Type  Label  Description 
Statement 

Theorem  fzof 9001 
Functionality of the halfopen integer set function. (Contributed by
Stefan O'Rear, 14Aug2015.)

..^ 

Theorem  elfzoel1 9002 
Reverse closure for halfopen integer sets. (Contributed by Stefan
O'Rear, 14Aug2015.)

..^ 

Theorem  elfzoel2 9003 
Reverse closure for halfopen integer sets. (Contributed by Stefan
O'Rear, 14Aug2015.)

..^ 

Theorem  elfzoelz 9004 
Reverse closure for halfopen integer sets. (Contributed by Stefan
O'Rear, 14Aug2015.)

..^ 

Theorem  fzoval 9005 
Value of the halfopen integer set in terms of the closed integer set.
(Contributed by Stefan O'Rear, 14Aug2015.)

..^


Theorem  elfzo 9006 
Membership in a halfopen finite set of integers. (Contributed by Stefan
O'Rear, 15Aug2015.)

..^ 

Theorem  elfzo2 9007 
Membership in a halfopen integer interval. (Contributed by Mario
Carneiro, 29Sep2015.)

..^ 

Theorem  elfzouz 9008 
Membership in a halfopen integer interval. (Contributed by Mario
Carneiro, 29Sep2015.)

..^ 

Theorem  fzolb 9009 
The left endpoint of a halfopen integer interval is in the set iff the
two arguments are integers with . This
provides an alternative
notation for the "strict upper integer" predicate by analogy to
the "weak
upper integer" predicate .
(Contributed by Mario
Carneiro, 29Sep2015.)

..^ 

Theorem  fzolb2 9010 
The left endpoint of a halfopen integer interval is in the set iff the
two arguments are integers with . This
provides an alternative
notation for the "strict upper integer" predicate by analogy to
the "weak
upper integer" predicate .
(Contributed by Mario
Carneiro, 29Sep2015.)

..^


Theorem  elfzole1 9011 
A member in a halfopen integer interval is greater than or equal to the
lower bound. (Contributed by Stefan O'Rear, 15Aug2015.)

..^ 

Theorem  elfzolt2 9012 
A member in a halfopen integer interval is less than the upper bound.
(Contributed by Stefan O'Rear, 15Aug2015.)

..^ 

Theorem  elfzolt3 9013 
Membership in a halfopen integer interval implies that the bounds are
unequal. (Contributed by Stefan O'Rear, 15Aug2015.)

..^ 

Theorem  elfzolt2b 9014 
A member in a halfopen integer interval is less than the upper bound.
(Contributed by Mario Carneiro, 29Sep2015.)

..^ ..^ 

Theorem  elfzolt3b 9015 
Membership in a halfopen integer interval implies that the bounds are
unequal. (Contributed by Mario Carneiro, 29Sep2015.)

..^ ..^ 

Theorem  fzonel 9016 
A halfopen range does not contain its right endpoint. (Contributed by
Stefan O'Rear, 25Aug2015.)

..^ 

Theorem  elfzouz2 9017 
The upper bound of a halfopen range is greater or equal to an element of
the range. (Contributed by Mario Carneiro, 29Sep2015.)

..^ 

Theorem  elfzofz 9018 
A halfopen range is contained in the corresponding closed range.
(Contributed by Stefan O'Rear, 23Aug2015.)

..^ 

Theorem  elfzo3 9019 
Express membership in a halfopen integer interval in terms of the "less
than or equal" and "less than" predicates on integers,
resp.
, ..^
.
(Contributed by Mario Carneiro, 29Sep2015.)

..^ ..^ 

Theorem  fzom 9020* 
A halfopen integer interval is inhabited iff it contains its left
endpoint. (Contributed by Jim Kingdon, 20Apr2020.)

..^ ..^ 

Theorem  fzossfz 9021 
A halfopen range is contained in the corresponding closed range.
(Contributed by Stefan O'Rear, 23Aug2015.) (Revised by Mario
Carneiro, 29Sep2015.)

