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Theorem List for Metamath Proof Explorer - 45201-45300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspd 45201 Specialization deduction, using implicit substitution. Based on the proof of spimed 2398. (Contributed by Emmett Weisz, 17-Jan-2020.)
(𝜒 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (∀𝑥𝜑𝜓))
 
Theoremspcdvw 45202* A version of spcdv 3544 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
(𝜑𝐴𝐵)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theoremtfis2d 45203* Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
(𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))    &   (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 𝜒𝜓)))       (𝜑 → (𝑥 ∈ On → 𝜓))
 
Theorembnd2d 45204* Deduction form of bnd2 9310. (Contributed by Emmett Weisz, 19-Jan-2021.)
(𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)       (𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜓))
 
Theoremdffun3f 45205* Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴       (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(𝑥𝐴𝑦𝑦 = 𝑧)))
 
20.42.2  Set Recursion
 
20.42.2.1  Basic Properties of Set Recursion

Symbols in this section:

All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 45207. The class 𝑌 is explained in the comment of setrec1lem1 45210. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.

 
Syntaxcsetrecs 45206 Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
 
Definitiondf-setrecs 45207* Define a class setrecs(𝐹) by transfinite recursion, where (𝐹𝑥) is the set of new elements to add to the class given the set 𝑥 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(𝐹). The goal of this definition is to construct a class fulfilling theorems setrec1 45214 and setrec2v 45219, which give a more intuitive idea of the meaning of setrecs. Unlike wrecs, setrecs is well-defined for any 𝐹 and meaningful for any function 𝐹.

For example, see theorem onsetrec 45230 for how the class On 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 𝑧, and consider the result when applying 𝐹 to any subset 𝑤𝑧. Remember that 𝐹 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 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements.

Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2o} would fail the condition since 2o is not an element of the successor of or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is , and suc ∅ does not have 1o as an element. However, we can let 𝑦 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(𝐹) was setrecs(𝐹) = ran recs((𝑔 ∈ V ↦ (𝐹 ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 45214 and setrec2v 45219 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, 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(𝐹) is that the necessary properties to prove are self-referential (see setrec1 45214 and setrec2v 45219), 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 8033, 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(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time.

Note that we cannot define setrecs(𝐹) 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(𝐹) as a union of a class abstraction.

If setrecs(𝐹) = 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹.

One necessary condition is that each element of a given 𝑥𝐴 must be generated by 𝐹 when applied to a previous element 𝑦𝐴. In symbols, 𝑥𝐴𝑦𝐴(𝑦𝑥𝑥 ⊆ (𝐹𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. 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 𝑦𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴.

The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 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, Fr. 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 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥¬ (𝐹𝑧) ⊆ 𝑦)).

On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝑦𝑧)}

In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. {𝑦 ∣ ∀𝑧 (𝑦𝑧 → ∃𝑤(𝑤𝑦 ∧ (𝐹𝑤) ∈ 𝑧 ∧ ¬ 𝑤𝑧))}

These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 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 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of 𝑧 with 𝑧, making 𝑧 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 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤𝑧, so the definition requires the additional constraint (𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) in order to work. This issue can also be avoided by replacing 𝑧 with 𝑧. The induction version of the result is used in the final definition.

Version 2: (18-Aug-2020) Induction: {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(𝑦𝑧 ≠ ∅ → 𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ (𝐹𝑤) ∩ 𝑧 ≠ ∅))}

In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹𝑤) for 𝑤 ⊆ (𝑦𝑧) even if those elements of (𝐹𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹𝑤) in the definition by ((𝐹𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦𝑧 would be false. We could then simply add all elements of (𝐹𝑤) outside of 𝑦 for any 𝑤𝑦, 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 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦𝑧), and in particular ((𝐹𝑤) ∩ 𝑦) ⊆ 𝑧.

The foundedness version is starting to look more like ax-reg 9044! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2.

Version 3 - Foundedness (5-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → ∃𝑣𝑧𝑤(𝑤𝑦𝑤𝑧 = ∅ ∧ 𝑣 ∈ (𝐹𝑤)))}

Now, if we replace (𝐹𝑤) by ((𝐹𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣𝑦 since 𝑣𝑧 and 𝑧𝑦. 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 5482, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧.

