(30-Dec-04) I think iunss2 is rather nice (even though its proof is
very short), and I might be able to use it to shorten oaass.
(25-Dec-04) oaass is a huge proof! This has been on my to-do list for a
long time, so getting it done is a relief. Before we only had nnaass,
which is limited to natural numbers and which I did a long time ago for
the real/complex number construction. To build the oaass proof I
basically copied nnaass, then added the induction hypothesis for limit
ordinals (step 164). But proving this hypothesis is 2/3 of the final
proof, or steps 44-164! (And this doesn't count oalimcl, which is
really a lemma for oaass with no other use I'm aware of.) For the other
steps, you may want to compare oaass to nnaass - they are identical
except for closure laws. (Eventually I'll use oaass to shorten nnaass.)
By the way to understand why nnaass was needed for our complex number
construction, you can follow its path as it propagates forward through
the stages of construction of complex numbers (and the associative law
for complex number addition in particular), which you can see in the
"referenced by" lists:
nnaass -> addasspi -> addasspq -> addasspr -> addasssr -> axaddass
BTW Merry Christmas!
(23-Dec-04) sumdmd is Holland's theorem (finally).
(21-Dec-04) sbequ5 is just a cleanup of an earlier version. By the way
the theorem eq5 used in its proof was the first "significant" theorem
(so I felt) that I proved from the unusual predicate calculus subsystem
ax-4 through ax-15, and it was the initial clue towards eventually
proving completeness for my "Finitely Axiomatized..." paper. I remember
feeling very excited that it held, some dozen years ago. I still think
its proof is strange and unintuitive. Axiom ax-11 is not used for its
proof and was added later. An open problem is whether ax-11 is
redundant; it can be proved as a "metamatheorem" (outside of Metamath)
from the others but I've never been able to prove it directly from the
others.
The Hilbert space stuff is more progress towards Holland's theorem.
(20-Dec-04) Q is strictly less equinumerous than R as promised. (Now,
that wasn't hard, was it?)
(19-Dec-04) qbtwnre, that Q is dense in R, is an important result about
real numbers. This is one of those counterintuitive results about
infinite sets: Q is strictly less equinumerous than R (I guess I should
add a theorem for that - yes, I'll do it for tomorrow), but on the other
hand, in between any two reals, no matter how close they are, you'll
find a rational! How can that be? Think about it. This used to bother
me when I was younger, and I suppose it still gives me an uneasy feeling
now and then.:)
(17-Dec-04) I finally received from the library Holland's 1969 paper
(listed in the Hilbert Space Explorer references) which turns out to
have a marvelous proof of subspace closure of the sums of dual modular
pairs. I've been looking for such a proof for a long time. Maeda's
version that I tried to use has a gap I haven't been able to figure out,
but Maeda died a couple of years ago so I can't ask him to clarify it
(and no one else seems to know). The Hilbert space results for the last
couple of days are some basic facts we'll eventually draw on for
Holland's proof.
(15-Dec-04) Some simple but frequently-used theorems that let me shorten
a bunch of proofs.
(14-Dec-04) The proof of Prop. 8.8 in Takeuti&Zaring p. 59 has a typo:
the index on the big union should be "delta < gamma", not "delta <
beta". (This brings to total typos I've found in the book to about 30.
In spite of this, it is by far the best book for the technical details
of many proofs that other books gloss over. Too bad it is out of
print.)
I put a lot of work into oawordeulem because I was afraid it was going
to be huge. The end result is that it is actually 1/3 smaller than the
finite version that it replaces, and I'm kind of proud of it. (The
finite version was nnordexlem, which I deleted but you can still find on
some of the mirrors.) The intersection of the temporary class S
corresponds to gamma in the T&K proof.
(13-Dec-04) These variations of modus ponens and syllogism are used to
shorten a number of proofs and will be handy to have available in the
future.
(12-Dec-04) In our database, ordinal arithmetic is not well developed.
The history is that I did just enough to construct real numbers, and
some operations (e.g. the associative law nnaass) are proved for
natural numbers only. Over time I'll be extending these to the
transfinite. I'm discovering that some textbook proofs have subtle
mistakes (or to be generous, they gloss over details) that only become
apparent when you try to work them out, making for a somewhat
frustrating task. Perhaps I'll mention some of them here as they come
up.
Today's oawordri is the commuted version of oaword of 8-Dec. Unlike the
natural number version it must be proved separately since ordinal
addition ceases to be commutative in the transfinite. Note that the
commuted version of oaord of 7-Dec does not hold in the transfinite.
Also, the converse direction of oawordri does not hold in the
transfinite.
(11-Dec-04) In ordunidif, note that we're not removing a single element
but an entire subset of elements up to that element. (Otherwise it
would be A\{B}, not A\B.) For example: if A is 5={0,1,2,3,4} and B is
4={0,1,2,3}, then A\B is {4}. Then the union of A is 4, and the union
of A\B is also 4. Notice that for finite (and other successor) ordinals,
the union operation subtracts 1.
(9-Dec-04) All the previous work leading to rebtwnz, involving heavy use
of the archimedean principle and well ordering of bounded integer sets,
finally pays off by letting us show the properties of the floor function
in flleltt. It is interesting that so much work is needed for such a
"trivial" thing as the floor function.
(7-Dec-04) I have been wanting to prove oaordi for some time but the
limit case of the transfinite induction (steps 32-51) kept proving
illusive. Finally I realized ssiun2s was the trick I needed, and we
don't even need the available "for all y less than x" antecedent (added
with a1d at step 51, and that proved to be a red herring in earlier
proof attempts).
(3-Dec-04) Look at how easy the definition of factorial df-fac becomes
using our new seq operation. I'm very pleased. (See
http://us2.metamath.org:8888/mpegif/mmset.html#function for why the
notation is (!`n) instead of n!.)
(2-Dec-04) I am torn whether to add a definition for the class of all
cardinal numbers. Well, it actually has been added - df-cardn - but I
consider it provisional and so far I've avoided using it. With the
definition, ondomcard could be simplified to {x e. On|...} e. Card. But
df-cardn saves only a few symbols, and already we have so many
definitions that it's hard to remember them all. I've never seen both
df-card and df-cardn in the same textbook, although textbooks have the
advantage of informal English like "is a cardinal number".
shsumval3 is nice, compared to the "official" subspace sum
value, shsumvalt. I'm tempted to redefine df-shsum accordingly,
although the current definition does capture the "intent" of subspace
sum even though it's more complicated.
(1-Dec-04) zmax shows an application of reuxfr that we proved a
few days ago. Note the use of reuhyp to eliminate one of reuxfr's
hypotheses.
I added a new "feature" for those people who check the most
recent proofs page every day. You can go directly to the new theorems
via http://us2.metamath.org:8888/mpegif/mmrecent.html#table without
having to scroll down.
(30-Nov-04) It is surprisingly difficult to generalize the integer B in
uzwo2 with the real B in uzwo3, when on the surface this seems so
obvious. Integers and reals don't mix naturally.
(19-Nov-04) euxfr, even though it is simple-looking, is a culmination of
a lot of results about existential uniqueness. As you backtrack through
its proof you will find mopick, moexex, 2moswap, 2euswap, and euxfr2,
all of which I find interesting as somewhat non-obvious, but none of
which appear in the literature to my knowledge. But euxfr is often used
in informal proofs implicitly; for example: if we know that there is
exactly one x such that f(x)=4, then we can conclude there is exactly
one y such that f(3y+5)=4. And vice-versa. The formalization of this
argument is euxfr. Who would have guessed that something so "obvious"
is apparently so hard to prove? (I don't know how hard it would be to
prove euxfr directly, though.)
(17-Nov-04) Takeuti/Zaring have a rather long proof of ac6s2 that I
wasn't looking forward to formalizing. Then it occurred to me that I
already have ac6s, which is proved rather simply from the AC variation
ac6 plus the Boundedness Axiom bnd2. The final proof of ac6s2 is very
simple.
(11-Nov-04) uzwo generalizes nnwo, whose proof has been shortened to
become a special case of uzwo.
Interesting tidbit: I did a quick count, and there are now 5177
theorems (including Hilbert space) in set.mm, and 1480 of them reference
969 bibliographic entries (theorems, exercises, etc.) on the
mmbiblio.html Bibliographic Cross Reference page. (This is more
bibliographic entries than I would have guessed offhand.)
(10-Nov-04) mapunen, with its long, tortuous proof, completes (whew) the
"basic" equinumerosity theorems needed for the 3 operations (addition,
multiplication, exponentiation) of cardinal arithmetic.
(5-Nov-04) syli is a little "discovery" that shortens 14 proofs, more
than paying for itself by reducing the total size of the database.
(4-Nov-04) I think Takeuti/Zaring's definition of omega, dfom2, is
rather strange and unintuitive. But this proof confirms it is correct.
(2-Nov-04) For infcntss, Takeuti/Zaring write after their Exercise 8 for
this: "Hint: Use AC". Our antecedent expresses "is infinite" in a
different way that allow us to avoid AC. (We also don't need the Axiom
of Infinity because the antecedent is false if omega doesn't exist.
This is a fortuitous quirk of our definition of dominance, and a
different definition of dominance might require Infinity for this
theorem.)
