Theorem List for Intuitionistic Logic Explorer - 9901-10000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | elfzmlbm 9901 |
Subtracting the lower bound of a finite set of sequential integers from an
element of this set. (Contributed by Alexander van der Vekens,
29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|
|
|
Theorem | elfzmlbp 9902 |
Subtracting the lower bound of a finite set of sequential integers from an
element of this set. (Contributed by Alexander van der Vekens,
29-Mar-2018.)
|
|
|
Theorem | fzctr 9903 |
Lemma for theorems about the central binomial coefficient. (Contributed
by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro,
2-Aug-2014.)
|
|
|
Theorem | difelfzle 9904 |
The difference of two integers from a finite set of sequential nonnegative
integers is also element of this finite set of sequential integers.
(Contributed by Alexander van der Vekens, 12-Jun-2018.)
|
|
|
Theorem | difelfznle 9905 |
The difference of two integers from a finite set of sequential nonnegative
integers increased by the upper bound is also element of this finite set
of sequential integers. (Contributed by Alexander van der Vekens,
12-Jun-2018.)
|
|
|
Theorem | nn0split 9906 |
Express the set of nonnegative integers as the disjoint (see nn0disj 9908)
union of the first values and the rest.
(Contributed by AV,
8-Nov-2019.)
|
|
|
Theorem | nnsplit 9907 |
Express the set of positive integers as the disjoint union of the first
values and the
rest. (Contributed by Glauco Siliprandi,
21-Nov-2020.)
|
|
|
Theorem | nn0disj 9908 |
The first elements of the set of nonnegative integers are
distinct from any later members. (Contributed by AV, 8-Nov-2019.)
|
|
|
Theorem | 1fv 9909 |
A function on a singleton. (Contributed by Alexander van der Vekens,
3-Dec-2017.)
|
|
|
Theorem | 4fvwrd4 9910* |
The first four function values of a word of length at least 4.
(Contributed by Alexander van der Vekens, 18-Nov-2017.)
|
|
|
Theorem | 2ffzeq 9911* |
Two functions over 0 based finite set of sequential integers are equal
if and only if their domains have the same length and the function
values are the same at each position. (Contributed by Alexander van der
Vekens, 30-Jun-2018.)
|
|
|
4.5.6 Half-open integer ranges
|
|
Syntax | cfzo 9912 |
Syntax for half-open integer ranges.
|
..^ |
|
Definition | df-fzo 9913* |
Define a function generating sets of integers using a half-open range.
Read ..^ as the integers from up to, but not
including, ;
contrast with df-fz 9784, which
includes . Not
including the endpoint simplifies a number of
formulas related to cardinality and splitting; contrast fzosplit 9947 with
fzsplit 9824, for instance. (Contributed by Stefan
O'Rear,
14-Aug-2015.)
|
..^
|
|
Theorem | fzof 9914 |
Functionality of the half-open integer set function. (Contributed by
Stefan O'Rear, 14-Aug-2015.)
|
..^ |
|
Theorem | elfzoel1 9915 |
Reverse closure for half-open integer sets. (Contributed by Stefan
O'Rear, 14-Aug-2015.)
|
..^ |
|
Theorem | elfzoel2 9916 |
Reverse closure for half-open integer sets. (Contributed by Stefan
O'Rear, 14-Aug-2015.)
|
..^ |
|
Theorem | elfzoelz 9917 |
Reverse closure for half-open integer sets. (Contributed by Stefan
O'Rear, 14-Aug-2015.)
|
..^ |
|
Theorem | fzoval 9918 |
Value of the half-open integer set in terms of the closed integer set.
(Contributed by Stefan O'Rear, 14-Aug-2015.)
|
..^
|
|
Theorem | elfzo 9919 |
Membership in a half-open finite set of integers. (Contributed by Stefan
O'Rear, 15-Aug-2015.)
|
..^ |
|
Theorem | elfzo2 9920 |
Membership in a half-open integer interval. (Contributed by Mario
Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | elfzouz 9921 |
Membership in a half-open integer interval. (Contributed by Mario
Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | fzodcel 9922 |
Decidability of membership in a half-open integer interval. (Contributed
by Jim Kingdon, 25-Aug-2022.)
|
DECID ..^ |
|
Theorem | fzolb 9923 |
The left endpoint of a half-open integer interval is in the set iff the
two arguments are integers with . This
provides an alternate
notation for the "strict upper integer" predicate by analogy to
the "weak
upper integer" predicate .
(Contributed by Mario
Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | fzolb2 9924 |
The left endpoint of a half-open integer interval is in the set iff the
two arguments are integers with . This
provides an alternate
notation for the "strict upper integer" predicate by analogy to
the "weak
upper integer" predicate .
