  B )
 ) )
 
Theoremelico2 8907 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <-> 
 ( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )
 
Theoremelicc2 8908 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,] B )  <-> 
 ( C  e.  RR  /\  A  <_  C  /\  C  <_  B ) ) )
 
Theoremelicc2i 8909 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <_  B )
 )
 
Theoremelicc4 8910 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C 
 /\  C  <_  B ) ) )
 
Theoremiccss 8911 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C [,] D ) 
 C_  ( A [,] B ) )
 
Theoremiccssioo 8912 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <  B ) )  ->  ( C [,] D ) 
 C_  ( A (,) B ) )
 
Theoremicossico 8913 Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C [,) D )  C_  ( A [,) B ) )
 
Theoremiccss2 8914 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( C  e.  ( A [,] B ) 
 /\  D  e.  ( A [,] B ) ) 
 ->  ( C [,] D )  C_  ( A [,] B ) )
 
Theoremiccssico 8915 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A [,) B ) )
 
Theoremiccssioo2 8916 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( C  e.  ( A (,) B ) 
 /\  D  e.  ( A (,) B ) ) 
 ->  ( C [,] D )  C_  ( A (,) B ) )
 
Theoremiccssico2 8917 Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( C  e.  ( A [,) B ) 
 /\  D  e.  ( A [,) B ) ) 
 ->  ( C [,] D )  C_  ( A [,) B ) )
 
Theoremioomax 8918 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
 |-  ( -oo (,) +oo )  =  RR
 
Theoremiccmax 8919 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( -oo [,] +oo )  =  RR*
 
Theoremioopos 8920 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,) +oo )  =  { x  e.  RR  |  0  < 
 x }
 
Theoremioorp 8921 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( 0 (,) +oo )  =  RR+
 
Theoremiooshf 8922 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  B )  e.  ( C (,) D )  <->  A  e.  (
 ( C  +  B ) (,) ( D  +  B ) ) ) )
 
Theoremiocssre 8923 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
 
Theoremicossre 8924 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  ( A [,) B )  C_  RR )
 
Theoremiccssre 8925 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  C_  RR )
 
Theoremiccssxr 8926 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  ( A [,] B )  C_  RR*
 
Theoremiocssxr 8927 An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  ( A (,] B )  C_  RR*
 
Theoremicossxr 8928 A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  ( A [,) B )  C_  RR*
 
Theoremioossicc 8929 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( A (,) B )  C_  ( A [,] B )
 
Theoremicossicc 8930 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 8931 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 8932 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 8933 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 8934 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B ) )  ->  ( C [,) D ) 
 C_  ( A (,) B ) )
 
Theoremioossioo 8935 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A 
 <_  C  /\  D  <_  B ) )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremiccsupr 8936* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  y  <_  x ) )
 
Theoremelioopnf 8937 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( A (,) +oo )  <->  ( B  e.  RR  /\  A  <  B ) ) )
 
Theoremelioomnf 8938 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  RR*  ->  ( B  e.  ( -oo (,) A )  <->  ( B  e.  RR  /\  B  <  A ) ) )
 
Theoremelicopnf 8939 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( A  e.  RR  ->  ( B  e.  ( A [,) +oo )  <->  ( B  e.  RR  /\  A  <_  B ) ) )
 
Theoremrepos 8940 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
 |-  ( A  e.  (
 0 (,) +oo )  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremioof 8941 The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 (,) : ( RR*  X.  RR* )
 --> ~P RR
 
Theoremiccf 8942 The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |- 
 [,] : ( RR*  X.  RR* )
 --> ~P RR*
 
Theoremunirnioo 8943 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
 |- 
 RR  =  U. ran  (,)
 
Theoremdfioo2 8944* Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |- 
 (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { w  e.  RR  |  ( x  <  w  /\  w  <  y ) } )
 
Theoremioorebasg 8945 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ran  (,) )
 
Theoremelrege0 8946 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( A  e.  (
 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrge0ssre 8947 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
 |-  ( 0 [,) +oo )  C_  RR
 
Theoremelxrge0 8948 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A  e.  (
 0 [,] +oo )  <->  ( A  e.  RR*  /\  0  <_  A ) )
 
Theorem0e0icopnf 8949 0 is a member of  ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,) +oo )
 
Theorem0e0iccpnf 8950 0 is a member of  ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,] +oo )
 
Theoremge0addcl 8951 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  +  B )  e.  ( 0 [,) +oo ) )
 
Theoremge0mulcl 8952 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |-  ( ( A  e.  ( 0 [,) +oo )  /\  B  e.  (
 0 [,) +oo ) ) 
 ->  ( A  x.  B )  e.  ( 0 [,) +oo ) )
 
Theoremlbicc2 8953 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B ) )
 
Theoremubicc2 8954 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B ) )
 
Theorem0elunit 8955 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  0  e.  ( 0 [,] 1 )
 
Theorem1elunit 8956 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  1  e.  ( 0 [,] 1 )
 