(1-Nov-04) qexpclt nicely illustrates the use of our general-purpose
exponentiation closure lemma expcllem.
(29-Oct-04) nominpos was written specifically to refute an incorrect
statement on Usenet:
http://groups.google.com/groups?selm=XY-dnRTa4ISmyhzcRVn-vw%40rcn.net
A somewhat significant change was put into the site today. The old
object "ded", used for the weak deduction theorem, has been renamed to
"if", and it is now called the "conditional operator" instead of the
"deduction class". This is more in keeping with its true function,
especially since it is being used more and more for
non-deduction-theorem things. df-ded was renamed to df-if, along with
some other renaming. The Deduction Theorem page was revised to reflect
this change. You may want to look at the quick example at the beginning
of the Deduction Theorem page mmdeduction.html to see if it makes more
sense now.
(25-Oct-04) I forgot to mention that on 19-Oct I added onpwsuc
specifically to answer this Usenet question:
http://groups.google.com/groups?threadm=Ts6dnc3uTY-bLejcRVn-qQ%40rcn.net
(23-Oct-04) infdif was quite frustrating, particularly because Enderton
mentions it in passing almost as an obvious, trivial fact. His one-line
proof gives no hint of the difficulties lurking underneath. I could see
no shorter way to prove it.
(22-Oct-04) The new, elegant df-exp, that makes use of our new infinite
sequence builder df-seq, now completely replaces the old awkward one.
Decisions, decisions. I don't like that the new df-exp requires the
artificial complex number axiom axmulex. I might redefine it so that in
place of "( x. seq ( NN X. { x } ) )" it would use
"( ( x. |` ( CC X. CC ) ) seq ( NN X. { x } ) )", so that the sethood
of multiplication "x." becomes irrelevant. But that complexifies
df-exp, and I'm also thinking about future applications of df-seq.
So I'm not sure what I'll do.
(20-Oct-04) By using the Axiom of Choice, qnnen has the shortest proof
I'm aware of. The standard proof uses somewhat complex (from Metamath's
point of view) algebraic manipulations. Of course most authors want to
avoid AC when possible, but I thought it interesting to show how it can
be shortened with AC.
(18-Oct-04) I finally completed what turned out to be a bigger project
that I expected, which is to prove that the reals are uncountable.
Today's proof ruc is the culmination of several days' work. It actually
involves 39 lemmas, ruclem1 through ruclem39.
(I artificially backdated the lemmas ruclem1 through ruclem39 so they
won't clutter up the "most recent proofs" page.)
See us2.metamath.org:8888/mpegif/mmcomplex.html#uncountable for a
detailed description.
(17-Oct-04) I think df-seq is going to be extremely useful in the
future. For a long time I've been stuggling with a good way to
represent finite and infinites sequences, series, products, etc. which
are universally prevalent in the study of limits, calculus, and pretty
much all of higher analysis. Each one seemed to require its own messy
development - see our ugly df-exp for example, and that's just the tip
of the iceberg. Well, df-seq seems to be the answer! For example,
df-exp can be alternately defined so its value becomes (in set.mm
notation):
( A ^ B ) = ( ( x. seq ( NN X. { A } ) ) ` B )
where NN X. { A } is the constant function with value A. The seq
operation produces the sequence A, A x. A, (A x. A) x. A,... that we
evaluate at B. How much simpler could it get? Eventually I'll redefine
df-exp accordingly. df-seq will even be useful for showing that real
numbers can be represented by an infinite decimal expansion (a mundane
thing in everyday mathematics but quite difficult formally). So expect
to see a lot more of the seq object!
Fortunately you don't have to be concerned with df-seq itself (with its
horrendously complex definition) but only with its consequences seq1 and
seqsuc, which should be enough to handle most work with sequences and
series. In fact the only purpose of df-seq is to provide us with an
object that has those simple properties.
I defined seq to start a 1, not 0, to make it compatible with our
df-clim. A long time ago I initially defined df-clim to start at 0
but ran into messy problems avoiding divide-by-zero when actually
computing limits, and 1 makes certain things a lot simpler. This is
possibly one reason most analysts have the natural numbers start at 1.
Anyway, if the need for starting at 0 or some other integer arises, I
can redefine a more general df-seq, but for now I wanted to keep it as
simple as possible.
Internally, df-seq uses an ordered pair to increment a
counter that is used to look up the value of the input sequence, so in a
way it is almost like it has a little "computer program" inside. Unlike
a computer program, though, the "counter" counts to infinity! The
incredible power of this concept almost boggles the mind.
By the way, with regard to "computer program like" devices, it is
becoming clear that the df-ded object, now used almost exclusively for
the weak deduction theorem, is in fact a far more applicable kind of
general-purpose "if statement". Raph Levien pointed this out to me, and
I've already used it as such in unxpdomlem. ded(phi,A,B) means "if phi
is true, return A, otherwise return B" and is almost exactly analogous
to the C-language conditional operator "condition ? A : B". It can be
useful for defining functions such as
/
| 1 / x, if x =/= 0
f(x) = <
| 0, otherwise
\
and if you look at Takeuti/Zaring's proof of unxpdomlem you'll see that
this is exactly what I am doing, with nested ded's to represent 3
possible alternatives. Maybe I should rename "ded" to be "sel" for
selector or even "if". What do you think?
The "seq", "ded", and "rec" objects in set.mm apparently do not appear
in the literature, so there is no standard notation for them, which I
think is unfortunate.
(12-Oct-04) Prof. Nambiar communicated his paper to me earlier this
year. gch-kn shows that the Generalized Continuum Hypothesis is
equivalent to a generalized version of his Axiom of Combinatorial Sets,
from which the result in his paper falls out as the special case of A=0.
(10-Oct-04) ssenen doesn't seem to appear in any book but is apparently
assumed implicitly in the proof in Lemma 6.2 of Jech's _Set Theory_ p.
43 (otherwise I can't see how that proof could work). At first I
thought the proof of ssenen would be trivial - it seems intuitive enough
on the surface. But I soon discovered it's a rather difficult thing to
prove, and I hope you like the proof I came up with.
(9-Oct-04) The theorem cdaval has a comment explaining how we do
cardinal arithmetic, which you may find confusing. The only new
operation we actually introduce is +c (disjoint union), and we re-use
existing set operations for the others. Even +c is not really an
operation from cardinal numbers to cardinal numbers, but just a general
set operation. Here is the mapping:
+c is used to represent cardinal number addition
cross product is used to represent cardinal number multiplication
set exponentiation is used to represent cardinal number exponentiation
equinumerosity is used to represent cardinal number equality
With this mapping, we can easily obtain the "real" cardinal arithmetic
operations, in the sense of operations that take actual cardinal numbers
and produce new actual cardinal numbers. To get the actual cardinal
number (which is the smallest ordinal equinumerous to it) corresponding
to any set, we use the "card" function. A cardinal number is its own
cardinal number as shown by cardid and cardcard.
A cardinal number is nothing more than a measure of the "size" of a set.
For finite sets, it is easy to see how the set operations correspond.
For example, the cross product of a set with 3 elements and a set with
5 elements will result in a set with 15 ordered pairs, corresponding to
3 x 5 = 15.
Our approach is exactly the approach used by Mendelson, and it saves us
having to have a whole bunch of new operations and theorems on them that
would essentially just be window dressing disguising what we already
have.
(5-Oct-04) infxpidm is a deep and important theorem that is typically
one of the most difficult in elementary set theory textbooks, and some
books don't even get that far (or omit the proof). The only other
formal proof of this I'm aware of is on the Isabelle prover. About half
of the paper "Mechanizing set theory: cardinal arithmetic and the axiom
of choice" http://www.cl.cam.ac.uk/users/lcp/papers/Sets/AC.pdf is
devoted to the Isabelle proof of infxpidm. Of course that was in 1996,
so Metamath is about 8 years behind the times.:)
(2-Oct-04) Some important milestones today!
In our development there are two completely different sets, omega (a
subset of ordinal numbers) and N (a subset of complex numbers), that are
both called "natural numbers," undoubtedly causing a great deal of
confusion. The former are the natural numbers of set theorists, the
latter the natural numbers of analysts, and both sets satisfy Peano's
axioms. nnenom relates these two sets, that live in completely different
worlds, by showing that they are equinumerous.
xpomen is a milestone theorem showing the cross product of omega with
itself is countable, and will be used as a key stepping stone for the
multiplication of infinite cardinal numbers.
xpomen is derived directly from xpnnen with nnenom. xpnnen in turn is
essentially dependent on Raph Levien's nn0opth. It is pleasant to see
everything finally all fits together, with one thing building on
another.
(1-Oct-04) Right now, the proof of sucdom uses some heavy-duty AC
(entri2) and Infinity (omex, nnsdom) stuff for such a trivial theorem
(that is casually used, without mention of how it's proved, in
textbooks). I could not for the life of me figure out how to prove it
without AC and Infinity. Does anyone know?
(30-Sep-04) As unrelated as they may seem, these are all preliminaries
towards getting into transfinite cardinal arithmetic. dedex was a
surprise: I expected a somewhat involved proof, and behold! it is just
a special case of keepel.
(29-Sep-04) An interesting curiosity is that xp2cda is an actual
equality, not just an equinumerosity relation.