(Contributed by Mario
Carneiro, 29-Sep-2015.)
|
..^
|
|
Theorem | elfzole1 9925 |
A member in a half-open integer interval is greater than or equal to the
lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^ |
|
Theorem | elfzolt2 9926 |
A member in a half-open integer interval is less than the upper bound.
(Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^ |
|
Theorem | elfzolt3 9927 |
Membership in a half-open integer interval implies that the bounds are
unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^ |
|
Theorem | elfzolt2b 9928 |
A member in a half-open integer interval is less than the upper bound.
(Contributed by Mario Carneiro, 29-Sep-2015.)
|
..^ ..^ |
|
Theorem | elfzolt3b 9929 |
Membership in a half-open integer interval implies that the bounds are
unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
|
..^ ..^ |
|
Theorem | fzonel 9930 |
A half-open range does not contain its right endpoint. (Contributed by
Stefan O'Rear, 25-Aug-2015.)
|
..^ |
|
Theorem | elfzouz2 9931 |
The upper bound of a half-open range is greater or equal to an element of
the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | elfzofz 9932 |
A half-open range is contained in the corresponding closed range.
(Contributed by Stefan O'Rear, 23-Aug-2015.)
|
..^ |
|
Theorem | elfzo3 9933 |
Express membership in a half-open integer interval in terms of the "less
than or equal" and "less than" predicates on integers,
resp.
, ..^
.
(Contributed by Mario Carneiro, 29-Sep-2015.)
|
..^ ..^ |
|
Theorem | fzom 9934* |
A half-open integer interval is inhabited iff it contains its left
endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
|
..^ ..^ |
|
Theorem | fzossfz 9935 |
A half-open range is contained in the corresponding closed range.
(Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario
Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | fzon 9936 |
A half-open set of sequential integers is empty if the bounds are equal or
reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
|
..^ |
|
Theorem | fzonlt0 9937 |
A half-open integer range is empty if the bounds are equal or reversed.
(Contributed by AV, 20-Oct-2018.)
|
..^
|
|
Theorem | fzo0 9938 |
Half-open sets with equal endpoints are empty. (Contributed by Stefan
O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | fzonnsub 9939 |
If then is a positive integer.
(Contributed by Mario
Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
|
..^ |
|
Theorem | fzonnsub2 9940 |
If then is a positive integer.
(Contributed by Mario
Carneiro, 1-Jan-2017.)
|
..^ |
|
Theorem | fzoss1 9941 |
Subset relationship for half-open sequences of integers. (Contributed
by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro,
29-Sep-2015.)
|
..^ ..^ |
|
Theorem | fzoss2 9942 |
Subset relationship for half-open sequences of integers. (Contributed by
Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|
..^ ..^ |
|
Theorem | fzossrbm1 9943 |
Subset of a half open range. (Contributed by Alexander van der Vekens,
1-Nov-2017.)
|
..^ ..^ |
|
Theorem | fzo0ss1 9944 |
Subset relationship for half-open integer ranges with lower bounds 0 and
1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
|
..^ ..^ |
|
Theorem | fzossnn0 9945 |
A half-open integer range starting at a nonnegative integer is a subset of
the nonnegative integers. (Contributed by Alexander van der Vekens,
13-May-2018.)
|
..^ |
|
Theorem | fzospliti 9946 |
One direction of splitting a half-open integer range in half.
(Contributed by Stefan O'Rear, 14-Aug-2015.)
|
..^
..^ ..^ |
|
Theorem | fzosplit 9947 |
Split a half-open integer range in half. (Contributed by Stefan O'Rear,
14-Aug-2015.)
|
..^ ..^ ..^ |
|
Theorem | fzodisj 9948 |
Abutting half-open integer ranges are disjoint. (Contributed by Stefan
O'Rear, 14-Aug-2015.)
|
..^ ..^
|
|
Theorem | fzouzsplit 9949 |
Split an upper integer set into a half-open integer range and another
upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
|
..^ |
|
Theorem | fzouzdisj 9950 |
A half-open integer range does not overlap the upper integer range
starting at the endpoint of the first range. (Contributed by Mario
Carneiro, 21-Sep-2016.)
|
..^
|
|
Theorem | lbfzo0 9951 |
An integer is strictly greater than zero iff it is a member of .
(Contributed by Mario Carneiro, 29-Sep-2015.)
|
..^
|
|
Theorem | elfzo0 9952 |
Membership in a half-open integer range based at 0. (Contributed by
Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | fzo1fzo0n0 9953 |
An integer between 1 and an upper bound of a half-open integer range is
not 0 and between 0 and the upper bound of the half-open integer range.
