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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlincmb01cmp 8601 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.)
 RR  RR  <  T  0 [,] 1  1  -  T  x.  +  T  x.  [,]
 
Theoremiccf1o 8602* Describe a bijection from  0 ,  1 to an arbitrary nontrivial closed interval  , . (Contributed by Mario Carneiro, 8-Sep-2015.)
 F  0 [,] 1  |->  x.  +  1  -  x.    =>     RR  RR  <  F : 0 [,] 1 -1-1-onto->
 [,]  `' F  [,]  |-> 
 -  -
 
Theoremunitssre 8603  0 [,] 1 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
 0 [,] 1  C_  RR
 
3.5.4  Finite intervals of integers
 
Syntaxcfz 8604 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."
 ...
 
Definitiondf-fz 8605* 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 8606 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)

 ...  m  ZZ ,  n 
 ZZ  |->  { k  ZZ  |  m  <_  k  k  <_  n }
 
Theoremfzval 8606* 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  ZZ  N  ZZ  M ... N  { k  ZZ  |  M 
 <_  k  k  <_  N }
 
Theoremfzval2 8607 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
 M  ZZ  N  ZZ  M ... N  M [,] N 
 i^i  ZZ
 
Theoremfzf 8608 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 8609 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
 M  ZZ  N  ZZ  K  M ... N  K  ZZ  M  <_  K  K  <_  N
 
Theoremelfz 8610 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
 K  ZZ  M  ZZ  N  ZZ  K  M ... N  M 
 <_  K  K  <_  N
 
Theoremelfz2 8611 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  ZZ and  N  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  M ... N  M  ZZ  N  ZZ  K  ZZ  M  <_  K  K  <_  N
 
Theoremelfz5 8612 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
 K  ZZ>= `  M  N  ZZ  K  M ... N  K  <_  N
 
Theoremelfz4 8613 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 M  ZZ  N  ZZ  K  ZZ  M 
 <_  K  K  <_  N  K  M ... N
 
Theoremelfzuzb 8614 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  M ... N  K  ZZ>= `  M  N  ZZ>=
 `  K
 
Theoremeluzfz 8615 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  ZZ>= `  M  N  ZZ>= `  K  K  M ... N
 
Theoremelfzuz 8616 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  M ... N 
 K  ZZ>= `  M
 
Theoremelfzuz3 8617 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  M ... N 
 N  ZZ>= `  K
 
Theoremelfzel2 8618 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  M ... N 
 N  ZZ
 
Theoremelfzel1 8619 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  M ... N 
 M  ZZ
 
Theoremelfzelz 8620 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  M ... N 
 K  ZZ
 
Theoremelfzle1 8621 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  M ... N 
 M  <_  K
 
Theoremelfzle2 8622 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  M ... N 
 K  <_  N
 
Theoremelfzuz2 8623 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  M ... N 
 N  ZZ>= `  M
 
Theoremelfzle3 8624 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  M ... N 
 M  <_  N
 
Theoremeluzfz1 8625 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 N  ZZ>=
 `  M 
 M  M
 ... N
 
Theoremeluzfz2 8626 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 N  ZZ>=
 `  M 
 N  M
 ... N
 
Theoremeluzfz2b 8627 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
 N  ZZ>=
 `  M  N  M ... N
 
Theoremelfz3 8628 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
 N  ZZ  N  N
 ... N
 
Theoremelfz1eq 8629 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
 K  N ... N 
 K  N
 
Theoremelfzubelfz 8630 If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
 K  M ... N 
 N  M
 ... N
 
Theorempeano2fzr 8631 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
 K  ZZ>= `  M  K  +  1  M
 ... N  K  M
 ... N
 
Theoremfzm 8632* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
 M ... N  N  ZZ>= `  M
 
Theoremfztri3or 8633 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 K  ZZ  M  ZZ  N  ZZ  K  <  M  K  M
 ... N  N  <  K
 
Theoremfzdcel 8634 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 K  ZZ  M  ZZ  N  ZZ DECID  K  M ... N
 
Theoremfznlem 8635 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
 M  ZZ  N  ZZ  N  <  M  M ... N  (/)
 
Theoremfzn 8636 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
 M  ZZ  N  ZZ  N  <  M  M ... N  (/)
 
Theoremfzen 8637 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
 M  ZZ  N  ZZ  K  ZZ  M ... N  ~~  M  +  K ... N  +  K
 