I added a note to yesterday's cdaval, explaining our approach to
cardinal arithmetic.
(28-Sep-04) Today, with cdacomen, we take the first baby step in what
promises to be a fascinating journey into the deeply profound world of
cardinal arithmetic, where we will uncover the mysterious properties the
objects in "Cantor's paradise" (transfinite cardinal numbers). The new
definition added is df-cda.
Some books define cardinal arithmetic operations explicitly, whereas
other books use cardinality and equinumerosity applied to ordinary set
operations. We will use Mendelson's hybrid approach, where cardinal sum
is defined as a kind of "disjoint union" operation, and where cross
product and set exponentiation serve the roles of product and powers.
So, equinumerosity will mean "equals", and dominance will mean "less
than or equal to".
(27-Sep-04) I changed the turnstile |- color from green to gray. The
color had no purpose other than to make it stand out, and with so many
other colors now (especially with the new rainbow-colored little
numbers) it seems like a gratuitous and pointless use of color. In the
old days I suppose it added a splash of color to an otherwise dreary
page, but now it just makes the page busy. In addition, it is
inconsistent since color otherwise distinguishes variables from
constants (and |- is a constant). I hope the gray is a sufficient
distinction to impress the reader that it is a meta symbol that somehow
transcends the "normal" math. The indentation indicator next to it is
also gray, but I don't think it will cause confusion.
It seems another proof system called "DC Proof" is comparing itself to
Metamath. I suppose I should take that as flattery. :)
http://groups.google.com/groups?selm=OQh5d.2688%24KF.21330%40tor-nn1.netcom.ca
(24-Sep-04) Our proof of undom is much simpler (when formalized) than
Mendelson's, who defines a complicated one-to-one function from the
left-hand to right-hand side of the dominance relation in the
conclusion. Basically we exploit unen, domen, and ssdom2g, avoiding any
use of functions at all.
(17-Sep-04) dmsnsnsn allows us to complete the "theory" of domains of
iterated singletons of the empty set 0. Thus:
dom 0 = 0
dom {0} = 0
dom {{0}} = 0
dom {{{0}}} = {0}
dom {{{{0}}}} = {{0}}
dom {{{{{0}}}}} = {{{0}} etc.
The first 3 are dm0, dmsn0, dmsnsn0. By the way this has absolutely
no practical value whatsoever. :)
Historical trivia: The singleton {0} reminded me of the theorem pwpw0,
which is considered too obvious even to mention as a separate theorem in
most books. Suppes leaves its proof as an exercise. Even when a book
gives a "proof" it goes something like "{0} has 2 possible subsets, 0
and {0}, so its power set is {0,{0}}." In the early days of Metamath I
was completely baffled how to formalize this and spent several
frustrating days with this theorem. I finally stumbled upon exintr,
which I don't think is in any book, as the key to its proof. As a kind
of cynical joke, in its description I said (as many books do) that we
"compute" its power set, although the proof seems to shed little light
on what the algorithm might be. :)
(16-Sep-04) The key theorem for qaddclt is divadddivt at step 26, which
allows us to prove that the sum of two ratios of integers is a ratio of
integers. The other stuff is the messy overhead, made more so by having
to avoid dividing by zero and having to work with N, Z, Q, and C in the
same proof.
(14-Sep-04) I like abrexex2. Simple to state yet intrinsically quite
deep and powerful. The Axiom of Replacement plays an important role in
its proof, and this theorem might be equivalent to it (although I'm not
sure). Note that it is essentially derived from iunex, which in turn
derived from abrexex and the Axiom of Union. On the other hand, abrexex
can be easily recovered from abrexex2 by replacing phi with y=B, since
the 2nd hypothesis {y|y=B} always exists (it is either a singleton, or
the empty set when B is a proper class). Wow! I wonder if this has
ever been published.
(13-Sep-04) I think uni0b would make a nice textbook exercise. By the
way sssn can be used to show that A in uni0b is either the empty set or
the singleton of the empty set. sssn is another nice little theorem that
seems obvious but whose proof is not quite as simple as one might think.
(12-Sep-04) zltle1 is one of those unfuriatingly long proofs of
something seemingly extremely trivial. This is in spite of the fact we
already have nnlelt1 (with an insanely long proof that has so far defied
attempts to shorten it), on which zltle1 is ultimately based.
elab2 and elab2g replace the venerable vtoclab and vtoclabg,
generalizing them so as not to require that x and B be distinct
variables. This is important in cases where B is a class variable in a
hypothesis, usually as part of an intermediate lemma, that will
eventually be eliminated with cleqid. By removing this restriction I
was able to shorten a couple of additional proofs that previously used
elab.
(7-Sep-04) Our proof of Abian's "A most fundamental fixed point theorem"
(abianfp) is, as far as I know, the first credible validation of this
interesting and not-so-trivial theorem. In his last years, Abian
espoused some rather unconventional theories on Usenet, causing him to
be considered a crank and not taken seriously. The following Usenet
post appears to be the original one announcing this theorem:
http://groups.google.com/groups?selm=68v5ij%24bar%241%40news.iastate.edu&output=gplain
Abian claimed for this theorem, "The fundamental significance of the
Theorem lies in the fact that a great many fixed point theorems can be
reduced to the special cases of the Theorem." After some effort I
located a post where Abian seems to have proved Tarski's classical Fixed
Point Theorem as a corollary to this one:
http://groups.google.com/groups?selm=6972ve%24t96%241%40news.iastate.edu&output=gplain
This seems correct (although I need to work out the details to be
completely convinced). If so, I think this is significant, but
apparently no one on Usenet took it seriously nor indicated they
understood it.
iunconst, fnresdm, and abianfplem are prerequisites for abianfp.
(6-Sep-04) tfinds2 should produce shorter proofs than tfinds for some
problems (analogous to finds vs. finds2). I derived it from tfindes to
show it can be done (i.e. going from explicit back to implicit
substitution), but it required the new (and somewhat unusual) theorems
sbralie (what a brain-teaser), sbcco2 (allows us to obtain
[ suc x / x ] phi where suc x and x share a free variable), and sbcie
added today. An interesting exercise, and the technique should be
useful in the future.
(2-Sep-04) dedlem1 and dedlem2 are not new but revisions of older
versions. The "ded" operation can be useful for more than just the weak
deduction theorem - as these lemmas show, it acts as a "selector" from
two classes depending on the truth or falsity of a wff.
(31-Aug-04) undm may be the shortest proof in the database. ("V \ A"
means the "complement of A", and most authors introduce a symbol
abbreviating this expression. We need it so rarely that I decided not
to define it separately, at least not yet.)
Mr. O'Cat provides us with a shorter proof for ja.
difun was used to shorten the proof of dif23 of Aug 27.
(30-Aug-04) Yesterday's cfub is used to simplify the proof of cf0.
(29-Aug-04) A neat thing about snsspr is that neither A nor B have to be
sets (which we accomplish with the use of snprc in its proof). I was
actually surprised that I didn't have this in the database, and was able
to shorten a couple of proofs with it (see the "referenced by" list).
cfub is (for me) an important breakthrough that will let us finally get
some cofinality stuff proved efficiently. I haven't seen this in a
textbook.
(28-Aug-04) fconstfv is tricky - the cases of A = empty and A =
non-empty must be proved separately. The reason is the non-emptiness
requirement of r19.9rzv in step 23. This is one of those little
surprises you don't think of until you actually try to prove it. But
it's neat the theorem still holds when A is empty. It's also neat the
theorem still works when B is a proper class, and it is a pleasant
coincidence that the theorem still works when generalized to these
cases.
By contrast, fconst2 (without its hypothesis) fails when B is a proper
class. Specifically, if F is empty and A is non-empty, the left-hand
side will be false but the right-hand side will be true. See snprc for
what happens to a singleton of a proper class, and see f00 for what
happens when a mapping has an empty codomain.
(26-Aug-04) In dminss, R"A is the image of R under A - it "maps" a
subset of the domain to a subset of the range. Taking the converse
image of this, we go back and get something at least as big as the
original subset of the domain. Example: consider the constant
function; the converse image of any (non-empty) subset of the range
(i.e. the whole of the range, which is a singleton) will be the entire
domain. And this of course is at least as big as the original subset of
the domain. Suppes calls this "somewhat surprising" but doesn't say
why.
(21-Aug-04) I saw iuniin for the first time on the web page
http://en.wikipedia.org/wiki/Union_(set_theory) (go to bottom of the
page) and thought it was neat, so here is the proof. Does anyone
know what book it comes from?
(20-Aug-04) A long time ago when I was first learning logic, I was so
enamored with dfor2 that I wrote it on an index card for my "curious
math facts" collection. I can't remember if I read it somewhere or
stumbled across it on my own.
At one point I thought looinv was intuitionistic and sent Mr. O'Cat on a
wild goose chase trying to prove it from ax-1 and ax-2. It turns out
it's as nonintuitionistic as it could possibly be, being equivalent in
fact to Peirce's axiom (theorem peirce).
(17-Aug-04) ac6s had me stumped for a long time. It seemed reasonable
to me that there was no reason to require that B be a set in ac6, but
proving it was a different matter. It now seems we need the very deep
and strong Boundedness Axiom (whose proof is the culmination of our
development of rank and the cumulative hierarchy of sets). The proof of
ac6s is a nice example of an application of the Boundedness Axiom.