..^ 

Theorem  fzon 9022 
A halfopen set of sequential integers is empty if the bounds are equal or
reversed. (Contributed by Alexander van der Vekens, 30Oct2017.)

..^ 

Theorem  fzonlt0 9023 
A halfopen integer range is empty if the bounds are equal or reversed.
(Contributed by AV, 20Oct2018.)

..^


Theorem  fzo0 9024 
Halfopen sets with equal endpoints are empty. (Contributed by Stefan
O'Rear, 15Aug2015.) (Revised by Mario Carneiro, 29Sep2015.)

..^ 

Theorem  fzonnsub 9025 
If then is a positive integer.
(Contributed by Mario
Carneiro, 29Sep2015.) (Revised by Mario Carneiro, 1Jan2017.)

..^ 

Theorem  fzonnsub2 9026 
If then is a positive integer.
(Contributed by Mario
Carneiro, 1Jan2017.)

..^ 

Theorem  fzoss1 9027 
Subset relationship for halfopen sequences of integers. (Contributed
by Stefan O'Rear, 15Aug2015.) (Revised by Mario Carneiro,
29Sep2015.)

..^ ..^ 

Theorem  fzoss2 9028 
Subset relationship for halfopen sequences of integers. (Contributed by
Stefan O'Rear, 15Aug2015.) (Revised by Mario Carneiro, 29Sep2015.)

..^ ..^ 

Theorem  fzossrbm1 9029 
Subset of a half open range. (Contributed by Alexander van der Vekens,
1Nov2017.)

..^ ..^ 

Theorem  fzo0ss1 9030 
Subset relationship for halfopen integer ranges with lower bounds 0 and
1. (Contributed by Alexander van der Vekens, 18Mar2018.)

..^ ..^ 

Theorem  fzossnn0 9031 
A halfopen integer range starting at a nonnegative integer is a subset of
the nonnegative integers. (Contributed by Alexander van der Vekens,
13May2018.)

..^ 

Theorem  fzospliti 9032 
One direction of splitting a halfopen integer range in half.
(Contributed by Stefan O'Rear, 14Aug2015.)

..^
..^ ..^ 

Theorem  fzosplit 9033 
Split a halfopen integer range in half. (Contributed by Stefan O'Rear,
14Aug2015.)

..^ ..^ ..^ 

Theorem  fzodisj 9034 
Abutting halfopen integer ranges are disjoint. (Contributed by Stefan
O'Rear, 14Aug2015.)

..^ ..^


Theorem  fzouzsplit 9035 
Split an upper integer set into a halfopen integer range and another
upper integer set. (Contributed by Mario Carneiro, 21Sep2016.)

..^ 

Theorem  fzouzdisj 9036 
A halfopen integer range does not overlap the upper integer range
starting at the endpoint of the first range. (Contributed by Mario
Carneiro, 21Sep2016.)

..^


Theorem  lbfzo0 9037 
An integer is strictly greater than zero iff it is a member of .
(Contributed by Mario Carneiro, 29Sep2015.)

..^


Theorem  elfzo0 9038 
Membership in a halfopen integer range based at 0. (Contributed by
Stefan O'Rear, 15Aug2015.) (Revised by Mario Carneiro, 29Sep2015.)

..^ 

Theorem  fzo1fzo0n0 9039 
An integer between 1 and an upper bound of a halfopen integer range is
not 0 and between 0 and the upper bound of the halfopen integer range.
(Contributed by Alexander van der Vekens, 21Mar2018.)

..^ ..^ 

Theorem  elfzo0z 9040 
Membership in a halfopen range of nonnegative integers, generalization of
elfzo0 9038 requiring the upper bound to be an integer
only. (Contributed by
Alexander van der Vekens, 23Sep2018.)

..^ 

Theorem  elfzo0le 9041 
A member in a halfopen range of nonnegative integers is less than or
equal to the upper bound of the range. (Contributed by Alexander van der
Vekens, 23Sep2018.)

..^ 

Theorem  elfzonn0 9042 
A member of a halfopen range of nonnegative integers is a nonnegative
integer. (Contributed by Alexander van der Vekens, 21May2018.)