(Contributed by Alexander van der Vekens, 21-Mar-2018.)
|
..^ ..^ |
|
Theorem | elfzo0z 9954 |
Membership in a half-open range of nonnegative integers, generalization of
elfzo0 9952 requiring the upper bound to be an integer
only. (Contributed by
Alexander van der Vekens, 23-Sep-2018.)
|
..^ |
|
Theorem | elfzo0le 9955 |
A member in a half-open range of nonnegative integers is less than or
equal to the upper bound of the range. (Contributed by Alexander van der
Vekens, 23-Sep-2018.)
|
..^ |
|
Theorem | elfzonn0 9956 |
A member of a half-open range of nonnegative integers is a nonnegative
integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
|
..^
|
|
Theorem | fzonmapblen 9957 |
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,
19-May-2018.)
|
..^
..^
|
|
Theorem | fzofzim 9958 |
If a nonnegative integer in a finite interval of integers is not the upper
bound of the interval, it is contained in the corresponding half-open
integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
|
..^ |
|
Theorem | fzossnn 9959 |
Half-open integer ranges starting with 1 are subsets of NN. (Contributed
by Thierry Arnoux, 28-Dec-2016.)
|
..^ |
|
Theorem | elfzo1 9960 |
Membership in a half-open integer range based at 1. (Contributed by
Thierry Arnoux, 14-Feb-2017.)
|
..^ |
|
Theorem | fzo0m 9961* |
A half-open integer range based at 0 is inhabited precisely if the upper
bound is a positive integer. (Contributed by Jim Kingdon,
20-Apr-2020.)
|
..^ |
|
Theorem | fzoaddel 9962 |
Translate membership in a half-open integer range. (Contributed by Stefan
O'Rear, 15-Aug-2015.)
|
..^
..^
|
|
Theorem | fzoaddel2 9963 |
Translate membership in a shifted-down half-open integer range.
(Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^
..^ |
|
Theorem | fzosubel 9964 |
Translate membership in a half-open integer range. (Contributed by Stefan
O'Rear, 15-Aug-2015.)
|
..^
..^ |
|
Theorem | fzosubel2 9965 |
Membership in a translated half-open integer range implies translated
membership in the original range. (Contributed by Stefan O'Rear,
15-Aug-2015.)
|
..^
..^ |
|
Theorem | fzosubel3 9966 |
Membership in a translated half-open integer range when the original range
is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^
..^ |
|
Theorem | eluzgtdifelfzo 9967 |
Membership of the difference of integers in a half-open range of
nonnegative integers. (Contributed by Alexander van der Vekens,
17-Sep-2018.)
|
..^ |
|
Theorem | ige2m2fzo 9968 |
Membership of an integer greater than 1 decreased by 2 in a half-open
range of nonnegative integers. (Contributed by Alexander van der Vekens,
3-Oct-2018.)
|
..^ |
|
Theorem | fzocatel 9969 |
Translate membership in a half-open integer range. (Contributed by
Thierry Arnoux, 28-Sep-2018.)
|
..^ ..^
..^ |
|
Theorem | ubmelfzo 9970 |
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 half-open range of nonnegative integers with the same
upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV,
30-Oct-2018.)
|
..^ |
|
Theorem | elfzodifsumelfzo 9971 |
If an integer is in a half-open 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 half-open range of nonnegative integers
containing the minuend of the difference. (Contributed by AV,
13-Nov-2018.)
|
..^
..^ |
|
Theorem | elfzom1elp1fzo 9972 |
Membership of an integer incremented by one in a half-open range of
nonnegative integers. (Contributed by Alexander van der Vekens,
24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
|
..^ ..^ |
|
Theorem | elfzom1elfzo 9973 |
Membership in a half-open range of nonnegative integers. (Contributed by
Alexander van der Vekens, 18-Jun-2018.)
|
..^
..^ |
|
Theorem | fzval3 9974 |
Expressing a closed integer range as a half-open integer range.
(Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^
|
|
Theorem | fzosn 9975 |
Expressing a singleton as a half-open range. (Contributed by Stefan
O'Rear, 23-Aug-2015.)
|
..^ |
|
Theorem | elfzomin 9976 |
Membership of an integer in the smallest open range of integers.
