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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | digexp 42401 | The th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.) |
digit | ||
Theorem | dig1 42402 | All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.) |
digit | ||
Theorem | 0dig1 42403 | The th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.) |
digit | ||
Theorem | 0dig2pr01 42404 | The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.) |
digit | ||
Theorem | dig2nn0 42405 | A digit of a nonnegative integer in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.) |
digit | ||
Theorem | 0dig2nn0e 42406 | The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.) |
digit | ||
Theorem | 0dig2nn0o 42407 | The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.) |
digit | ||
Theorem | dig2bits 42408 | The th digit of a nonnegative integer in a binary system is its th bit. (Contributed by AV, 24-May-2020.) |
digit bits | ||
Theorem | dignn0flhalflem1 42409 | Lemma 1 for dignn0flhalf 42412. (Contributed by AV, 7-Jun-2012.) |
Theorem | dignn0flhalflem2 42410 | Lemma 2 for dignn0flhalf 42412. (Contributed by AV, 7-Jun-2012.) |
Theorem | dignn0ehalf 42411 | The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.) |
digit digit | ||
Theorem | dignn0flhalf 42412 | The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.) |
digit digit | ||
Theorem | nn0sumshdiglemA 42413* | Lemma for nn0sumshdig 42417 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.) |
#b ..^digit #b ..^ digit | ||
Theorem | nn0sumshdiglemB 42414* | Lemma for nn0sumshdig 42417 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.) |
#b ..^digit #b ..^ digit | ||
Theorem | nn0sumshdiglem1 42415* | Lemma 1 for nn0sumshdig 42417 (induction step). (Contributed by AV, 7-Jun-2020.) |
#b ..^digit #b ..^ digit | ||
Theorem | nn0sumshdiglem2 42416* | Lemma 2 for nn0sumshdig 42417. (Contributed by AV, 7-Jun-2020.) |
#b ..^digit | ||
Theorem | nn0sumshdig 42417* | A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.) |
..^#bdigit | ||
Theorem | nn0mulfsum 42418* | Trivial algorithm to calculate the product of two nonnegative integers and by adding up times. (Contributed by AV, 17-May-2020.) |
Theorem | nn0mullong 42419* | Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers and by multiplying the multiplicand by each digit of the multiplier and then add up all the properly shifted results. Here, the binary representation of the multiplier is used, i.e. the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 15215. (Contributed by AV, 7-Jun-2020.) |
..^#bdigit | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 42420 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
Theorem | nfiund 42421 | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) |
Theorem | iunord 42422* | The indexed union of a collection of ordinal numbers is ordinal. This proof is based on the proof of ssorduni 6985, but does not use it directly, since ssorduni 6985 does not work when is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
Theorem | iunordi 42423* | The indexed union of a collection of ordinal numbers is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
Theorem | rspcdf 42424* | Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.) |
Theorem | spd 42425 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2255. (Contributed by Emmett Weisz, 17-Jan-2020.) |
Theorem | spcdvw 42426* | A version of spcdv 3291 where and are direct substitutions of each other. This theorem is useful because it does not require and to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
Theorem | tfis2d 42427* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
Theorem | bnd2d 42428* | Deduction form of bnd2 8756. (Contributed by Emmett Weisz, 19-Jan-2021.) |
Theorem | dffun3f 42429* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
Symbols in this section: All the symbols used in the definition of setrecs are explained in the comment of df-setrecs 42431. The class is explained in the comment of setrec1lem1 42434. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 42430 | Extend class notation to include a set defined by transfinite recursion. |
setrecs | ||
Definition | df-setrecs 42431* |
Define a class setrecs by transfinite recursion,
where
is
the set of new elements to add to the class given the
set of elements
in the class so far. We do not need a base case,
because we can start with the empty set, which is vacuously a subset of
setrecs. The goal of this definition is to construct a
class
fulfilling theorems setrec1 42438 and setrec2v 42443, which give a more
intuitive idea of the meaning of setrecs. Unlike wrecs,
setrecs is well-defined for any and meaningful for any
function .