Theoremfz1n 8638 A 1-based finite set of sequential integers is empty iff it ends at index  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 N  NN0  1 ... N  (/)  N  0
 
Theorem0fz1 8639 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
 N  NN0  F  Fn  1
 ... N  F  (/)  N  0
 
Theoremfz10 8640 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  (/)
 
Theoremuzsubsubfz 8641 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 L  ZZ>= `  M  N  ZZ>= `  L  N  -  L  -  M  M ... N
 
Theoremuzsubsubfz1 8642 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 L  NN  N  ZZ>=
 `  L  N  -  L  -  1  1 ...
 N
 
Theoremige3m2fz 8643 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 N  ZZ>=
 `  3  N  -  2  1 ...
 N
 
Theoremfzsplit2 8644 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 K  +  1  ZZ>= `  M  N  ZZ>= `  K  M ... N  M ... K  u.  K  +  1
 ... N
 
Theoremfzsplit 8645 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
 K  M ... N  M ... N  M
 ... K  u.  K  +  1 ... N
 
Theoremfzdisj 8646 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 8647 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
 N  ZZ  0 ... N  -  1 
 ~~  1 ...
 N
 
Theoremelfznn 8648 A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
 K 
 1 ... N 
 K  NN
 
Theoremelfz1end 8649 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 NN  1 ...
 
Theoremfznn0sub 8650 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  M ... N  N  -  K  NN0
 
Theoremfzmmmeqm 8651 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
 M  L ... N  N  -  L  -  M  -  L  N  -  M
 
Theoremfzaddel 8652 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
 M  ZZ  N  ZZ  J  ZZ  K  ZZ  J  M ... N  J  +  K  M  +  K ... N  +  K
 
Theoremfzsubel 8653 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
 M  ZZ  N  ZZ  J  ZZ  K  ZZ  J  M ... N  J  -  K  M  -  K ... N  -  K
 
Theoremfzopth 8654 A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 N  ZZ>=
 `  M  M ... N  J ... K  M  J  N  K
 
Theoremfzass4 8655 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 ... D  C 
 ... D  ... C  C  ... D
 
Theoremfzss1 8656 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 K  ZZ>=
 `  M  K ... N  C_  M ... N
 
Theoremfzss2 8657 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
 N  ZZ>=
 `  K  M ... K  C_  M ... N
 
Theoremfzssuz 8658 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
 M ... N  C_  ZZ>= `  M
 
Theoremfzsn 8659 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
 M  ZZ  M ... M  { M }
 
Theoremfzssp1 8660 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 M ... N  C_  M ... N  +  1
 
Theoremfzsuc 8661 Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 N  ZZ>=
 `  M  M ... N  +  1  M
 ... N  u.  { N  +  1 }
 
Theoremfzpred 8662 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
 N  ZZ>=
 `  M  M ... N  { M }  u.  M  +  1 ... N
 
Theoremfzpreddisj 8663 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
 N  ZZ>=
 `  M  { M }  i^i  M  +  1 ... N  (/)
 
Theoremelfzp1 8664 Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 N  ZZ>=
 `  M  K  M ... N  +  1  K  M ... N  K  N  +  1
 
Theoremfzp1ss 8665 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 M  ZZ  M  +  1 ... N  C_  M ... N
 
Theoremfzelp1 8666 Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  M ... N 
 K  M
 ... N  +  1
 
Theoremfzp1elp1 8667 Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
 K  M ... N  K  +  1  M ... N  +  1
 
Theoremfznatpl1 8668 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
 N  NN  I 
 1 ... N  -  1  I  +  1  1 ...
 N
 
Theoremfzpr 8669 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
 M  ZZ  M ... M  +  1  { M ,  M  +  1 }
 
Theoremfztp 8670 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
 M  ZZ  M ... M  +  2  { M ,  M  +  1 ,  M  +  2 }
 
Theoremfzsuc2 8671 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
 M  ZZ  N  ZZ>=
 `  M  -  1  M ... N  +  1  M
 ... N  u.  { N  +  1 }
 
Theoremfzp1disj 8672  M ... N  +  1 is the disjoint union of  M ... N with  { N  +  1 }. (Contributed by Mario Carneiro, 7-Mar-2014.)
 M ... N  i^i  { N  +  1 }  (/)
 