(Contributed by Alexander van der Vekens, 22-Sep-2018.)
|
..^ |
|
Theorem | zpnn0elfzo 9977 |
Membership of an integer increased by a nonnegative integer in a half-
open integer range. (Contributed by Alexander van der Vekens,
22-Sep-2018.)
|
..^
|
|
Theorem | zpnn0elfzo1 9978 |
Membership of an integer increased by a nonnegative integer in a half-
open integer range. (Contributed by Alexander van der Vekens,
22-Sep-2018.)
|
..^
|
|
Theorem | fzosplitsnm1 9979 |
Removing a singleton from a half-open integer range at the end.
(Contributed by Alexander van der Vekens, 23-Mar-2018.)
|
..^ ..^ |
|
Theorem | elfzonlteqm1 9980 |
If an element of a half-open 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, 3-Nov-2018.)
|
..^ |
|
Theorem | fzonn0p1 9981 |
A nonnegative integer is element of the half-open range of nonnegative
integers with the element increased by one as an upper bound.
(Contributed by Alexander van der Vekens, 5-Aug-2018.)
|
..^
|
|
Theorem | fzossfzop1 9982 |
A half-open range of nonnegative integers is a subset of a half-open range
of nonnegative integers with the upper bound increased by one.
(Contributed by Alexander van der Vekens, 5-Aug-2018.)
|
..^ ..^ |
|
Theorem | fzonn0p1p1 9983 |
If a nonnegative integer is element of a half-open 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, 5-Aug-2018.)
|
..^
..^ |
|
Theorem | elfzom1p1elfzo 9984 |
Increasing an element of a half-open range of nonnegative integers by 1
results in an element of the half-open range of nonnegative integers with
an upper bound increased by 1. (Contributed by Alexander van der Vekens,
1-Aug-2018.)
|
..^ ..^ |
|
Theorem | fzo0ssnn0 9985 |
Half-open integer ranges starting with 0 are subsets of NN0.
(Contributed by Thierry Arnoux, 8-Oct-2018.)
|
..^ |
|
Theorem | fzo01 9986 |
Expressing the singleton of as a half-open integer range.
(Contributed by Stefan O'Rear, 15-Aug-2015.)
|
..^ |
|
Theorem | fzo12sn 9987 |
A 1-based half-open integer interval up to, but not including, 2 is a
singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
|
..^ |
|
Theorem | fzo0to2pr 9988 |
A half-open integer range from 0 to 2 is an unordered pair. (Contributed
by Alexander van der Vekens, 4-Dec-2017.)
|
..^ |
|
Theorem | fzo0to3tp 9989 |
A half-open integer range from 0 to 3 is an unordered triple.
(Contributed by Alexander van der Vekens, 9-Nov-2017.)
|
..^ |
|
Theorem | fzo0to42pr 9990 |
A half-open integer range from 0 to 4 is a union of two unordered pairs.
(Contributed by Alexander van der Vekens, 17-Nov-2017.)
|
..^ |
|
Theorem | fzo0sn0fzo1 9991 |
A half-open range of nonnegative integers is the union of the singleton
set containing 0 and a half-open range of positive integers. (Contributed
by Alexander van der Vekens, 18-May-2018.)
|
..^ ..^ |
|
Theorem | fzoend 9992 |
The endpoint of a half-open integer range. (Contributed by Mario
Carneiro, 29-Sep-2015.)
|
..^ ..^ |
|
Theorem | fzo0end 9993 |
The endpoint of a zero-based half-open range. (Contributed by Stefan
O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
|
..^ |
|
Theorem | ssfzo12 9994 |
Subset relationship for half-open integer ranges. (Contributed by
Alexander van der Vekens, 16-Mar-2018.)
|
..^ ..^ |
|
Theorem | ssfzo12bi 9995 |
Subset relationship for half-open integer ranges. (Contributed by
Alexander van der Vekens, 5-Nov-2018.)
|
..^ ..^
|
|
Theorem | ubmelm1fzo 9996 |
The result of subtracting 1 and an integer of a half-open range of
nonnegative integers from the upper bound of this range is contained in
this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV,
30-Oct-2018.)
|
..^
..^ |
|
Theorem | fzofzp1 9997 |
If a point is in a half-open range, the next point is in the closed range.
(Contributed by Stefan O'Rear, 23-Aug-2015.)
|
..^
|
|
Theorem | fzofzp1b 9998 |
If a point is in a half-open range, the next point is in the closed range.
(Contributed by Mario Carneiro, 27-Sep-2015.)
|
..^
|
|
Theorem | elfzom1b 9999 |
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 Mario Carneiro, 27-Sep-2015.)
|
..^
..^ |
|
Theorem | elfzonelfzo 10000 |
If an element of a half-open integer range is not contained in the lower
subrange, it must be in the upper subrange. (Contributed by Alexander van
der Vekens, 30-Mar-2018.)
|
..^ ..^
..^ |