For example, see theorem onsetrec 42451 for how the class is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set , and consider the result when applying to any subset . Remember that can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within . However, if we restrict the domain of to a given set , the resulting range will be a set. Therefore, with this restricted , it makes sense to consider sets that are closed under applied to its subsets. Now we can test whether a given set is recursively generated by . If every set that is closed under contains , that means that every member of must eventually be generated by . On the other hand, if some such does not contain a certain element of , then that element can be avoided even if we apply in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by , but not through the elements of . In this case, would fail the condition in the definition, but the omitted element would still be included in some larger . For example, if is the successor function, the set would fail the condition since is not an element of the successor of or . Remember that we are applying to subsets of , not elements of . In fact, even the set fails the condition, since the only subset of previously generated elements is , and does not have as an element. However, we can let be any ordinal, since each of its elements is generated by starting with and repeatedly applying the successor function. A similar definition I initially used for setrecs was setrecs recs . I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 42438 and setrec2v 42443 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Conway noted in the Appendix to Part Zero of ONAG, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant." Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the metamath framework. The difficulty in writing a definition in metamath for setrecs is that the necessary properties to prove are self-referential (see setrec1 42438 and setrec2v 42443), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 7506, this is not actually a requirement of the metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs, which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs, which "know" about multiple individual elements at a time. Note that we cannot define setrecs as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs as a union of a class abstraction. If setrecs , the elements of A must be subsets of setrecs which together include everything recursively generated by . We can do this by letting be the class of sets whose elements are all recursively generated by . One necessary condition is that each element of a given must be generated by when applied to a previous element . In symbols, . However, this is not sufficient. All fixed points of will satisfy this condition whether they should be in setrecs or not. If we replace the subset relation with the proper subset relation, cannot be the empty set, even though the empty set should be in . Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set to be generated by when applied only to sets of previously generated elements within . For example, taking to be the successor function, we can let rather than , and we will still have as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set is in without knowing about any of the other elements of . The definition I ended up using accomplishes this using induction: is defined as the class of sets for which a sort of induction on the elements of holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, . I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let , then . On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation . It is the contrapositive of the previous one with replaced by its complement. These definitions didn't work because the induction didn't "get off the ground." If does not contain the empty set, the condition fails, so doesn't get included in even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection of all individuals that have been generated so far. So one approach is to replace every occurence of with , making a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let be the power set function, then the induction in the inductive version works for being the class of transitive sets, restricted to subsets of . Therefore, must be transitive by definition of . This doesn't affect the union of all such , but it may or may not be desirable. The problem is that is only applied to transitive sets, because of the strong requirement , so the definition requires the additional constraint a ) C_ ( F in order to work. This issue can also be avoided by replacing with . The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: Foundedness: In the induction version, not only does include all the elements of , but it must include the elements of for even if those elements of are not in . We shouldn't care about any of the elements of outside , but this detail doesn't affect the correctness of the definition. If we replaced in the definition by , we would get the same class for setrecs. Suppose we could find a for which the condition fails for a given under the changed definition. Then the antecedent would be true, but would be false. We could then simply add all elements of outside of for any , which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded fails the condition under the metamath definition. The other direction is easier. If a certain fails the metamath definition, then all for , and in particular . The foundedness version is starting to look more like ax-reg 8497! We want to take advantage of the preexisting relation , which seems closely related to our foundedness definition. Since we only care about the elements of which are subsets of , we can restrict to in the foundedness definition. Furthermore, instead of quantifying over , quantify over the elements overlapping with . Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): Now, if we replace by , we do not change the definition. We already know that since and . All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5073, we add another variable representing the nonexistent element of in . Version 4 - Foundedness (6-Sep-2020): This is so close to df-fr 5073; the only change needed is to switch with . Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses , but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 8497. Rather than a disjoint element of , there's a disjoint coverer of an element of . Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep as a family of sets but allow to be any subset of in the induction. With this stronger induction, we can also allow for the stronger requirement rather than only . This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: Foundedness: ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of to be in . Version 2 does not have this issue. ) Similarly, we could allow to be any subset of any element of rather than any subset of . I think this has the same problem. We want to take advantage of the preexisting relation , which seems closely related to our foundedness definition. Since we only care about the elements of which are subsets of , we can restrict to in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: Now this looks more like df-fr 5073! The last step necessary to be able to use directly in our definition is to replace with its own setvar variable, corresponding to in df-fr 5073. This definition is incorrect, though, since there's nothing forcing the subset of an element of to be in . Version 1.3 (31-Aug-2020) Induction: Foundedness: must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on or , so this doesn't work. Let's try letting R be the covering relation to solve the transitivity issue (i.e. that if is the power set relation, consists only of transitive sets). The set corresponds to the variable in df-fr 5073. Thus, in our case, df-fr 5073 is equivalent to . Substituting our relation gives This doesn't work for non-injective because we need all to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show so we can replace one with the other. Negate both sides: If is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.) |
setrecs | ||
Theorem | setrecseq 42432 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
setrecs setrecs | ||
Theorem | nfsetrecs 42433 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
setrecs | ||
Theorem | setrec1lem1 42434* |
Lemma for setrec1 42438. This is a utility theorem showing the
equivalence
of the statement and its expanded form.
The proof uses
elabg 3351 and equivalence theorems.
Variable is the class of sets that are recursively generated by the function . In other words, iff by starting with the empty set and repeatedly applying to subsets of our set, we will eventually generate all the elements of . In this theorem, is any element of , and is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
Theorem | setrec1lem2 42435* | Lemma for setrec1 42438. If a family of sets are all recursively generated by , so is their union. In this theorem, is a family of sets which are all elements of , and is any class. Use dfss3 3592, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.) |
Theorem | setrec1lem3 42436* | Lemma for setrec1 42438. If each element of is covered by a set recursively generated by , then there is a single such set covering all of . The set is constructed explicitly using setrec1lem2 42435. It turns out that also works, i.e., given the hypotheses it is possible to prove that . I don't know if proving this fact directly using setrec1lem1 42434 would be any easier than the current proof using setrec1lem2 42435, and it would only slightly simplify the proof of setrec1 42438. Other than the use of bnd2d 42428, this is a purely technical theorem for rearranging notation from that of setrec1lem2 42435 to that of setrec1 42438. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
Theorem | setrec1lem4 42437* |
Lemma for setrec1 42438. If is recursively generated by , then
so is .
In the proof of setrec1 42438, the following is substituted for this theorem's : Therefore, we cannot declare to be a distinct variable from , since we need it to appear as a bound variable in . This theorem can be proven without the hypothesis , but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1762, making the antecedent of each line something more complicated than . The proof of setrec1lem2 42435 could similarly be made easier to read by adding the hypothesis , but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
Theorem | setrec1 42438 |
This is the first of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs is closed under . This effectively sets the
actual value of setrecs as a lower bound for
setrecs, as it implies that any set generated by
successive
applications of is a member of . This theorem "gets off the
ground" because we can start by letting , and the hypotheses
of the theorem will hold trivially.
Variable represents an abbreviation of setrecs or another name of setrecs (for an example of the latter, see theorem setrecon). Proof summary: Assume that , meaning that all elements of are in some set recursively generated by . Then by setrec1lem3 42436, is a subset of some set recursively generated by . (It turns out that itself is recursively generated by , but we don't need this fact. See the comment to setrec1lem3 42436.) Therefore, by setrec1lem4 42437, is a subset of some set recursively generated by . Thus, by ssuni 4459, it is a subset of the union of all sets recursively generated by . See df-setrecs 42431 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
setrecs | ||
Theorem | setrec2fun 42439* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs is a subclass of all classes that are closed
under . Taken
together, theorems setrec1 42438 and setrec2v 42443 say
that setrecs is the minimal class closed under .