(16-Aug-04) Using a technique inspired by Mel O'Cat, which is just
deleting the whole proof then letting the metamath program try to prove
it using "improve all/depth n" for n=1,2,3,4 in succession, I found
shorter proofs for the following 43 theorems:
pm2.21i pm2.21d pm2.18 pm2.65 orc jctil jctir imp42 imp44 imp45 adantl
adantr adantld anidm anidms ancom imdistanri abai anabs1 anabs7 anabsi5
anabsi8 anabss1 anabss3 anabss4 anabss5 anandis anandirs ibi bilimd iba
ibar 3impdi 3impdir pm5.1 baibr biantr niabn ninba ax6 19.33 euorv
exmoeu2
I tried this on the first 984 theorems in the database (everything up to
set theory). ("improve all" was not designed to find proofs, but just
to fill out syntax constructions.) Out of these, "improve all" was able
to find proofs for 237 of them! 134 of these were identical to the
existing proofs found by hand, 60 were longer, and the 43 above were
shorter.
(16-Aug-04) zorn2 is a simpler version of Zorn's Lemma than zorn, so I
made it the "official" Zorn's lemma on the Theorem Sampler of the
Metamath Proof Explorer Home Page.
(15-Aug-04) qlax* are the new theorems that directly correspond to the
Quantum Logic Explorer axioms.
Most of today's non-Hilbert-space theorems are just slight modifications
to older ones for better overall consistency. Sometimes the older ones
are still there if you click the "Next" link, until I slowly weed them
out of the theorems that use them.
If you have any comments on the new "Theorem list" links let me know.
For example, is the upper right corner the best place to put it? Do you
find it useful?
(14-Aug-04) Mr. O'Cat comes through for us again with loolin.
Well, well, it looks like _both_ directions of reluni hold! Neat. Only
the reverse direction is given in Takeuti/Zaring's exercise (and in
set.mm since 1994, in the older version of reluni, which was based on
the excercise). I wonder if they knew that both directions hold.
(13-Aug-04) fh1 is (one-half of) the very famous Foulis-Holland theorem
of orthomodular lattices, which was proved and published independently
by Foulis and Holland at almost exactly the same time. I'm not sure who
really came first but both are always credited. This proof is identical
in structure to the Quantum Logic Explorer version
http://us2.metamath.org:8888/qlegif/fh1.html which you may wish to
compare it to, and in fact I "borrowed" it from there. You can see why
the Quantum Logic Explorer is simpler to work with for these kinds of
things: we don't need "member of CH" hypotheses, and we don't have to
keep proving operation closure over and over. I put fh1 in the Hilbert
Space Explorer also because we will later need it for deeper Hilbert
space things that the Quantum Logic Explorer axioms can't do (e.g.
involving quantification).
A bit of trivia regarding this theorem: I used this theorem to
win a $2.00 bet with physicist John Baez, who was unaware of it. See
http://groups.google.com/groups?selm=MaVe6.4799%24JG.614310%40news.shore.net
(12-Aug-04) Compare the proofs of un00 and chj00 - they're very similar!
(8-Aug-04) The last half of my talk at the Argonne workshop
http://www-unix.mcs.anl.gov/~mccune/award-2004/ is related to theorem
mdsym (of 2-Aug-04). (As you can see, work on this site continued from
my hotel room thanks to Linux and ssh...:)
(4-Aug-04) Most of today's theorems prove the "theorem" form of the
result rather than the "inference" form of the result, and most steps do
nothing more that manipulate antecedents with propositional calculus.
Their proof displays give us excellent examples of the usefulness of the
little colored numbers.
Look, for example, at div23t. At first it seems to be a formidable,
hopelessly complicated mess. But if you ignore the steps with
red/orange numbers, there are only 5 steps left that have blue numbers.
Now, if you look at only those steps and ignore the rest, you can pretty
much see "in your head" how to connect the equalities to obtain an
informal proof. Doesn't it make more sense this way?
The other way to prove the "theorem" form is to use the weak deduction
theorm dedth and its variants. But as the number of antecedents grows,
so does the size of the proof using the weak deduction theorem.
Depending on the proof, sometimes you'll get a shorter proof proving it
directly, like with today's. Sometimes I would prove it both ways and
pick whichever way was shortest (in terms of size of the compressed
proof in set.mm).
(3-Aug-04) expcllem shortens the proof of reexpclt (closure of natural
number exponentiation of reals) and will later be reused for integers,
rationals, etc.
(2-Aug-04) mdsym (M-symmetry in Hilbert space) is the final goal of
the recent Hilbert space work. I will be giving a talk on this
remarkable theorem Aug. 5-7 at the Automated Reasoning workshop
http://www.mcs.anl.gov/~mccune/award-2004/ and I can now claim that I
understand its proof in complete detail.
(1-Aug-04) halfnz is a curiously long proof for such a simple fact.
If anyone sees a way to shorten it let me know. On the other hand
it may illustrate the implicit complexity underlying even trivial
arithmetic facts (even if we start with the axioms for arithmetic).
(31-Jul-04) 2cn is a long-overdue theorem that shortens the proof of the
venerable 2p2e4 (2+2=4) as well as a couple of dozen other theorems.
(28-Jul-04) oncardid and oncardon are used to reprove cardnn so that it
doesn't use the Axiom of Choice. I didn't like that overkill for
something as basic as cardnn. I also reproved cardom so it also doesn't
need the Axiom of Choice.
(26-Jul-04) dom2d will help us prove some equinumerosity dominance
theorems more easily, i.e. without directly exhibiting the 1-1 function
required by brdom.
(25-Jul-04) nthruz is the 4000th theorem added to the non-Hilbert-space
part of set.mm. (Hilbert space adds another 600 or so.) I'll have to
update the summaries that say the Metamath Proof Explorer has "over 3000
theorems" to say "over 4000 theorems", once the 4001st is added. The
date stamp was added to the metamath program on 5-Aug-93, which is the
earliest date in set.mm, at which point there were 450 theorems covering
propositional calculus, predicate calculus, and the beginnings of set
theory (subsets, union, intersection, abstraction classes, unordered and
ordered pairs).
(24-Jul-04) nn0enom provides an important link between between two very
different theories - complex numbers (nonnegative integer subset) and
ordinal numbers (finite subset). With it, we can exploit our many
complex number theorems to get some equinumerosity results that will
ultimately be used for cardinal arithmetic. I guess it's kind of
indirect and ugly but it will save a lot of work. (And formally it's
absolutely rigorous of course.) Actually we'll need the reals anyway to
study the continuum so maybe it's not that ugly.
(23-Jul-04) bnd2, like bnd, is one of those "obvious" things that is
actually quite deep. The proof (if you trace it back) ties together
key results about ranks, the Axiom of Regularity, and the Axiom of
Infinity.
It was quite a journey to get atom1d - whereas Beltrametti/Cassinelli
merely say "The one-dimensional subspaces of H are obviously the atoms
of C(H)." I guess this is the difference between formal and informal
math.
(22-Jul-04) ranksn, rankuni, rankuniss are some interesting little facts
about rank. Most textbooks seem to have them, so why not. (But we
don't have a use for them yet.)
(21-Jul-04) I am torn about the notation "Mod" for modular pair. Maeda
uses (A,B)M for our A Mod B, but a plain "M" just seems too unspecific
in a general set theory environment. On the other hand "Mod" suggests
the arithmetic modulo, but I'll use "mod" (lower case) for that when we
start to need that. But I still may change A Mod B to something else
after I think about it - it is nonstandard and I dislike nonstandard
notation in set.mm unless it is unavoidable.
(18-Jul-04 - 19-Jul-04) isfinite2 does not require the Axiom of Infinity
for its proof, whereas the stronger isfinite does require it.
(14-Jul-04) unbnn will eventually give us a proof that a set strictly
dominated by omega is finite, without invoking the axiom of infinity.
(This is the hard part of the proof.) See Suppes Th. 42 p. 151 for more
information. unbnn is also a useful general result in its own right; for
example it will become useful for proving certain cofinality results.
(12-Jul-04 - 13-Jul-04) Cardinal exponentiation, when we define it in
the future, will be the cardinality of set exponentiation. mapen and
xpmapen provide two fundamental theorems used to establish properties of
cardinal exponentiation. (xpmapen was tough to formalize - ugh - I
wonder if there is an easier way to do it.)
(11-Jul-04) canth3 gives us an unlimited supply of ever-bigger infinite
cardinal numbers.
(10-Jul-04) cardnn gives us a very convenient property of finite
cardinal numbers: the natural numbers and the finite cardinal numbers
are the same thing! The alternate definition of cardinals given by
karden (using the Axiom of Regularity instead of the Axiom of Choice)
does not have this property, which is one of their disadvantages. Once
I worked out the first few finite cardinals from karden (it is an
exercise in Enderton) and they become very messy and complicated very
quickly.
(9-Jul-04) sucprcreg is probably not useful for anything, but it was
fun. (We already have sucprc that sometimes helps prove more "general"
theorems where A does not have to be a set. And I like to avoid
theorems requiring the Axiom of Regularity whenever possible.)