..^


Theorem  fzonmapblen 9043 
The result of subtracting a nonnegative integer from a positive integer
and adding another nonnegative integer which is less than the first one is
less then the positive integer. (Contributed by Alexander van der Vekens,
19May2018.)

..^
..^


Theorem  fzofzim 9044 
If a nonnegative integer in a finite interval of integers is not the upper
bound of the interval, it is contained in the corresponding halfopen
integer range. (Contributed by Alexander van der Vekens, 15Jun2018.)

..^ 

Theorem  fzossnn 9045 
Halfopen integer ranges starting with 1 are subsets of NN. (Contributed
by Thierry Arnoux, 28Dec2016.)

..^ 

Theorem  elfzo1 9046 
Membership in a halfopen integer range based at 1. (Contributed by
Thierry Arnoux, 14Feb2017.)

..^ 

Theorem  fzo0m 9047* 
A halfopen integer range based at 0 is inhabited precisely if the upper
bound is a positive integer. (Contributed by Jim Kingdon,
20Apr2020.)

..^ 

Theorem  fzoaddel 9048 
Translate membership in a halfopen integer range. (Contributed by Stefan
O'Rear, 15Aug2015.)

..^
..^


Theorem  fzoaddel2 9049 
Translate membership in a shifteddown halfopen integer range.
(Contributed by Stefan O'Rear, 15Aug2015.)

..^
..^ 

Theorem  fzosubel 9050 
Translate membership in a halfopen integer range. (Contributed by Stefan
O'Rear, 15Aug2015.)

..^
..^ 

Theorem  fzosubel2 9051 
Membership in a translated halfopen integer range implies translated
membership in the original range. (Contributed by Stefan O'Rear,
15Aug2015.)

..^
..^ 

Theorem  fzosubel3 9052 
Membership in a translated halfopen integer range when the original range
is zerobased. (Contributed by Stefan O'Rear, 15Aug2015.)

..^
..^ 

Theorem  eluzgtdifelfzo 9053 
Membership of the difference of integers in a halfopen range of
nonnegative integers. (Contributed by Alexander van der Vekens,
17Sep2018.)

..^ 

Theorem  ige2m2fzo 9054 
Membership of an integer greater than 1 decreased by 2 in a halfopen
range of nonnegative integers. (Contributed by Alexander van der Vekens,
3Oct2018.)

..^ 

Theorem  fzocatel 9055 
Translate membership in a halfopen integer range. (Contributed by
Thierry Arnoux, 28Sep2018.)

..^ ..^
..^ 

Theorem  ubmelfzo 9056 
If an integer in a 1 based finite set of sequential integers is subtracted
from the upper bound of this finite set of sequential integers, the result
is contained in a halfopen range of nonnegative integers with the same
upper bound. (Contributed by AV, 18Mar2018.) (Revised by AV,
30Oct2018.)

..^ 

Theorem  elfzodifsumelfzo 9057 
If an integer is in a halfopen range of nonnegative integers with a
difference as upper bound, the sum of the integer with the subtrahend of
the difference is in the a halfopen range of nonnegative integers
containing the minuend of the difference. (Contributed by AV,
13Nov2018.)

..^
..^ 

Theorem  elfzom1elp1fzo 9058 
Membership of an integer incremented by one in a halfopen range of
nonnegative integers. (Contributed by Alexander van der Vekens,
24Jun2018.) (Proof shortened by AV, 5Jan2020.)

..^ ..^ 

Theorem  elfzom1elfzo 9059 
Membership in a halfopen range of nonnegative integers. (Contributed by
Alexander van der Vekens, 18Jun2018.)

..^
..^ 

Theorem  fzval3 9060 
Expressing a closed integer range as a halfopen integer range.
(Contributed by Stefan O'Rear, 15Aug2015.)

..^


Theorem  fzosn 9061 
Expressing a singleton as a halfopen range. (Contributed by Stefan
O'Rear, 23Aug2015.)