Theoremfzdifsuc 8673 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
 N  ZZ>=
 `  M  M ... N  M
 ... N  +  1  \  { N  +  1 }
 
Theoremfzprval 8674* Two ways of defining the first two values of a sequence on  NN. (Contributed by NM, 5-Sep-2011.)
 1 ... 2 F `  if  1 ,  ,  F `  1  F `  2
 
Theoremfztpval 8675* Two ways of defining the first three values of a sequence on  NN. (Contributed by NM, 13-Sep-2011.)
 1 ... 3 F `  if  1 ,  ,  if  2 ,  ,  C  F `  1  F `  2  F `  3  C
 
Theoremfzrev 8676 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 M  ZZ  N  ZZ  J  ZZ  K  ZZ  K  J  -  N ... J  -  M  J  -  K  M ... N
 
Theoremfzrev2 8677 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 M  ZZ  N  ZZ  J  ZZ  K  ZZ  K  M ... N  J  -  K  J  -  N ... J  -  M
 
Theoremfzrev2i 8678 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 J  ZZ  K  M ... N  J  -  K  J  -  N ... J  -  M
 
Theoremfzrev3 8679 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 K  ZZ  K  M ... N  M  +  N 
 -  K  M ... N
 
Theoremfzrev3i 8680 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 K  M ... N  M  +  N  -  K  M ... N
 
Theoremfznn 8681 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
 N  ZZ  K 
 1 ... N  K  NN  K  <_  N
 
Theoremelfz1b 8682 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
 N 
 1 ... M  N  NN  M  NN  N  <_  M
 
Theoremelfzm11 8683 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 M  ZZ  N  ZZ  K  M ... N  -  1  K  ZZ  M  <_  K  K  <  N
 
Theoremuzsplit 8684 Express an upper integer set as the disjoint (see uzdisj 8685) union of the first  N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
 N  ZZ>=
 `  M  ZZ>= `  M  M ... N  -  1  u.  ZZ>=
 `  N
 
Theoremuzdisj 8685 The first  N elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
 M ... N  -  1  i^i  ZZ>= `  N  (/)
 
Theoremfseq1p1m1 8686 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
 H  { <. N  +  1 ,  >. }   =>     N  NN0  F : 1 ... N -->  G  F  u.  H  G : 1 ... N  +  1
 -->  G `
  N  +  1  F  G  |`  1 ... N
 
Theoremfseq1m1p1 8687 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
 H  { <. N ,  >. }   =>     N  NN  F : 1 ... N  -  1 -->  G  F  u.  H  G : 1 ... N
 -->  G `
  N  F  G  |`  1
 ... N  -  1
 
Theoremfz1sbc 8688* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
 N  ZZ  k  N ... N  [. N  k ].
 
Theoremelfzp1b 8689 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 K  ZZ  N  ZZ  K  0 ... N  -  1  K  +  1  1 ...
 N
 
Theoremelfzm1b 8690 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 K  ZZ  N  ZZ  K  1 ... N  K  -  1  0 ... N  -  1
 
Theoremelfzp12 8691 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
 N  ZZ>=
 `  M  K  M ... N  K  M  K  M  +  1 ... N
 
Theoremfzm1 8692 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 N  ZZ>=
 `  M  K  M ... N  K  M ... N  -  1  K  N
 
Theoremfzneuz 8693 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
 N  ZZ>= `  M  K  ZZ  M ... N  ZZ>= `  K
 
Theoremfznuz 8694 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
 K  M ... N  K  ZZ>= `  N  +  1
 
Theoremuznfz 8695 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
 K  ZZ>=
 `  N  K  M
 ... N  -  1
 
Theoremfzp1nel 8696 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
 N  +  1  M
 ... N
 
Theoremfzrevral 8697* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 M  ZZ  N  ZZ  K  ZZ  j  M ... N  k  K  -  N ... K  -  M
 [. K  -  k  j ].
 
Theoremfzrevral2 8698* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 M  ZZ  N  ZZ  K  ZZ  j  K  -  N ... K  -  M  k  M ... N [. K  -  k  j ].
 
Theoremfzrevral3 8699* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 M  ZZ  N  ZZ  j  M ... N  k  M ... N [. M  +  N  -  k  j ].
 
Theoremfzshftral 8700* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
 M  ZZ  N  ZZ  K  ZZ  j  M ... N  k  M  +  K ... N  +  K
 [. k  -  K  j ].
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