(8-Jul-04) aleph1 is an important theorem piecing together some recent
results (alephnbtwn2, canth2, pw2en). The cardinality of reals is
2^aleph0 (which hopefully we will prove here someday), and in the
absence of the Continuum Hypothesis this theorem is the best we can do -
the reals are at least as numerous as aleph-one, but possibly more so.
(7-Jul-04) I had been wanting to prove spansncv for a long time but was
unable to (it is an exercise in Kalmbach with no answer given). Prof.
Eric Schechter (http://www.math.vanderbilt.edu/~schectex/) finally
provided me with the missing piece, which is the theorem spansnj.
(6-Jul-04) The proof of pw2en seems much longer than it should be.
I'll have to revisit it someday.
(1-Jul-04) ssext is interesting because A and B may be proper classes,
yet we need to compare only their (non-proper-class) subsets to
determine whether they are equal.
nssinpss is one of those neat little "it ought to be a theorem or
example or exercise in a book" things, but apparently it isn't. (Any of
our theorems that omits a bibliographical reference doesn't appear in
any book that I'm aware of. If you find one that does let me know!)
(30-Jun-04) elsuc vs. sucel - We have many theorems expanding membership
of a class in a defined object, such as elsuc, but hardly any expanding
membership of a defined object in a class. I'm not sure why that is,
but the need just doesn't seem to arise that often. Today I added two,
0el and sucel, and used them to shorten a couple of existing proofs.
These actually seem to be the only such theorems in the database - I
looked through set.mm for others, but I couldn't find any others of the
form "(defined object) in (class variable) <-> ..." whose right-hand
side didn't reference the defined object. On the other hand there are
probably over 100 theorems showing membership _in_ defined objects, like
elsuc. Strange. By the way there is no "membership in" version of 0el
in our database but rather we use the simpler noel.
(29-Jun-04) I used to be in awe of the fact that Cantor's theorem failed
in NF. With ncanth it now seems kind of obvious why it would fail,
because V is a set in NF. (The actual proof in NF is somewhat different
since the axioms are different, but the idea is the same.) Anyway
ncanth takes away some of the mystery, which is good, but unfortunately
it also means I'm not as awed as I used to be. I guess a little mystery
can sometimes make things more appealing.
(28-Jun-04) xpsnen shows another use of our friend en2 of 25-Jun. The
proof makes essential use of the interesting op1st that extracts the
first member of an order pair.
eqssd was used to shorten 11 proofs that previously used eqss, jca, and
sylibrd. The number of bytes trimmed off of their compressed proofs
exceeded the number of bytes in eqss, its comment, and its proof, so the
database size SHRUNK when eqssd was ADDED! I guess that's what you'd
call a positive ROI. :)
spanun and spansn are two important theorems of Hilbert space. spanun
allows us to work in set theory to take the union of two sets of
vectors, then take the span to get their subspace sum. This can be a
very powerful simplification over working directly with subspace sum.
spansn equates two very different ways of expressing a one-dimensional
subspace. (Familiar example: if the Hilbert space is the ordinary
3-dimensional RxRxR of real numbers, a one-dimensional subspace is a
line that goes through the origin. Any non-zero vector A on the line
essentially specifies the line. To get the line we take the span of the
singleton of A. Alternately, we take the orthocomplement of the
singleton of A, which becomes the plane (through the origin)
perpendicular to A. Then the second orthocomplement recovers the line
from that plane. By the way there are 4 kinds of objects in the Hilbert
lattice of R^3: the point at the origin, lines, planes, and the
entirety of R^3. These are such different things, like apples and
oranges, that it seems counterintuitive that they could all be "like"
each other in the sense of being the elements of a lattice. But it is
often useful to think of the R^3 example when trying to understand
Hilbert lattice theorems intuitively.)
(27-Jun-04) Normally rank is defined in terms of R1 (the cumulative
hierarchy of sets) - see df-rank. Monk takes the opposite approach - he
defines rank first, then defines R1. Here we prove, with r1val2, Monk's
definition as a theorem of our approach. R1 and rank are in a sense
equivalent concepts, although very different in their approach and uses,
analogous to say the time domain vs. frequency domain representations of
a signal in Fourier analysis: either representation can be recovered
from the other. And each representation can have advantages over the
other depending on the problem.
dfom2 is a traditional way of defining the set omega of natural numbers
(as a subset of the ordinal numbers, not the natural number subset of
the reals) that is used in many textbooks. However, dfom2 requires the
Axiom of Infinity in order to be valid, whereas our df-om does not. (If
you look under the proof of dfom2, you'll see that ax-inf is used, and
in fact it is needed to ensure that at least one set x in the class
abstraction exists.) This means that in these books, all theorems about
natural numbers presuppose the the Axiom of Infinity. The reason they
do this is that it makes their development easier. Our philosophy is to
avoid the Axiom of Infinity when it is not necessary, so our development
starting with df-om is quite a bit more complex than in these books.
(Takeuti/Zaring's book takes our approach. However they take a shortcut
by using the Axiom of Regularity, which is still undesireable. By
carefully reworking their proofs, I was able to avoid both Infinity and
Regularity in our development of natural numbers. For example, if you
look at say nnacom you'll see infinity is not used. T/Z also give a
somewhat confusing definition of omega. Our equivalent but simpler
definition df-om doesn't seem to occur in the literature, although it is
related to Bell/Machover's natural number predicate.)
(26-Jun-04) map1 is our first theorem using en2. See how easy the proof
is - no messy one-to-one onto function stuff! (There is a function that
comes from elmap, but that has to do with the definition of set
exponentiation, and we quickly get rid of it with fconst2.)
In Hilbert space I'm building up a collection of basic theorems that
will be needed to develop the theory of atoms.
(25-Jun-04) I think en2 is quite neat - it is not immediately obvious
(to me at least) that you can conclude equinumerosity from the
innocent-looking hypotheses. But look at its simple, beautiful proof:
the function stuff magically appears then vanishes! I don't think en2
is published anywhere. It should let us arrive at some equinumerosity
proofs much more quickly.
(24-Jun-04) omsmo is one of those "intuitively obvious" or "trivial"
facts that textbooks never bother to prove. Unfortunately for us we
must actually prove it to use it since Metamath will not let us overlook
such minor details. :)
(20-Jun-04) I haven't had any luck shortening zind. I am tired of
working on it and will put it aside for a while.
Today's theorems finish up the list of single-digit operations, where
the result is a single digit. As you probably have noticed we don't yet
have a standard way of expressing multiple digit numbers like they are
normally written down. I used to have a definition for it but it was
clumsy and I took it out. The interesting thing is that in abstract
math there is rarely a need for numbers larger than say 4 (which is the
largest number I have used so far outside of the list that we completed
today). Glancing through the proof of Fermat's Last Theorem - one of
the most difficult proofs ever - I don't think I saw a number larger
than 12. So I'll probably postpone multiple-digit numbers until it is
really needed. And even so we can represent them with expressions of
single-digit numbers; for example 42 can be expressed as 6 times 7. Now
that is an interesing problem: find the shortest representation of all
numbers from say 1 to 1000 in terms of operations on single digits
(addition, subtraction, multiplication, division, and exponentiation).
Any takers?
Raph will have multiple-digit numbers in Ghilbert.
(18-Jun-04) At first zind looked trivial. After all, we already have
induction on natural numbers (ind at step 290), and this theorem is just
"shifting" it to start at a different number. But as I got deeper into
it more and more unpleasant surprises kept popping up. After hours of
tedious work I ended up with one of the biggest and ugliest proofs in
set.mm that I see no way to simplify, and there are no "neat" lemmas
that stand out (if it were broken up it would only be for the purpose of
breaking it up and would not make the proof more efficient). It doesn't
make any sense why it should be so huge and it makes me depressed. I'll
have to look at it again after I recover.:)
(17-Jun-04) See isomin for an explanation of Takeuti and Zaring's
initial segment idiom in isoini. This theorem, though cryptic, is
copied symbol for symbol from T&Z.
oprprc1 is philosophically ugly but will allow us to "cheat" and shorten
a few proofs in the future, by not having to worry so much about whether
a class is proper or not.
(16-Jun-04) erthi is the "key" theorem involved in the proof of erdisj.
(14-Jun-04) zrevaddclt basically says, if B is an integer and A is a
complex number and A+B is an integer, then A is an integer. I think
reverse closure laws are interesting twists on closure laws.
(13-Jun-04) isotr and isotrALT are two different proofs of the same
thing, because I wanted to see which approach is shorter. Well, one is
shorter in set.mm and the other produces a shorter web page. So which
do I pick?
f1owe uses the new isomorphism stuff to dramatically simplify its proof.
Compare the old version (that will eventually be deleted), f1oweOBS.
(12-Jun-04) The definitions df-un and df-in have been changed to
be more traditional. What used to be df-un is now dfun2 and vice
versa; what used to be df-in is now dfin2 and vice versa. The price
we pay is a dummy variable, but I finally decided that dfun2 and dfin2
are just too unconventional.
The proof of isowe was left as an exercise for the reader in Takeuti and
Zaring, and thankfully I was able to figure it out! It will provide a
great reduction in size of the enormous proof for f1owe - stay tuned...
The new definition df-cv for covering allows us to have a much simpler
definition for atoms, and df-at is the new version of the defintion.
Theorems ch0psst, atcv0, and atelt are supporting theorems for the new
df-at.
