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Theorem List for Metamath Proof Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiccssico2 10601 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 ) )
 
Theoremioomax 10602 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
 |-  (  -oo (,)  +oo )  =  RR
 
Theoremiccmax 10603 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  (  -oo [,]  +oo )  =  RR*
 
Theoremioopos 10604 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,)  +oo )  =  { x  e.  RR  |  0  < 
 x }
 
Theoremioorp 10605 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( 0 (,)  +oo )  =  RR+
 
Theoremiooshf 10606 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 10607 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 10608 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 10609 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 10610 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 10611 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 10612 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 10613 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( A (,) B )  C_  ( A [,] B )
 
Theoremiccsupr 10614* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9594). (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 10615 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 10616 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 10617 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 10618 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 10619 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 10620 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 10621 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 10622* 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 ) } )
 
Theoremioorebas 10623 Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)
 |-  ( A (,) B )  e.  ran  (,)
 
Theoremelrege0 10624 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 ) )
 
Theoremelxrge0 10625 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A  e.  (
 0 [,]  +oo )  <->  ( A  e.  RR*  /\  0  <_  A ) )
 
Theoremge0addcl 10626 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 10627 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 ) )
 
Theoremge0xaddcl 10628 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
 0 [,]  +oo ) ) 
 ->  ( A + e B )  e.  (
 0 [,]  +oo ) )
 
Theoremge0xmulcl 10629 The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
 0 [,]  +oo ) ) 
 ->  ( A x e B )  e.  (
 0 [,]  +oo ) )
 
Theoremlbicc2 10630 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 10631 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 10632 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  0  e.  ( 0 [,] 1 )
 
Theorem1elunit 10633 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  1  e.  ( 0 [,] 1 )
 
Theoremiooneg 10634 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 10635 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 10636 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 10637* 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 )
 ) )
 
Theoremicoun 10638 The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C )
 )  ->  ( ( A [,) B )  u.  ( B [,) C ) )  =  ( A [,) C ) )
 
Theoremicodisj 10639 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 ) )  =  (/) )
 
Theoremsnunioo 10640 The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
 
Theoremsnunico 10641 The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
 ( A [,) B )  u.  { B }
 )  =  ( A [,] B ) )
 
Theoremprunioo 10642 The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
 ( A (,) B )  u.  { A ,  B } )  =  ( A [,] B ) )
 
Theoremioodisj 10643 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 ) )  =  (/) )
 
Theoremioojoin 10644 Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  ( (
 ( A (,) B )  u.  { B }
 )  u.  ( B (,) C ) )  =  ( A (,) C ) )
 
Theoremdifreicc 10645 The class difference of  RR and a closed interval. (Contributed by FL, 18-Jun-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( RR  \  ( A [,] B ) )  =  ( ( 
 -oo (,) A )  u.  ( B (,)  +oo ) ) )
 
Theoremiccsplit 10646 Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  ( A [,] B ) ) 
 ->  ( A [,] B )  =  ( ( A [,] C )  u.  ( C [,] B ) ) )
 
Theoremiccshftr 10647 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 10648 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 10649 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 10650 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 10651 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 10652 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 10653 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 10654 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 ) )
 
Theoremlincmb01cmp 10655 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 10656* 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 ) ) ) ) )
 
Theoremiccen 10657 Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  ( 0 [,] 1
 )  ~~  ( A [,] B ) )
 
Theoremxov1plusxeqvd 10658 A complex number  X is positive real iff  X  /  (
1  +  X ) is in  ( 0 (,) 1 ). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  X  =/=  -u 1 )   =>    |-  ( ph  ->  ( X  e.  RR+  <->  ( X  /  ( 1  +  X ) )  e.  (
 0 (,) 1 ) ) )
 
Theoremunitssre 10659  ( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( 0 [,] 1
 )  C_  RR
 
5.5.5  Finite intervals of integers
 
Syntaxcfz 10660 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 10661* 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 10662 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 10662* 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 10663 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 10664 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 10665 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 10666 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 10667 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 10668 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 10669 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 10670 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 10671 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 10672 A member of a finite set of sequential integers belongs to a set of upper integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M ) )
 
Theoremelfzuz3 10673 Membership in a finite set of sequential integers implies membership in a set of upper integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K ) )
 
Theoremelfzel2 10674 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 10675 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 10676 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 10677 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 10678 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 10679 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 10680 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 10681 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 10682 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 10683 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 10684 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
 |-  ( N  e.  ZZ  ->  N  e.  ( N
 ... N ) )
 
Theoremelfz1eq 10685 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
 |-  ( K  e.  ( N ... N )  ->  K  =  N )
 
Theorempeano2fzr 10686 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M
 ... N ) ) 
 ->  K  e.  ( M
 ... N ) )
 
Theoremfzn0 10687 Properties of a finite interval of integers which is non-empty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
 `  M ) )
 
Theoremfzn 10688 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <-> 
 ( M ... N )  =  (/) ) )
 
Theoremfzen 10689 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M ... N )  ~~  ( ( M  +  K ) ... ( N  +  K ) ) )
 
Theoremfz1n 10690 A 1-based finite set of sequential integers is empty iff it ends at index  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( ( 1 ...
 N )  =  (/)  <->  N  =  0 ) )
 
Theorem0fz1 10691 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( F  =  (/)  <->  N  =  0 ) )
 
Theoremfz10 10692 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( 1 ... 0
 )  =  (/)
 
Theoremfzsplit2 10693 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( ( K  +  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzsplit 10694 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzdisj 10695 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( K  <  M  ->  ( ( J ... K )  i^i  ( M
 ... N ) )  =  (/) )
 
Theoremfz01en 10696 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( N  e.  ZZ  ->  ( 0 ... ( N  -  1 ) ) 
 ~~  ( 1 ...
 N ) )
 
Theoremelfznn 10697 A member of a finite set of sequential integers starting at 1 is a natural number. (Contributed by NM, 24-Aug-2005.)
 |-  ( K  e.  (
 1 ... N )  ->  K  e.  NN )
 
Theoremelfz1end 10698 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN  <->  A  e.  ( 1 ... A ) )
 
Theoremelfz2nn0 10699 Membership in a finite set of sequential integers starting at 0. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  <->  ( K  e.  NN0  /\  N  e.  NN0  /\  K  <_  N ) )
 
Theoremelfznn0 10700 A member of a finite set of sequential integers starting at 0 is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  K  e.  NN0 )
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