Version 4 - Foundedness (6-Sep-2020): {𝑦 ∣ ∀𝑧((𝑧𝑦𝑧 ≠ ∅) → 𝑣𝑧𝑤𝑢𝑧(𝑤𝑦 ∧ ¬ 𝑢𝑤𝑣 ∈ (𝐹𝑤))

This is so close to df-fr 5482; the only change needed is to switch 𝑤 with 𝑢𝑧. 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 Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9044. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧.

Finally, here's a different dead end I followed:

To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦𝑧 rather than only 𝑦𝑧. This will help improve the foundedness version.

Version 1.1 (28-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤 𝑧 → (𝐹𝑤) ∈ 𝑧)) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧(∃𝑎(𝑎𝑦𝑎𝑧) → ∃𝑤(𝑤𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. )

Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of 𝑧. I think this has the same problem.

We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition:

Version 1.2 (31-Aug-2020) Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦𝑤 𝑧 = ∅ ∧ (𝐹𝑤) ∈ 𝑧))}

Now this looks more like df-fr 5482! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5482.

This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧.

Version 1.3 (31-Aug-2020) Induction: {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤 𝑧 → (𝑤𝑧 ∧ (𝐹𝑤) ∈ 𝑧))) → 𝒫 𝑦𝑧)} Foundedness: {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 𝑤 𝑧 = ∅ ∧ (𝑤𝑧 ∨ (𝐹𝑤) ∈ 𝑧)))}

𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work.

Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹𝑤) corresponds to the variable 𝑦 in df-fr 5482. Thus, in our case, df-fr 5482 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧𝑣𝑅(𝐹𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → 𝑤((𝐹𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣𝑧(𝐹𝑤) ∈ (𝐹𝑣)))

This doesn't work for non-injective 𝐹 because we need all 𝑧 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 ¬ 𝑤𝑧 ↔ ∀𝑣𝑧¬ 𝑣𝑅(𝐹𝑤) so we can replace one with the other. Negate both sides: 𝑤𝑧 ↔ ∃𝑣𝑧𝑣𝑅(𝐹𝑤)

If 𝐹 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 𝑤 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(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
 
Theoremsetrecseq 45208 Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
(𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
 
Theoremnfsetrecs 45209 Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
𝑥𝐹       𝑥setrecs(𝐹)
 
Theoremsetrec1lem1 45210* Lemma for setrec1 45214. This is a utility theorem showing the equivalence of the statement 𝑋𝑌 and its expanded form. The proof uses elabg 3617 and equivalence theorems.

Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)

𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
 
Theoremsetrec1lem2 45211* Lemma for setrec1 45214. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3906, 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.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝑋𝑉)    &   (𝜑𝑋𝑌)       (𝜑 𝑋𝑌)
 
Theoremsetrec1lem3 45212* Lemma for setrec1 45214. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 45211. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴𝑌. I don't know if proving this fact directly using setrec1lem1 45210 would be any easier than the current proof using setrec1lem2 45211, and it would only slightly simplify the proof of setrec1 45214. Other than the use of bnd2d 45204, this is a purely technical theorem for rearranging notation from that of setrec1lem2 45211 to that of setrec1 45214. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌))       (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
 
Theoremsetrec1lem4 45213* Lemma for setrec1 45214. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 45214, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis 𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1836, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 45211 could similarly be made easier to read by adding the hypothesis 𝑧𝜑, 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.)

𝑧𝜑    &   𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝑋)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)
 
Theoremsetrec1 45214 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(𝐹) is closed under 𝐹. This effectively sets the actual value of setrecs(𝐹) as a lower bound for setrecs(𝐹), as it implies that any set generated by successive applications of 𝐹 is a member of 𝐵. This theorem "gets off the ground" because we can start by letting 𝐴 = ∅, and the hypotheses of the theorem will hold trivially.

Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon).

Proof summary: Assume that 𝐴𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 45212, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 45212.) Therefore, by setrec1lem4 45213, (𝐹𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4828, it is a subset of the union of all sets recursively generated by 𝐹.

See df-setrecs 45207 for a detailed description of how the setrecs definition works.

(Contributed by Emmett Weisz, 9-Oct-2020.)