(11-Jun-04) bitr2d, psseq2i, psseq2d shorten some existing proofs.
znegclt, zaddclt, zsubclt will be needed when we get into proofs about
integers. cvbr, cvbr2 express the basic property of the new df-cv
(covers) definition. There is much literature on the properties related
to covering in Hilbert space, and these properties are ultimately tied
to the superposition of states (Schroedinger's cat, etc.)
(10-Jun-04) funimass2 arose in a proof in a to-be-published logic book
I'm reviewing. I thought it was very neat that the domain and range of
F are completely irrevelevant to the result, nor does F have to be
one-to-one or onto. funimass1 is similar but depends on the range of F.
(9-Jun-04) elnnz1 could be used as a definition for natural numbers if
we started with integers. nn0subt is a kind of closure law for
subtraction of nonnegative integers. It is kind of surprising that its
proof is so long, given that we already have the almost-identical
nnsubt, but I couldn't find a simpler proof. An example of shintclt is
that the intersection of two planes (linear subspaces) is a line (a
linear subspace). spanclt shows that the span function "grows" a
arbitrary set of vectors into (the smallest) linear space containing
them.
(8-Jun-04) anadis, anandirs shorten a bunch of proofs previously using
anandi, anandir plus sylbir. By the way the "s" suffix on anandis,
anandirs means "eliminates a syllogism". ssel2 is a trivial variant of
ssel but it is needed so often that I decided to put it in - it makes a
bunch of proofs slightly shorter. The proof of isofr was left as an
exercise for the reader in Takeuti and Zaring - I always worry about
these because figuring out the answers to exercises is often the
hardest part of doing these proofs, and if I can't do it the whole
project comes to a standstill because there can't be any missing
pieces the database. hvm1negt, hvsubcan1t, hvsubcan2t - some
more elementary Hilbert space elementary results; by the way the suffix
"t" means "theorem form" (vs. "inference form" where the membership
requirements are hypotheses).
(7-Jun-04) If you think isomin is incomprehensibly cryptic, don't blame
me. :) It is stated exactly as written in Takeuti and Zaring's book,
symbol for symbol. I did add a note explaining the initial segment
idiom.
I was surprised there's never been a need for something as basic
as unss12 yet. funimacnv tells us that if A is a subset of the
range, the image of the converse image of A is A - neat and in a
sense intuitive, but its proof took some thought... elznn0
and elnn0z express some more number class relationships.
(3-Jun-04) df-span is the same as the span of a vector space you learned
about in linear algebra. It does not require any special properties of
Hilbert space.
(2-Jun-04) onxpdisj may ultimately find use in extending the set of
complex numbers with +infinity, -infinity (for reals) that is useful in
analysis. Basically, in our definition df-c (and most textbook
definitions) complex numbers are ordered pairs, so per onxpdisj any
ordinal provides a disjoint set that could extend C with artificial
things like +infinity.
(1-Jun-04) sqr9 extends the series sqr0, sqr1, sqr4.
r1pwcl is a new version of the May 30 version that now has Lim B
as an antecedent (instead of a hypothesis).
(31-May-04) h1dle simplifies the proof of hatomic and should be useful
for other things too. Note: "atom" has nothing to do with physical
atoms but is just a mathematical term describing a smallest non-zero
element of a lattice, where there is nothing smaller than it except the
zero element. In the lattice of closed subspaces of Hilbert space, all
atoms are one-dimensional subspaces (I hope to show the proof of this
soon). To get a one-dimensional subspace, we start with a singleton
containing any non-zero vector, then take the double orthogonal
complement to grow the singleton to a closed subspace (which in this
case would just be the set of all vectors that differ by a scalar
factor, i.e. a one-dimensional "line"). I haven't seen the idiom
_|_ _|_ {B} for a one-dimensional subspace in the literature, but it is
convenient. (The antecedent of h1dle doesn't require that B be nonzero,
but that's because the theorem also works when B is zero - although
_|_ _|_ {B} is not a 1-dimensional subspace in that case but instead the
zero subspace.)
(30-May-04) Raph contributed 3 really nice results - rankr1a, r1pw,
and r1pwcl - on rank and the cumulative hierarchy of sets (R1). I
haven't seen any of them in a textbook.
(28-May-04) exp0 and expp1 are the final goals of exponentiation of
complex numbers to nonnegative integer powers. These effectively
provide the recursive definition found in most textbooks (which
typically do not go through the tedious work of justifying the
recursion). We could have started with exp0 and expp1 instead of the
direct definition df-exp, but then we would have had to introduce expp1
as an axiom because it is self-referential. Instead, we want to ensure
direct traceability back to ZF set theory axioms. From this point
forward we will develop all further properties from exp0 and expp1 only,
never referring back to the development leading up to them (and never
referring directly to df-exp again). The work we did to develop
recursion on nonnegative integers - nn0rzer and nn0rsuc - will be
reusuable in the future for other recursive definitions.
It turns out that nn0rone and nn0rfnnn0 are not needed. In particular,
we can prove closure of exponentiation from exp0 and expp1 using
induction, eliminating the need for nn0rfnnn0. However I'll leave them
in because they may useful in future applications.
(24-May-04) Raph Levien developed the definition of exponentiation of
complex numbers to nonnegative integer powers in Ghilbert and translated
it to Metamath. (He has a Ghilbert to Metamath translator now.) I am
adapting it for the official set.mm. It will replace the current
"dummy" definition df-expOBS that only works for powers of 2.
(18-May-04) None of these function-related theorems appear in any
textbook I'm aware of (if there is no bibliographical reference), but
most will be needed later to prove some theorems (about cardinality
etc.) that do appear in textbooks. I guess that once set theory
advances beyond a certain point, these kinds of theorems are expected to
be "obvious" to the reader and are not explicitly mentioned. Some of
them might make nice homework exercises for a set theory class. I
wonder if any teacher has ever borrowed from our collection for that
purpose.
(15-May-04) I've often thought Takeuti/Zaring's initial segment notation
in today's eliniseg, iniseg, dffr3 is a rather cryptic and convoluted
idiom just to avoid the dummy variable in {y|y show labels *a4*
The assertions that match are shown with statement number, label, and type.
1971 a4i $p 1979 a4s $p 1983 a4sd $p 2402 a4a $p
2407 a4c $p 2413 a4c1 $p 2533 sbea4 $p 2534 sbia4 $p
2535 sbba4 $p 2643 a4b $p 2648 a4b1 $p 2653 a4w $p
2659 a4w1 $p 3905 ra4 $p 3906 ra4e $p 3907 ra42 $p
4642 cla4gf $p 4643 cla4egf $p 4649 cla4gv $p 4650 cla4egv $p
4657 cla4e2gv $p 4658 cla42gv $p 4665 cla4v $p 4666 cla4ev $p
4673 rcla4v $p 4674 rcla4ev $p 4686 rcla42v $p 4687 rcla42ev $p
4696 cla4e2v $p 4891 a4sbc $p
A few people have commented that the theorem labels are obscure, and in
general I'm not thrilled with them either. Every now and then I revise
a few for better uniformity. But no one has been able to suggest a
better approach that still keeps the names short. For example, I use
"sbth" instead of "SchroederBernsteinTheorem" because the latter would
be annoying to type while entering a proof, and lists like the above
would not be very compact. In general, an "English-like"
descriptive label would be have to be very long to be meaningful - how
would you label cla42gv? I wonder if a breakdown for each label like
the "cla42gv" example above would be useful enough to be worth the
effort of maintaining it. Maybe this could be done with an annotated
label in each description, something like:
cl+a4+2:4+g+v
where cl, a4, etc. are looked up in a list and 2:4 refers to the
4th meaning of "2" in that list.
(10-May-04 - 12-May-04) Some miscellaneous stuff and simple arithmetic
facts we'll need later on. A bold attempt to venture beyond 2+2=4...
Someday of course we'll have to complete the elementary school addition
and multiplication tables. (At least Raph Levien will do this for his
Ghilbert database, and has ideas for decimal numbers - I used to have
them but took them out because I thought the notation was ugly and I
wasn't happy with it.) However it is interesting that "higher math"
rarely uses numbers beyond 2. For example scanning the proof of Fermat's
last theorem http://math.stanford.edu/~lekheng/flt/wiles.pdf the largest
number I see for most of the proof is 4 (although the last paragraph
mentions some larger numbers, but this may be material beyond FLT, I'm
not sure).
(9-May-04) exists2 is another one from the existential uniqueness list I
made several years ago. I think it is interesting because it shows a
way to "talk" about multiple objects without using set theory.
(7-May-04) rdglem1 was a final loose end that was cleaned up. zfreg* are
cleanups of older versions that didn't use restricted quantification.
Other stuff you don't see behind the scenes: over 2 dozen theorems were
deleted. These were obsolete versions have been completely eliminated
from all proofs using them, and there are now no *OBS theorems in
set.mm.
zfregs (the strong version) is curious because the Axiom of Infinity
is apparently needed for its proof (at step 5 of tz9.1 used in its
proof) yet there is no obvious component of Infinity in the result. No
one, including some prominent mathematicians, has been able to tell me
why or even whether Infinity is _required_ for a proof of this theorem.