Theoremiooneg 8957 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  -u C  e.  ( -u B (,) -u A ) ) )
 
Theoremiccneg 8958 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  -u C  e.  ( -u B [,] -u A ) ) )
 
Theoremicoshft 8959 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( X  e.  ( A [,) B )  ->  ( X  +  C )  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
 
Theoremicoshftf1o 8960* Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  ( A [,) B )  |->  ( x  +  C ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  F :
 ( A [,) B )
 -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
 ) )
 
Theoremicodisj 8961 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A [,) B )  i^i  ( B [,) C ) )  =  (/) )
 
Theoremioodisj 8962 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
 |-  ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  /\  B  <_  C )  ->  ( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
 
Theoremiccshftr 8963 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  +  R )  =  C   &    |-  ( B  +  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  +  R )  e.  ( C [,] D ) ) )
 
Theoremiccshftri 8964 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR   &    |-  ( A  +  R )  =  C   &    |-  ( B  +  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  +  R )  e.  ( C [,] D ) )
 
Theoremiccshftl 8965 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  -  R )  =  C   &    |-  ( B  -  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  -  R )  e.  ( C [,] D ) ) )
 
Theoremiccshftli 8966 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR   &    |-  ( A  -  R )  =  C   &    |-  ( B  -  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  -  R )  e.  ( C [,] D ) )
 
Theoremiccdil 8967 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  x.  R )  =  C   &    |-  ( B  x.  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )
 
Theoremiccdili 8968 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR+   &    |-  ( A  x.  R )  =  C   &    |-  ( B  x.  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  x.  R )  e.  ( C [,] D ) )
 
Theoremicccntr 8969 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  /  R )  =  C   &    |-  ( B  /  R )  =  D   =>    |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
 
Theoremicccntri 8970 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  R  e.  RR+   &    |-  ( A  /  R )  =  C   &    |-  ( B  /  R )  =  D   =>    |-  ( X  e.  ( A [,] B )  ->  ( X  /  R )  e.  ( C [,] D ) )
 
Theoremdivelunit 8971 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  (
 0 [,] 1 )  <->  A  <_  B ) )
 
Theoremlincmb01cmp 8972 A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1
 ) )  ->  (
 ( ( 1  -  T )  x.  A )  +  ( T  x.  B ) )  e.  ( A [,] B ) )
 
Theoremiccf1o 8973* Describe a bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B ) 
 /\  `' F  =  (
 y  e.  ( A [,] B )  |->  ( ( y  -  A )  /  ( B  -  A ) ) ) ) )
 
Theoremunitssre 8974  ( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( 0 [,] 1
 )  C_  RR
 
Theoremzltaddlt1le 8975 The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) ) 
 ->  ( ( M  +  A )  <  N  <->  ( M  +  A )  <_  N ) )
 
3.5.4  Finite intervals of integers
 
Syntaxcfz 8976 Extend class notation to include the notation for a contiguous finite set of integers. Read " M ... N " as "the set of integers from  M to  N inclusive."
 class  ...
 
Definitiondf-fz 8977* Define an operation that produces a finite set of sequential integers. Read " M ... N " as "the set of integers from  M to  N inclusive." See fzval 8978 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
 |- 
 ...  =  ( m  e.  ZZ ,  n  e. 
 ZZ  |->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) } )
 
Theoremfzval 8978* The value of a finite set of sequential integers. E.g.,  2 ... 5 means the set  { 2 ,  3 ,  4 ,  5 }. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where  NN_k means our  1 ... k; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  { k  e.  ZZ  |  ( M 
 <_  k  /\  k  <_  N ) } )
 
Theoremfzval2 8979 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  ( ( M [,] N )  i^i  ZZ ) )
 
Theoremfzf 8980 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 ... : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfz1 8981 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
 ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
 
Theoremelfz 8982 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( M  <_  K 
 /\  K  <_  N ) ) )
 
Theoremelfz2 8983 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) ) )
 
Theoremelfz5 8984 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  K  <_  N ) )
 
Theoremelfz4 8985 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuzb 8986 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) ) )
 
Theoremeluzfz 8987 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuz 8988 A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M ) )
 
Theoremelfzuz3 8989 Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K ) )
 
Theoremelfzel2 8990 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
 
Theoremelfzel1 8991 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
 
Theoremelfzelz 8992 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
 
Theoremelfzle1 8993 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  K )
 
Theoremelfzle2 8994 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  <_  N )
 
Theoremelfzuz2 8995 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  M ) )
 
Theoremelfzle3 8996 Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  N )
 
Theoremeluzfz1 8997 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  e.  ( M ... N ) )
 
Theoremeluzfz2 8998 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  ( M ... N ) )
 
Theoremeluzfz2b 8999 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  <->  N  e.  ( M ... N ) )
 
Theoremelfz3 9000 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
 |-  ( N  e.  ZZ  ->  N  e.  ( N
 ... N ) )
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