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Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremubioog 9301 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  B  e.  ( A (,) B ) )
 
Theoremiooval2 9302* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  <  x  /\  x  <  B ) } )
 
Theoremiooss1 9303 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
 |-  ( ( A  e.  RR*  /\  A  <_  B )  ->  ( B (,) C )  C_  ( A (,) C ) )
 
Theoremiooss2 9304 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( C  e.  RR*  /\  B  <_  C )  ->  ( A (,) B )  C_  ( A (,) C ) )
 
Theoremiocval 9305* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,] B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) } )
 
Theoremicoval 9306* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A 
 <_  x  /\  x  <  B ) } )
 
Theoremiccval 9307* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,] B )  =  { x  e.  RR*  |  ( A 
 <_  x  /\  x  <_  B ) } )
 
Theoremelioo2 9308 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR  /\  A  <  C  /\  C  <  B ) ) )
 
Theoremelioc1 9309 Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  RR*  /\  A  <  C  /\  C  <_  B ) ) )
 
Theoremelico1 9310 Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
 
Theoremelicc1 9311 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B ) ) )
 
Theoremiccid 9312 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
 |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A }
 )
 
Theoremicc0r 9313 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  ( A [,] B )  =  (/) ) )
 
Theoremeliooxr 9314 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  ( B (,) C )  ->  ( B  e.  RR*  /\  C  e.  RR* ) )
 
Theoremeliooord 9315 Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( A  e.  ( B (,) C )  ->  ( B  <  A  /\  A  <  C ) )
 
Theoremubioc1 9316 The upper bound belongs to an open-below, closed-above interval. See ubicc2 9371. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  e.  ( A (,] B ) )
 
Theoremlbico1 9317 The lower bound belongs to a closed-below, open-above interval. See lbicc2 9370. (Contributed by FL, 29-May-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  A  e.  ( A [,) B ) )
 
Theoremiccleub 9318 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 9319 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 9320 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 9321 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 9322 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
 |-  ( A (,) B )  C_  RR
 
Theoremelioc2 9323 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 9324 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 9325 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 9326 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 9327 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 9328 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 9329 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 9330 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 9331 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 9332 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 9333 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 9334 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 9335 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
 |-  ( -oo (,) +oo )  =  RR
 
Theoremiccmax 9336 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
 |-  ( -oo [,] +oo )  =  RR*
 
Theoremioopos 9337 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
 |-  ( 0 (,) +oo )  =  { x  e.  RR  |  0  < 
 x }
 
Theoremioorp 9338 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
 |-  ( 0 (,) +oo )  =  RR+
 
Theoremiooshf 9339 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 9340 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 9341 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 9342 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 9343 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 9344 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 9345 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 9346 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( A (,) B )  C_  ( A [,] B )
 
Theoremicossicc 9347 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 9348 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 9349 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 9350 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 9351 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 9352 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 9353* 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 9354 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 9355 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 9356 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 9357 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 9358 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 9359 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 9360 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 9361* 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 9362 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 9363 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 9364 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 9365 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 9366 0 is a member of  ( 0 [,) +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,) +oo )
 
Theorem0e0iccpnf 9367 0 is a member of  ( 0 [,] +oo ) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  e.  ( 0 [,] +oo )
 
Theoremge0addcl 9368 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 9369 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 9370 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 9371 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 9372 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  0  e.  ( 0 [,] 1 )
 
Theorem1elunit 9373 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  1  e.  ( 0 [,] 1 )
 
Theoremiooneg 9374 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 9375 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 9376 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 9377* 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 9378 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 9379 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 9380 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 9381 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 9382 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 9383 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 9384 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 9385 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 9386 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 9387 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 9388 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 9389 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 9390* 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 9391  ( 0 [,] 1 ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( 0 [,] 1
 )  C_  RR
 
Theoremzltaddlt1le 9392 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 9393 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 9394* 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 9395 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 9395* 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 9396 An alternate 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 9397 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 9398 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 9399 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 9400 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 ) ) )
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