𝐵 = setrecs(𝐹)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)
 
Theoremsetrec2fun 45215* 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(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 45214 and setrec2v 45219 say that setrecs(𝐹) is the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7552) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   Fun 𝐹    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetrec2lem1 45216* Lemma for setrec2 45218. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
 
Theoremsetrec2lem2 45217* Lemma for setrec2 45218. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
 
Theoremsetrec2 45218* 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(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, theorems setrec1 45214 and setrec2v 45219 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7552) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

𝑎𝐹    &   𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetrec2v 45219* Version of setrec2 45218 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
𝐵 = setrecs(𝐹)    &   (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))       (𝜑𝐵𝐶)
 
Theoremsetis 45220* Version of setrec2 45218 expressed as an induction schema. This theorem is a generalization of tfis3 7556. (Contributed by Emmett Weisz, 27-Feb-2022.)
𝐵 = setrecs(𝐹)    &   (𝑏 = 𝐴 → (𝜓𝜒))    &   (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))       (𝜑 → (𝐴𝐵𝜒))
 
20.42.2.2  Examples and properties of set recursion
 
Theoremelsetrecslem 45221* Lemma for elsetrecs 45222. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 45219. To see why this lemma also requires setrec1 45214, consider what would happen if we replaced 𝐵 with {𝐴}. 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.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
 
Theoremelsetrecs 45222* A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 45214 and setrec2 45218, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 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.)
𝐵 = setrecs(𝐹)       (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
 
Theoremsetrecsss 45223 The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐹𝐺)       (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))
 
Theoremsetrecsres 45224 A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.)
𝐵 = setrecs(𝐹)    &   (𝜑 → Fun 𝐹)       (𝜑𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵)))
 
Theoremvsetrec 45225 Construct V using set recursion. The proof indirectly uses trcl 9158, which relies on rec, but theoretically 𝐶 in trcl 9158 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.)
𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)       setrecs(𝐹) = V
 
Theorem0setrec 45226 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.)
(𝜑 → (𝐹‘∅) = ∅)       (𝜑 → setrecs(𝐹) = ∅)
 
Theoremonsetreclem1 45227* Lemma for onsetrec 45230. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝐹𝑎) = { 𝑎, suc 𝑎}
 
Theoremonsetreclem2 45228* Lemma for onsetrec 45230. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
 
Theoremonsetreclem3 45229* Lemma for onsetrec 45230. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
 
Theoremonsetrec 45230 Construct On using set recursion. When 𝑥 ∈ On, the function 𝐹 constructs the least ordinal greater than any of the elements of 𝑥, which is 𝑥 for a limit ordinal and suc 𝑥 for a successor ordinal.

For example, (𝐹‘{1o, 2o}) = { {1o, 2o}, suc {1o, 2o}} = {2o, 3o} which contains 3o, and (𝐹‘ω) = { ω, suc ω} = {ω, ω +o 1o}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated.

Any function 𝐹 fulfilling lemmas onsetreclem2 45228 and onsetreclem3 45229 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 33151.

The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.)

𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})       setrecs(𝐹) = On
 
20.42.3  Construction of Games and Surreal Numbers

Model organization after organization of reals - see TOC

 
Syntaxcpg 45231 Extend class notation to include the class of partisan game forms.
class Pg
 
Definitiondf-pg 45232 Define the class of partisan games. More precisely, this is the class of partisan game forms, many of which represent equal partisan games. In Metamath, equality between partisan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.)
Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
 
Theoremelpglem1 45233* Lemma for elpg 45236. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
 
Theoremelpglem2 45234* Lemma for elpg 45236. (Contributed by Emmett Weisz, 28-Aug-2021.)
(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
 
Theoremelpglem3 45235* Lemma for elpg 45236. (Contributed by Emmett Weisz, 28-Aug-2021.)
(∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
 
Theoremelpg 45236 Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿𝑅}. 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.)
(𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
 
20.43  Mathbox for David A. Wheeler

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/.

 
20.43.1  Natural deduction
 
Theoremsbidd 45237 An identity theorem for substitution. See sbid 2256. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
(𝜑 → [𝑥 / 𝑥]𝜓)       (𝜑𝜓)
 
Theoremsbidd-misc 45238 An identity theorem for substitution. See sbid 2256. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑𝜓))
 
20.43.2  Greater than, greater than or equal to.

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.