We express this by saying that if respects the relation and is closed under , then . By substituting strategically constructed classes for , we can easily prove many useful properties. Although this theorem cannot show equality between and , if we intend to prove equality between and some particular class (such as ), we first apply this theorem, then the relevant induction theorem (such as tfi 7053) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
setrecs | ||
Theorem | setrec2lem1 42440* | Lemma for setrec2 42442. The functional part of has the same values as . (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
Theorem | setrec2lem2 42441* | Lemma for setrec2 42442. The functional part of is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
Theorem | setrec2 42442* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs is a subclass of all classes that are closed
under . Taken
together, theorems setrec1 42438 and setrec2v 42443
uniquely determine setrecs to be the minimal class
closed
under .
We express this by saying that if respects the relation and is closed under , then . By substituting strategically constructed classes for , we can easily prove many useful properties. Although this theorem cannot show equality between and , if we intend to prove equality between and some particular class (such as ), we first apply this theorem, then the relevant induction theorem (such as tfi 7053) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
setrecs | ||
Theorem | setrec2v 42443* | Version of setrec2 42442 with a dv condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
setrecs | ||
Theorem | elsetrecslem 42444* | Lemma for elsetrecs 42445. Any element of setrecs is generated by some subset of setrecs. This is much weaker than setrec2v 42443. To see why this lemma also requires setrec1 42438, consider what would happen if we replaced with . The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
setrecs | ||
Theorem | elsetrecs 42445* | A set is an element of setrecs iff is generated by some subset of setrecs. The proof requires both setrec1 42438 and setrec2 42442, but this theorem is not strong enough to uniquely determine setrecs. If respects the subset relation, the theorem still holds if both occurrences of are replaced by for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
setrecs | ||
Theorem | vsetrec 42446 | Construct using set recursion. The proof indirectly uses trcl 8604, which relies on , but theoretically in trcl 8604 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable rather than to avoid a distinct variable requirement between and . (Contributed by Emmett Weisz, 23-Jun-2021.) |
setrecs | ||
Theorem | 0setrec 42447 | If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
setrecs | ||
Theorem | onsetreclem1 42448* | Lemma for onsetrec 42451. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
Theorem | onsetreclem2 42449* | Lemma for onsetrec 42451. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
Theorem | onsetreclem3 42450* | Lemma for onsetrec 42451. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
Theorem | onsetrec 42451 |
Construct using set
recursion. When , the function
constructs the
least ordinal greater than any of the elements of
, which is for a limit ordinal and for a
successor ordinal.
For example, which contains , and , which contains . If we start with the empty set and keep applying transfinitely many times, all ordinal numbers will be generated. Any function fulfilling lemmas onsetreclem2 42449 and onsetreclem3 42450 will recursively generate ; for example, also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let , based on dfon2 31697. The proof of this theorem uses the dummy variable rather than to avoid a distinct variable condition between and . (Contributed by Emmett Weisz, 22-Jun-2021.) |
setrecs | ||
Model organization after organization of reals - see TOC | ||
Syntax | cpg 42452 | Extend class notation to include the class of partizan game forms. |
Pg | ||
Definition | df-pg 42453 | Define the class of partizan games. More precisely, this is the class of partizan game forms, many of which represent equal partisan games. In metamath, equality between partizan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.) |
Pg setrecs | ||
Theorem | elpglem1 42454* | Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.) |
Pg Pg Pg | ||
Theorem | elpglem2 42455* | Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.) |
Pg Pg Pg | ||
Theorem | elpglem3 42456* | Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.) |
Pg Pg | ||
Theorem | elpg 42457 | Membership in the class of partizan games. In ONAG this is stated as "If and are any two sets of games, then there is a game . All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
Pg Pg Pg | ||
This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/. | ||
Theorem | 19.8ad 42458 | If a wff is true, it is true for at least one instance. Deductive form of 19.8a 2052. (Contributed by DAW, 13-Feb-2017.) |
Theorem | sbidd 42459 | An identity theorem for substitution. See sbid 2114. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
Theorem | sbidd-misc 42460 | An identity theorem for substitution. See sbid 2114. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems. | ||
Syntax | cge-real 42461 | Extend wff notation to include the 'greater than or equal to' relation, see df-gte 42463. |
Syntax | cgt 42462 | Extend wff notation to include the 'greater than' relation, see df-gt 42464. |
Definition | df-gte 42463 |
Define the 'greater than or equal' predicate over the reals. Defined in
ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the
"NIST Digital Library of Mathematical Functions" , front
introduction,
"Common Notations and Definitions" section at
http://dlmf.nist.gov/front/introduction#Sx4.
