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Type | Label | Description |
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Statement | ||
Theorem | fzdisj 9401 | Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Theorem | fz01en 9402 | 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.) |
Theorem | elfznn 9403 | A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.) |
Theorem | elfz1end 9404 | A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
Theorem | fznn0sub 9405 | 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.) |
Theorem | fzmmmeqm 9406 | 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.) |
Theorem | fzaddel 9407 | Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
Theorem | fzsubel 9408 | Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
Theorem | fzopth 9409 | A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzass4 9410 | Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Theorem | fzss1 9411 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzss2 9412 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Theorem | fzssuz 9413 | A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.) |
Theorem | fzsn 9414 | A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fzssp1 9415 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzsuc 9416 | 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.) |
Theorem | fzpred 9417 | Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
Theorem | fzpreddisj 9418 | A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.) |
Theorem | elfzp1 9419 | Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzp1ss 9420 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzelp1 9421 | Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fzp1elp1 9422 | 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.) |
Theorem | fznatpl1 9423 | Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.) |
Theorem | fzpr 9424 | A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fztp 9425 | A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzsuc2 9426 | Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzp1disj 9427 | is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzdifsuc 9428 | Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.) |
Theorem | fzprval 9429* | Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.) |
Theorem | fztpval 9430* | Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.) |
Theorem | fzrev 9431 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2 9432 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2i 9433 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev3 9434 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzrev3i 9435 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fznn 9436 | Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Theorem | elfz1b 9437 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) |
Theorem | elfzm11 9438 | Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | uzsplit 9439 | Express an upper integer set as the disjoint (see uzdisj 9440) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | uzdisj 9440 | The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | fseq1p1m1 9441 | 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.) |
Theorem | fseq1m1p1 9442 | Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | fz1sbc 9443* | Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.) |
Theorem | elfzp1b 9444 | 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.) |
Theorem | elfzm1b 9445 | 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.) |
Theorem | elfzp12 9446 | Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
Theorem | fzm1 9447 | Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fzneuz 9448 | No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.) |
Theorem | fznuz 9449 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.) |
Theorem | uznfz 9450 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
Theorem | fzp1nel 9451 | 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.) |
Theorem | fzrevral 9452* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral2 9453* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral3 9454* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzshftral 9455* | Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.) |
Theorem | ige2m1fz1 9456 | Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers (Contributed by Alexander van der Vekens, 14-Sep-2018.) |
Theorem | ige2m1fz 9457 | Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.) |
Theorem | fz01or 9458 | An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.) |
Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: , usually abbreviated by "fz0". | ||
Theorem | elfz2nn0 9459 | Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fznn0 9460 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
Theorem | elfznn0 9461 | A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | elfz3nn0 9462 | The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | 0elfz 9463 | 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.) |
Theorem | nn0fz0 9464 | A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.) |
Theorem | elfz0add 9465 | An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Theorem | fz0tp 9466 | An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Theorem | elfz0ubfz0 9467 | An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
Theorem | elfz0fzfz0 9468 | A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.) |
Theorem | fz0fzelfz0 9469 | If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.) |
Theorem | fznn0sub2 9470 | Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | uzsubfz0 9471 | Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Theorem | fz0fzdiffz0 9472 | The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.) |
Theorem | elfzmlbm 9473 | 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 9474 | 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 9475 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
Theorem | difelfzle 9476 | 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 9477 | 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 9478 | Express the set of nonnegative integers as the disjoint (see nn0disj 9480) union of the first values and the rest. (Contributed by AV, 8-Nov-2019.) |
Theorem | nnsplit 9479 | 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 9480 | The first elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.) |
Theorem | 1fv 9481 | A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | 4fvwrd4 9482* | The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.) |
Theorem | 2ffzeq 9483* | 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.) |
Syntax | cfzo 9484 | Syntax for half-open integer ranges. |
..^ | ||
Definition | df-fzo 9485* | 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 9360, which includes . Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 9519 with fzsplit 9400, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | fzof 9486 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoel1 9487 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoel2 9488 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoelz 9489 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | fzoval 9490 | Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzo 9491 | Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzo2 9492 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | elfzouz 9493 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzodcel 9494 | Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.) |
DECID ..^ | ||
Theorem | fzolb 9495 | 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 9496 | 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 9497 | 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 9498 | A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzolt3 9499 | Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzolt2b 9500 | A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ |
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