 
Syntaxcge-real 45239 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 45241.
class
 
Syntaxcgt 45240 Extend wff notation to include the 'greater than' relation, see df-gt 45242.
class >
 
Definitiondf-gte 45241 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 10674.

We do not write this as (𝑥𝑦𝑦𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦𝑦 < 𝑥), 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: > = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦 < 𝑥)} and ≥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑦𝑥)} but these are very complicated. This definition of , and the similar one for > (df-gt 45242), 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 45243 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.)

≥ =
 
Definitiondf-gt 45242 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 10543. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 45241 for a discussion on why this approach is used for the definition. See gt-lt 45244 and gt-lth 45246 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.)

> = <
 
Theoremgte-lte 45243 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
 
Theoremgt-lt 45244 Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
 
Theoremgte-lteh 45245 Relationship between and using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵𝐵𝐴)
 
Theoremgt-lth 45246 Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 > 𝐵𝐵 < 𝐴)
 
Theoremex-gt 45247 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.)
¬ 0 > 0
 
Theoremex-gte 45248 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.)
0 ≥ 0
 
20.43.3  Hyperbolic trigonometric functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cos‘(i · 𝑥)). 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.

 
Syntaxcsinh 45249 Extend class notation to include the hyperbolic sine function, see df-sinh 45252.
class sinh
 
Syntaxccosh 45250 Extend class notation to include the hyperbolic cosine function. see df-cosh 45253.
class cosh
 
Syntaxctanh 45251 Extend class notation to include the hyperbolic tangent function, see df-tanh 45254.
class tanh
 
Definitiondf-sinh 45252 Define the hyperbolic sine function (sinh). We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). See sinhval-named 45255 for a simple way to evaluate it. We define this function by dividing by i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional 45258 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 = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
 
Definitiondf-cosh 45253 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
 
Definitiondf-tanh 45254 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). (Contributed by David A. Wheeler, 10-May-2015.)
tanh = (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
 
Theoremsinhval-named 45255 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 45252. See sinhval 15503 for a theorem to convert this further. See sinh-conventional 45258 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.)
(𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))
 
Theoremcoshval-named 45256 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 45253. See coshval 15504 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
 
Theoremtanhval-named 45257 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 45254. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ (cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))
 
Theoremsinh-conventional 45258 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 set.mm. (Contributed by David A. Wheeler, 10-May-2015.)
(𝐴 ∈ ℂ → (sinh‘𝐴) = (-i · (sin‘(i · 𝐴))))
 
Theoremsinhpcosh 45259 Prove that (sinh‘𝐴) + (cosh‘𝐴) = (exp‘𝐴) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
(𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴))
 
20.43.4  Reciprocal trigonometric functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 45260 Extend class notation to include the secant function, see df-sec 45263.
class sec
 
Syntaxccsc 45261 Extend class notation to include the cosecant function, see df-csc 45264.
class csc
 
Syntaxccot 45262 Extend class notation to include the cotangent function, see df-cot 45265.
class cot
 
Definitiondf-sec 45263* Define the secant function. We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). 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 5113. (Contributed by David A. Wheeler, 14-Mar-2014.)
sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
 
Definitiondf-csc 45264* Define the cosecant function. We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). 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 5113. (Contributed by David A. Wheeler, 14-Mar-2014.)
csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
 
Definitiondf-cot 45265* Define the cotangent function. We define it this way for cmpt 5113, which requires the form (𝑥𝐴𝐵). 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 5113. (Contributed by David A. Wheeler, 14-Mar-2014.)
cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
 
Theoremsecval 45266 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴)))
 
Theoremcscval 45267 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴)))
 
Theoremcotval 45268 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴)))
 
Theoremseccl 45269 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) ∈ ℂ)
 
Theoremcsccl 45270 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℂ)
 
Theoremcotcl 45271 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) ∈ ℂ)
 
Theoremreseccl 45272 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) ∈ ℝ)
 
Theoremrecsccl 45273 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℝ)
 
Theoremrecotcl 45274 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) ∈ ℝ)
 