This relation is merely
the converse of the 'less than or equal to' relation defined by df-le 10080.
We do not write this as , and similarly we do not write ` > ` as , because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: and but these are very complicated. This definition of , and the similar one for (df-gt 42464), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 42465 for a more conventional expression of the relationship between and . As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
Definition | df-gt 42464 |
The 'greater than' relation is merely the converse of the 'less than or
equal to' relation defined by df-lt 9949. Defined in ISO 80000-2:2009(E)
operation 2-7.12. See df-gte 42463 for a discussion on why this approach is
used for the definition. See gt-lt 42466 and gt-lth 42468 for more conventional
expression of the relationship between and .
As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
Theorem | gte-lte 42465 | Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
Theorem | gt-lt 42466 | Simple relationship between and . (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
Theorem | gte-lteh 42467 | Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
Theorem | gt-lth 42468 | Relationship between and using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
Theorem | ex-gt 42469 | Simple example of , in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
Theorem | ex-gte 42470 | Simple example of , in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as . However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved. | ||
Syntax | csinh 42471 | Extend class notation to include the hyperbolic sine function, see df-sinh 42474. |
sinh | ||
Syntax | ccosh 42472 | Extend class notation to include the hyperbolic cosine function. see df-cosh 42475. |
cosh | ||
Syntax | ctanh 42473 | Extend class notation to include the hyperbolic tangent function, see df-tanh 42476. |
tanh | ||
Definition | df-sinh 42474 | Define the hyperbolic sine function (sinh). We define it this way for cmpt 4729, which requires the form . See sinhval-named 42477 for a simple way to evaluate it. We define this function by dividing by , which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 42480 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
sinh | ||
Definition | df-cosh 42475 | Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4729, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.) |
cosh | ||
Definition | df-tanh 42476 | Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4729, which requires the form . (Contributed by David A. Wheeler, 10-May-2015.) |
tanh cosh | ||
Theorem | sinhval-named 42477 | Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 42474. See sinhval 14884 for a theorem to convert this further. See sinh-conventional 42480 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
sinh | ||
Theorem | coshval-named 42478 | Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 42475. See coshval 14885 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.) |
cosh | ||
Theorem | tanhval-named 42479 | Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 42476. (Contributed by David A. Wheeler, 10-May-2015.) |
cosh tanh | ||
Theorem | sinh-conventional 42480 | Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.) |
sinh | ||
Theorem | sinhpcosh 42481 | Prove that sinh cosh using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.) |
sinh cosh | ||
Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them. | ||
Syntax | csec 42482 | Extend class notation to include the secant function, see df-sec 42485. |
Syntax | ccsc 42483 | Extend class notation to include the cosecant function, see df-csc 42486. |
Syntax | ccot 42484 | Extend class notation to include the cotangent function, see df-cot 42487. |
Definition | df-sec 42485* | Define the secant function. We define it this way for cmpt 4729, which requires the form . The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) |
Definition | df-csc 42486* | Define the cosecant function. We define it this way for cmpt 4729, which requires the form . The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) |
Definition | df-cot 42487* | Define the cotangent function. We define it this way for cmpt 4729, which requires the form . The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | secval 42488 | Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | cscval 42489 | Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | cotval 42490 | Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | seccl 42491 | The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | csccl 42492 | The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | cotcl 42493 | The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
Theorem | reseccl 42494 | The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
Theorem | recsccl 42495 | The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
Theorem | recotcl 42496 | The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
Theorem | recsec 42497 | The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | reccsc 42498 | The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Theorem | reccot 42499 | The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.) |
Theorem | rectan 42500 | The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.) |
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