..^ 

Theorem  elfzomin 9062 
Membership of an integer in the smallest open range of integers.
(Contributed by Alexander van der Vekens, 22Sep2018.)

..^ 

Theorem  zpnn0elfzo 9063 
Membership of an integer increased by a nonnegative integer in a half
open integer range. (Contributed by Alexander van der Vekens,
22Sep2018.)

..^


Theorem  zpnn0elfzo1 9064 
Membership of an integer increased by a nonnegative integer in a half
open integer range. (Contributed by Alexander van der Vekens,
22Sep2018.)

..^


Theorem  fzosplitsnm1 9065 
Removing a singleton from a halfopen integer range at the end.
(Contributed by Alexander van der Vekens, 23Mar2018.)

..^ ..^ 

Theorem  elfzonlteqm1 9066 
If an element of a halfopen integer range is not less than the upper
bound of the range decreased by 1, it must be equal to the upper bound of
the range decreased by 1. (Contributed by AV, 3Nov2018.)

..^ 

Theorem  fzonn0p1 9067 
A nonnegative integer is element of the halfopen range of nonnegative
integers with the element increased by one as an upper bound.
(Contributed by Alexander van der Vekens, 5Aug2018.)

..^


Theorem  fzossfzop1 9068 
A halfopen range of nonnegative integers is a subset of a halfopen range
of nonnegative integers with the upper bound increased by one.
(Contributed by Alexander van der Vekens, 5Aug2018.)

..^ ..^ 

Theorem  fzonn0p1p1 9069 
If a nonnegative integer is element of a halfopen range of nonnegative
integers, increasing this integer by one results in an element of a half
open range of nonnegative integers with the upper bound increased by one.
(Contributed by Alexander van der Vekens, 5Aug2018.)

..^
..^ 

Theorem  elfzom1p1elfzo 9070 
Increasing an element of a halfopen range of nonnegative integers by 1
results in an element of the halfopen range of nonnegative integers with
an upper bound increased by 1. (Contributed by Alexander van der Vekens,
1Aug2018.)

..^ ..^ 

Theorem  fzo0ssnn0 9071 
Halfopen integer ranges starting with 0 are subsets of NN0.
(Contributed by Thierry Arnoux, 8Oct2018.)

..^ 

Theorem  fzo01 9072 
Expressing the singleton of as a halfopen integer range.
(Contributed by Stefan O'Rear, 15Aug2015.)

..^ 

Theorem  fzo12sn 9073 
A 1based halfopen integer interval up to, but not including, 2 is a
singleton. (Contributed by Alexander van der Vekens, 31Jan2018.)

..^ 

Theorem  fzo0to2pr 9074 
A halfopen integer range from 0 to 2 is an unordered pair. (Contributed
by Alexander van der Vekens, 4Dec2017.)

..^ 

Theorem  fzo0to3tp 9075 
A halfopen integer range from 0 to 3 is an unordered triple.
(Contributed by Alexander van der Vekens, 9Nov2017.)

..^ 

Theorem  fzo0to42pr 9076 
A halfopen integer range from 0 to 4 is a union of two unordered pairs.
(Contributed by Alexander van der Vekens, 17Nov2017.)

..^ 

Theorem  fzo0sn0fzo1 9077 
A halfopen range of nonnegative integers is the union of the singleton
set containing 0 and a halfopen range of positive integers. (Contributed
by Alexander van der Vekens, 18May2018.)

..^ ..^ 

Theorem  fzoend 9078 
The endpoint of a halfopen integer range. (Contributed by Mario
Carneiro, 29Sep2015.)

..^ ..^ 

Theorem  fzo0end 9079 
The endpoint of a zerobased halfopen range. (Contributed by Stefan
O'Rear, 27Aug2015.) (Revised by Mario Carneiro, 29Sep2015.)

..^ 

Theorem  ssfzo12 9080 
Subset relationship for halfopen integer ranges. (Contributed by
Alexander van der Vekens, 16Mar2018.)

..^ ..^ 

Theorem  ssfzo12bi 9081 
Subset relationship for halfopen integer ranges. (Contributed by
Alexander van der Vekens, 5Nov2018.)