Theoremrecsec 45275 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) = (1 / (sec‘𝐴)))
 
Theoremreccsc 45276 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (sin‘𝐴) = (1 / (csc‘𝐴)))
 
Theoremreccot 45277 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0 ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = (1 / (cot‘𝐴)))
 
Theoremrectan 45278 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0 ∧ (cos‘𝐴) ≠ 0) → (cot‘𝐴) = (1 / (tan‘𝐴)))
 
Theoremsec0 45279 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
(sec‘0) = 1
 
Theoremonetansqsecsq 45280 Prove the tangent squared secant squared identity (1 + ((tan A ) ^ 2 ) ) = ( ( sec 𝐴)↑2)). (Contributed by David A. Wheeler, 25-May-2015.)
((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (1 + ((tan‘𝐴)↑2)) = ((sec‘𝐴)↑2))
 
Theoremcotsqcscsq 45281 Prove the tangent squared cosecant squared identity (1 + ((cot A ) ^ 2 ) ) = ( ( csc 𝐴)↑2)). (Contributed by David A. Wheeler, 27-May-2015.)
((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (1 + ((cot‘𝐴)↑2)) = ((csc‘𝐴)↑2))
 
20.43.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 45282 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4437 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 
20.43.6  Logarithms generalized to arbitrary base using ` logb `

Most of this subsection was moved to main set.mm, section "Logarithms to an arbitrary base".

 
Theoremlogb2aval 45283 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb𝐵, 𝑋 (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ( logb ‘⟨𝐵, 𝑋⟩) = ((log‘𝑋) / (log‘𝐵)))
 
20.43.7  Logarithm laws generalized to an arbitrary base - log_

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

This supports the notational form ((log_‘𝐵)‘𝑋); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.g., ((log_‘10)‘100) = 2.

This form is less convenient to work with inside set.mm as compared to the (𝐵 logb 𝑋) form defined separately.

 
Syntaxclog- 45284 Extend class notation to include the logarithm generalized to an arbitrary base.
class log_
 
Definitiondf-logbALT 45285* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as ((log_‘𝐵)‘𝑋) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.) (New usage is discouraged.)
log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
 
20.43.8  Formally define terms such as Reflexivity

EXPERIMENTAL. Several terms are used in comments but not directly defined in set.mm. For example, there are proofs that a number of specific relations are reflexive, but there is no formal definition of what being reflexive actually *means*. Stating the relationships directly, instead of defining a broader property such as being reflexive, can reduce proof size (because the definition of that property does not need to be expanded later). A disadvantage, however, is that there are several terms that are widely used in comments but do not have a clear formal definition.

Here we define wffs that formally define some of these key terms. The intent isn't to use these directly, but to instead provide a clear formal definition of widely-used mathematical terminology (we even use this terminology within the comments of set.mm itself).

We could define these using extensible structures, but doing so appears overly restrictive. These definitions don't require the use of extensible structures; requiring something to be in an extensible structure to use them is too restrictive. Even if an extensible structure is already in use, it may in use for other things. For example, in geometry, there is a "less-than" relation, but while the geometry itself is an extensible structure, we would have to build a new structure to state "the geometric less-than relation is transitive" (which is more work than it's probably worth). By creating definitions that aren't tied to extensible structures we create definitions that can be applied to anything, including extensible structures, in whatever way we'd like.

Benoit suggests that it might be better to define these as functions. There are many advantages to doing that, but they won't work for proper classes. I'm currently trying to also support proper classes, so I have not taken that approach, but if that turns out to be unreasonable then Benoit's approach is very much worth considering. Examples would be: BinRel = (𝑥 ∈ V ↦ {𝑟𝑟 ⊆ (𝑥 × 𝑥)}), ReflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel 𝑥) ∣ ( I ↾ 𝑥) ⊆ 𝑟}), and IrreflBinRel = (𝑥 ∈ V ↦ {𝑟 ∈ ( BinRel 𝑥) ∣ (𝑟 ∩ ( I ↾ 𝑥)) = ∅}).