All textbook proofs I've seen just implicitly use Infinity (in the form
of a finite recursion) at this step without further comment. In other
proofs I've been able to avoid the Axiom of Infinity (vs. textbook
proofs) by using transfinite recursion instead - interestingly
transfinite recursion does not require the Axiom of Infinity, whereas
finite recursion (the textbook version) does. This is because the
latter takes a shortcut by just showing the existence of the necessary
function over omega, without showing what it looks like, whereas the
former (with far more difficulty) must state explicitly the required
function over ordinals because it is proper class and does not exist as
a set. In set.mm we don't take these shortcuts, and in most cases we
use finite recursion (frfnom, frzer, frsuc) without invoking the Axiom
of Infinity.
(6-May-04) ackm completes the proof of Maes' axiom by establishing
the final link to our Axiom of Choice ax-ac.
(4-May-04) More work on the new transfinite recursion proof. Done with
all the lemmas now. By the way, when I revise existing proofs, I
usually suffix the old one with OBS (obsolete), so the old tfrlem1
becomes tfrlem1OBS. The OBS version will stay until all references to
it are removed. This way I can ensure that, at any point in time, the
set.mm database as a whole is consistent and complete, even while things
are being modified.
(3-May-04) More work on the new transfinite recursion proof.
(2-May-04) isoid, cflem, and cfval are simple theorems to try out the
new df-iso and df-cf definitions. The proof of transfinite recursion
(tfrlem*) is being redone in order to be slightly shorter.
(1-May-04) fv2, fv3, tz6.12*, etc. are part of a cleanup of function
value stuff to use the more compact xFy (df-br) in place of e. F.
(30-Apr-04) Finally, aceq5.
(28-Apr-04 - 29-Apr-04) We need aceq3 and aceq4 to get the final piece
of aceq5, and they will also fill in part of an AC equivalence series I
have in mind. (The eventual goal here is to get a link from Mae's AC to
ours.)
(27-Apr-04) I figured I might as well complete the equality/subclass
series with eqsstrd, eqsstr3d, etc. These are named for their
resemblance to bitr (simple chaining), bitr3 (useful for eliminating
definitions), and bitr4 (useful for introducing definitions).
limelon is cleaned up from an older version and is needed for some
future stuff related to df-cf.
(26-Apr-04) I cleaned up some function value stuff to improve the proofs
leading to the very useful abrexex, which is one of those simple-looking
theorems that hides a lot of power.
After noticing several times the "need" for syl5ss, syl5ssr, syl6ss,
syl6ssr, I finally added them in. (These, like others in the syl5xx,
syl6xx "family", are named after their resemblance to syl5 and syl6.) I
was able to shorten over 2 dozen proofs with them, so I was curious what
this actually meant in terms of set.mm file size. It turns out that a
total of about 300 bytes were trimmed from the (compressed) shortened
proofs, vs. about 1000 additional bytes for the 4 new theorems. So they
haven't quite "payed" for themselves yet. Perhaps they will when set.mm
is 3 times as big...
(23-Apr-04 - 24-Apr-04) These are some function and function value
theorems that we will need later. And a step closer to aceq5...
(17-Apr-04 - 22-Apr-04) I have a list of uniqueness and "most one"
theorems I worked out several years ago, and I decided to add some of
them to the database. Here is an excerpt from an email I wrote.
Let us consider ax-1 through ax-16 (plus ax-mp, ax-gen). We will ignore
ax-17 (which, external to set.mm, is a metatheorem derivable from the
others).
I mentioned earlier that ax-1 through ax-15 (the $d-free fragment)
cannot prove all $d-free theorems, and indeed by Andreka's theorem a
complete $d-free fragment is impossible. For completeness we need to
add ax-16, which states:
A. x x = y -> ( phi -> A. x phi ) where $d x y
This is the axiom that, one way or another, ultimately allows us to
eliminate dummy variables from a proof. But it has a $d condition.
There is a trivial theorem (exists1 in the current set.mm) that states:
E! x x = x <-> A. x x = y where $d x y
E! means "there exists exactly one." With this we can replace the
antecedent of ax-16 and restate it so that it does not have a $d
condition. E! effectively "hides" the distinct variable y otherwise
needed for ax-16.
Of course E! is a defined connective, and when we eliminate the
definition the distinct variable returns. I would be surprised if, by
treating E! as a primitive connective and adding $d-free axioms for it,
we could exploit its "hiding" feature to defeat Andreka's theorem. But
I don't have a proof that we couldn't defeat it.
In any case I think E! is an interesting connective, and it seems poorly
developed in the literature, usually mentioned briefly in passing and
sometimes not (formally) at all. Maybe it's considered too trivial.
But its theorems are sometimes not that obvious, and in fact E! x E! y
has been mistakenly used to mean "there exists exactly one x and exactly
one y." (Nowhere, to my knowledge, is the correct expression for this
found, except in set.mm).
I have sometimes wondered what a complete axiomatization of E! might
look like under (standard, not set.mm's) predicate calculus without
equality (that would allow one to prove all theorems mentioning E! but
not mentioning =). In set.mm there are many theorems of this sort such
as
E! x E. y phi -> E. y E! x phi
(15-Apr-04) I want to link Maes' AC all the way to our Axiom of Choice
ax-ac, but I'm missing some standard equivalents needed to do that, and
I'll be working on those as I can fit them in. aceq5lem1 is a start at
this. When it is complete, aceq5 will prove:
aceq5 $p |- ( A. x E. f ( f (_ x /\ f Fn dom x ) <->
A. x ( ( A. z e. x -. z = (/) /\
A. z e. x A. w e. x ( -. z = w -> ( z i^i w ) = (/) ) ) ->
E. y A. z e. x E! v v e. ( z i^i y ) ) ) $=
(14-Apr-04) Here, finally, we present Maes shorter version of his
5-quantifier AC. (The longer one has been deleted from the database,
since it's no longer "interesting.") Not only does this version have
only 5 quantifiers (our ax-ac has 7 or 8 if put in prenex form), but if
we expand out the biconditional in our ax-ac, Maes' axiom is the same
length as ours, tying my claim that ax-ac is the shortest possible.
This result refutes a conjecture of Harvey Friedman that a 5-quantifier
AC was not possible (and he thanked me for verifying Maes' result).
(7-Apr-04) Maes discovered a SHORTER version of his 5-quantifier Axiom
of Choice. That means the old one is obsolete, so we'll
scrap it and reprove some lemmas to derive the shorter one.
(6-Apr-04) exists1 is an interesting little observation. If we treat E!
as a primitive connective, we can use exists1 to replace the antecedent
of ax-16. Now, disregarding ax-17 (which is techically redundant), the
only axiom requiring distinct variables is ax-16, so exists1 makes that
requirement go away. Of course we would need to add axioms for E!, and
I'm not sure what they would be.
(5-Apr-04) eldm2, elrn2 shorten several proofs by eliminating the need
for df-br.
(3-Apr-04) Some more r19.* stuff, while we're on a roll...
(2-Apr-04) I thought I'd need these for Maes' theorem - I didn't - but
they still might be handy in the future.
(1-Apr-04) As you can guess, I've been busy formalizing the proof of
Kurt Maes' theorem. It's now done! AC with only 5 quantifiers in
prenex normal form is a new record and has not been published yet. The
Metamath proof now confirms with absolute certainty that there is not a
mistake in Kurt Maes' proof.
References:
http://www.cs.nyu.edu/pipermail/fom/2003-November/007653.html
http://www.cs.nyu.edu/pipermail/fom/2003-November/007690.html
(25-Mar-04) Miscellaneous stuff.
(25-Mar-04) 19.28v shortens some proofs that used to use 19.28.
uni0 revises an earlier version. funfvima3 might be useful in the
future to help prove Axiom of Choice equivalents.
(24-Mar-04) inf5 - Now, is this neat, or what?
(23-Mar-04) unidif0 and yesterday's disj4 will find use in a neat
theorem tomorrow - check back!
(22-Mar-04) mprgbir, even though kind of specialized, found use in
shortening 6 proofs. I haven't seen disj4 in books, but it is kind of
interesting - in particular it implies the right-hand side is
symmetrical when A and B are swapped.
(17-Mar-04 - 21-Mar-04) More odds and ends for future use.
(13-Mar-04 - 16-Mar-04) I thought dmuni is kind of interesting; I'd
never seen it before I glanced at it in Enderton so I put it in;
don't know if it has any use yet. In addition to residm, resabs1,2,
we also have rescom (commutative) law proved earlier. elimasn
is often used implicitly by Takeuti/Zaring to represent minimal
elements - I don't think it is intuitive but it does save a bound
variable the way they use it e.g. in their Def. 6.21 p. 30.
(12-Mar-04) hta is a nice theorem - it tells us the closest we can get
to Hilbert's epsilon in ZFC, and provides a way to eliminate it from
proofs. For a discussion see http://ghilbert.org/choice.txt
(11-Mar-04) Theorem schemes can be represented equivalently with class
variables and wff variables in set theory. scottexs and scott0s provide
excellent examples of converting theorems with class variables
(scottex, scott0) to theorem schemes with wff variables.