..^ ..^


Theorem  ubmelm1fzo 9082 
The result of subtracting 1 and an integer of a halfopen range of
nonnegative integers from the upper bound of this range is contained in
this range. (Contributed by AV, 23Mar2018.) (Revised by AV,
30Oct2018.)

..^
..^ 

Theorem  fzofzp1 9083 
If a point is in a halfopen range, the next point is in the closed range.
(Contributed by Stefan O'Rear, 23Aug2015.)

..^


Theorem  fzofzp1b 9084 
If a point is in a halfopen range, the next point is in the closed range.
(Contributed by Mario Carneiro, 27Sep2015.)

..^


Theorem  elfzom1b 9085 
An integer is a member of a 1based finite set of sequential integers iff
its predecessor is a member of the corresponding 0based set.
(Contributed by Mario Carneiro, 27Sep2015.)

..^
..^ 

Theorem  elfzonelfzo 9086 
If an element of a halfopen integer range is not contained in the lower
subrange, it must be in the upper subrange. (Contributed by Alexander van
der Vekens, 30Mar2018.)

..^ ..^
..^ 

Theorem  elfzomelpfzo 9087 
An integer increased by another integer is an element of a halfopen
integer range if and only if the integer is contained in the halfopen
integer range with bounds decreased by the other integer. (Contributed by
Alexander van der Vekens, 30Mar2018.)

..^
..^ 

Theorem  peano2fzor 9088 
A Peanopostulatelike theorem for downward closure of a halfopen integer
range. (Contributed by Mario Carneiro, 1Oct2015.)

..^
..^ 

Theorem  fzosplitsn 9089 
Extending a halfopen range by a singleton on the end. (Contributed by
Stefan O'Rear, 23Aug2015.)

..^ ..^ 

Theorem  fzosplitprm1 9090 
Extending a halfopen integer range by an unordered pair at the end.
(Contributed by Alexander van der Vekens, 22Sep2018.)

..^ ..^ 

Theorem  fzosplitsni 9091 
Membership in a halfopen range extended by a singleton. (Contributed by
Stefan O'Rear, 23Aug2015.)

..^ ..^ 

Theorem  fzisfzounsn 9092 
A finite interval of integers as union of a halfopen integer range and a
singleton. (Contributed by Alexander van der Vekens, 15Jun2018.)

..^ 

Theorem  fzostep1 9093 
Two possibilities for a number one greater than a number in a halfopen
range. (Contributed by Stefan O'Rear, 23Aug2015.)

..^ ..^


Theorem  fzoshftral 9094* 
Shift the scanning order inside of a quantification over a halfopen
integer range, analogous to fzshftral 8970. (Contributed by Alexander van
der Vekens, 23Sep2018.)

..^ ..^ 

Theorem  fzind2 9095* 
Induction on the integers from to
inclusive. The first four
hypotheses give us the substitution instances we need; the last two are
the basis and the induction step. Version of fzind 8353 using integer
range definitions. (Contributed by Mario Carneiro, 6Feb2016.)

..^ 

Theorem  fvinim0ffz 9096 
The function values for the borders of a finite interval of integers,
which is the domain of the function, are not in the image of the
interior of the interval iff the intersection of the images of the
interior and the borders is empty. (Contributed by Alexander van der
Vekens, 31Oct2017.) (Revised by AV, 5Feb2021.)

..^
..^ ..^ 

Theorem  subfzo0 9097 
The difference between two elements in a halfopen range of nonnegative
integers is greater than the negation of the upper bound and less than the
upper bound of the range. (Contributed by AV, 20Mar2021.)

..^ ..^


3.5.7 Rational numbers (cont.)


Theorem  qtri3or 9098 
Rational trichotomy. (Contributed by Jim Kingdon, 6Oct2021.)



Theorem  qletric 9099 
Rational trichotomy. (Contributed by Jim Kingdon, 6Oct2021.)



Theorem  qlelttric 9100 
Rational trichotomy. (Contributed by Jim Kingdon, 7Oct2021.)