For more discussion see: https://github.com/metamath/set.mm/pull/1286

 
Syntaxwreflexive 45286 Extend wff definition to include "Reflexive" applied to a class, which is true iff class R is a reflexive relation over the set A. See df-reflexive 45287. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅Reflexive𝐴
 
Definitiondf-reflexive 45287* Define reflexive relation; relation 𝑅 is reflexive over the set 𝐴 iff 𝑥𝐴𝑥𝑅𝑥. (Contributed by David A. Wheeler, 1-Dec-2019.)
(𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴 𝑥𝑅𝑥))
 
Syntaxwirreflexive 45288 Extend wff definition to include "Irreflexive" applied to a class, which is true iff class R is an irreflexive relation over the set A. See df-irreflexive 45289. (Contributed by David A. Wheeler, 1-Dec-2019.)
wff 𝑅Irreflexive𝐴
 
Definitiondf-irreflexive 45289* Define irreflexive relation; relation 𝑅 is irreflexive over the set 𝐴 iff 𝑥𝐴¬ 𝑥𝑅𝑥. Note that a relation can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.)
(𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴 ¬ 𝑥𝑅𝑥))
 
20.43.9  Algebra helpers

This is an experimental approach to make it clearer (and easier) to do basic algebra in set.mm.

These little theorems support basic algebra on equations at a slightly higher conceptual level. Instead of always having to "build up" equivalent expressions for one side of an equation, these theorems allow you to directly manipulate an equality. These higher-level steps lead to easier to understand proofs when they can be used, as well as proofs that are slightly shorter (when measured in steps).

There are disadvantages. In particular, this approach requires many theorems (for many permutations to provide all of the operations). It can also only handle certain cases; more complex approaches must still be approached by "building up" equalities as is done today.

However, I expect that we can create enough theorems to make it worth doing. I'm trying this out to see if this is helpful and if the number of permutations is manageable.

To commute LHS for addition, use addcomli 10825. We might want to switch to a naming convention like addcomli 10825.

 
Theoremcomraddi 45290 Commute RHS addition. See addcomli 10825 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       𝐴 = (𝐶 + 𝐵)
 
Theoremmvlraddi 45291 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐴 = (𝐶𝐵)
 
Theoremmvrladdi 45292 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       (𝐴𝐵) = 𝐶
 
Theoremassraddsubi 45293 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐴 = ((𝐵 + 𝐶) − 𝐷)       𝐴 = (𝐵 + (𝐶𝐷))
 
Theoremjoinlmuladdmuli 45294 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷       ((𝐴 + 𝐶) · 𝐵) = 𝐷
 
Theoremjoinlmulsubmuld 45295 Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴𝐶) · 𝐵) = 𝐷)
 
Theoremjoinlmulsubmuli 45296 Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   ((𝐴 · 𝐵) − (𝐶 · 𝐵)) = 𝐷       ((𝐴𝐶) · 𝐵) = 𝐷
 
Theoremmvlrmuld 45297 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐴 = (𝐶 / 𝐵))
 
Theoremmvlrmuli 45298 Move LHS right multiplication to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0    &   (𝐴 · 𝐵) = 𝐶       𝐴 = (𝐶 / 𝐵)
 
20.43.10  Algebra helper examples

Examples using the algebra helpers.

 
Theoremi2linesi 45299 Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use inference form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝑋 ∈ ℂ    &   𝑌 = ((𝐴 · 𝑋) + 𝐵)    &   𝑌 = ((𝐶 · 𝑋) + 𝐷)    &   (𝐴𝐶) ≠ 0       𝑋 = ((𝐷𝐵) / (𝐴𝐶))
 
Theoremi2linesd 45300 Solve for the intersection of two lines expressed in Y = MX+B form (note that the lines cannot be vertical). Here we use deduction form. We just solve for X, since Y can be trivially found by using X. This is an example of how to use the algebra helpers. Notice that because this proof uses algebra helpers, the main steps of the proof are higher level and easier to follow by a human reader. (Contributed by David A. Wheeler, 15-Oct-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑌 = ((𝐴 · 𝑋) + 𝐵))    &   (𝜑𝑌 = ((𝐶 · 𝑋) + 𝐷))    &   (𝜑 → (𝐴𝐶) ≠ 0)       (𝜑𝑋 = ((𝐷𝐵) / (𝐴𝐶)))
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454 45301-45326
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