(10-Mar-04) A surprising result communicated by Gregory Bush. If "B"
represents df-bi, then in D-notation the proof reads:
DDD3DD2D1D3D1B11B. George wrote: 'It's strange for two reasons:
first, it's radically more direct than the listed proof (17 steps
instead of 859). Second, it doesn't matter what axiom (or definition)
goes in the second-to-last place. When I first found this, it was
actually coming up with df-or instead of ax-1 in that spot, even though
"or" never appears in the expression, which led me through hours of
debugging.'
(9-Mar-04) Strengthen some class substitution theorems. elabs
(which used to be called sbc7) now does not have any distinct variable
restrictions.
(8-Mar-04) These will be needed to work with abstraction classes
substituted for class variables.
(7-Mar-04) heplem is something I'm working on with Raph Levien and needs
to be cleaned up at some point. The second hypothesis is not very
intuitive. Basically I took a shortcut and reused zornlem1 to get this
result fast, in order to prove that it could be done, but the result is
a very complicated Hilbert epsilon formula (although mathematically it's
perfectly sound). Raph indicated he would post some stuff on Hilbert
epsilons soon at ghilbert.org.
(6-Mar-04) The old transfinite induction theorems were converted so that
they now use restricted quantification.
(5-Mar-04) find updates an older version of finite induction by using
restricted quantifiers. dmeqi, dmeqd, rneqi, and rneqd shorten a bunch
of proofs by eliminating ax-mp or syl needed for these commonly used
variants of dmeq and rneq. Trivial as they are, last night I shortened
32 proofs with them.
(4-Mar-04) r19.20i2 will help us in some future proofs. zfinf shortens
an older version by using restricted quantification. The proof of omex
was shortened by using r19.20i2, the new version of zfinf, and the new
version of peano5 (proved a couple of days ago). ralxp is kind of neat,
letting us convert from a single quantification over a cross product to
a double quantification; it should be useful for some future theorems
involving operation values.
(3-Mar-04) reuuni, reuunis clean up older versions. dfchj3 is a simpler
definition of closed subspace join in terms of supremum; I might make
it the official definition some day.
(2-Mar-04) peano5 has been restated with restricted quantification and
reproved with a 25% shorter proof (although it is still somewhat long
and difficult). Unlike many textbooks, we prove peano5 without invoking
either the Axiom of Infinity nor the Axiom of Replacement. This makes
the proof longer than those textbook proofs. However, I think it is
philosophically nicer to be able to work with natural numbers without
having to assume that an infinite set exists.
(1-Mar-04) tfis, tfis2f, tfis2 update older versions of these theorems
by using restricted quantifiers. Their proofs are shorter as a result
(30% shorter for tfis), and they will also shorten proofs that use them
as they are phased in.
(29-Feb-04) tfi updates an older version of the same theorem by using a
restricted quantifier.
(28-Feb-04) r1val1 is one of several alternate definitions of R1 in the
literature.
(27-Feb-04) cardaleph not only shows us that there exists an aleph for
every transfinite cardinal, but it gives us an explicit expression for
that aleph! The textbook theorems I've seen only show existence, but I
think it's nicer to have a closed expression for the actual thing. The
proof is quite long and brings together a lot of the cardinal/aleph
results we've proved up to this point. The idea behind the proof is to
show it separately for 0, successor, and limit alephs. Miraculously,
each of the three cases evaluate to the exact same expression. Then we
combine the 3 cases with jaod at the end, then eliminate the triple OR
with ordzsl.
(26-Feb-04) alephord2i finally completes all the aleph ordering stuff we
should need. alephle may seem "obvious" - after all, 0 is certainly less
than aleph`0 (countable infinity), 1 is less than aleph`1 (first
uncountable infinity), and so on. In fact you'd almost think that it
should be "less than" instead of "less than or equal to". Well, think
again - it turns out that A=aleph`A when A is big enough! I think
that's amazing, and hopefully we'll prove it eventually.
(25-Feb-04) hbeleq nicely shortens proofs when it can used in place of
hbeq - I don't know why I didn't think of it before! I like this little
theorem. I shortened 12 proofs with it as you can see in the
"referenced by" list. Exercise: Does hbeleq hold if y and A are not
distinct?
(24-Feb-04) These are some boring utility theorems that will come in
handy later and also shorten some existing proofs. fnopab2 has already
"paid" for itself (meaning it shortens enough proofs so that the net
effect of adding it to the database is to reduce the database size)
(look at its "referenced by" list).
(23-Feb-04) elxp2 "modernizes" elxp with restricted quantification.
I think elxp4 is really cool - no dummy variables - it exploits
op1sta and op2nda we proved on 17-Feb.
(22-Feb-04) r19.22dv2, r19.26m, and ralcom give us some tools to use
when two different quantifier bounds are involved. This will shorten
some future and existing proofs. eusn and euuni are revisions of
earlier versions of these. Someday I might introduce an iota, but
so far it wouldn't be used much. I think Raph uses iota a lot in
his Ghilbert HOL stuff. There is a person who _may_ have volunteered
to write a gh2mm translator; if that happens we'll have a lot of
new stuff to put here.
(21-Feb-04) onuninsuc, orduninsuc, nnsuc restrict the existential
quantifier of older versions to strengthen them.
(20-Feb-04) nndiv may become the basis for a new defined relation x|y or
"x divides y" for number theory. divcan2t and divcan3t are needed for
nndiv. divcan2t and divcan3t actually were already proven as divcan2z
and divcan3z; we just use the weak deduction theorem to get the "t"
versions. It would have been nicer to prove them directly but we would
need divmult and divasst, and we only have divmulz and divassz.
Maybe someday I'll prove divmult and divasst, then I can shorten the
proofs of divcan2t and divcan3t. The weak deduction theorem produces
horrible-looking proofs but it is a quick and dirty way to get stronger
theorems.
(19-Feb-04) Added some elementary theorems for the ordinal number 2.
(18-Feb-04) disj is a version of disj1 with restricted quantification
that will shorten some proofs. wefrc, tz7.5 are new versions of older
theorems with restricted quantification and shorter proofs. alephgeom
shows all alephs are infinite; the converse arises because (with our
definition) a function's value outside its domain is the empty set per
ndmfv.
(17-Feb-04) r19.23aivv lets us shorten several proofs. dffr2, frc,
dfepfr, and epfrc are new versions of older theorems that have been
shortened with restricted quantifiers and re-proved with shorter proofs.
op1sta is from Raph's HOL Ghilbert stuff, where he calls it fst, and
op2nda is a modified version of his snd.
(16-Feb-04) birex2i, r19.21ad, cleqrabi are simple but we'll use them
later. fvres is a revision of an older version. dmsn0 completes the
series: dm0, dmsn0, dmsnsn0. Curiously dom 0 = dom {0} = dom{{0}} = 0,
but dom {{{0}}} and beyond are not 0. For example dom {{{0}}} = {0}.
(15-Feb-04) Enderton p.222 gives an interesting discussion of kard vs.
card. One thing I don't like about kard (that Enderton doesn't mention)
is that kard A is not always equinumerous to A, a property that card has
by cardid. For example, kard of 0, 1, and 2 are the singletons {0},
{1}, and {2}, which are all equinumerous to 1 by ensn1. (kard 3 is not
{3} but a messy expression, and they get messier as n grows.)
(14-Feb-04) We prove kardex using Scott's trick. This is a little
different than the proof suggested in Enderton.
(13-Feb-04) I've never seen alephsuc2 in a book, but it looks like a
nice way to define the aleph function, and it may be the shortest
possible. It exploits yesterday's cardval2.
(12-Feb-04) alephord3 is yet another ordering theorem. (In the
end we will need all of these.) I think cardval2 is kind of nice; I
came across it in Suppes only after I started, and maybe one day I'll
change df-card. Suppes is the only place I've seen it. The equivalence
of cardval and cardval2 is a very long journey and uses the Axiom of
Choice.
(11-Feb-04) alephord2 finally gives us a real textbook theorem (and its
converse as well). alephsucdom is a consequence of the ordering stuff;
actually one of the earlier lemmas, alephordlem1, is more convenient to
prove it, and alephnbtwn2 also gets used.
(10-Feb-04) alephord generalizes alephordi to include the converse. I
decided not to combine alephord and alephordi because the latter has one
less antecedent, making it slightly easier to use. It is also
referenced twice in the alephord proof, so combining would cause its
proof to be redundantly expanded twice.
(9-Feb-04) alephordi is a key theorem essential for further aleph
results. The proof provides a nice example of applying transfinite
induction. This version (using strict dominance) doesn't seem to appear
in textbooks but later we will derive the textbook version that uses
cardinal ordering.
(8-Feb-04) alephnbtwn2 is the equinumerosity version of alephnbtwn.
(7-Feb-04) sneqi,sneqd are trivial but convenient and will shorten some
proofs. ordsucun will be needed later to compute the rank of an
unordered pair.
(6-Feb-04) fvreseq shortens an earlier version with wral
notation. inv and unv are trivial and probably useless things to
do when feeling lazy...
(5-Feb-04) sbcel1 and sbcel2 complete class substitution into a
membership relation; equality is defined in terms of membership so now
we can do that too in principle. I can't think of a use for dmco2 yet
but I thought it was neat and couldn't resist proving it. It wasn't
immediately obvious (to me at least) but the proof is relatively simple.
(4-Feb-04) With sbci, sbcn, and sbcal we now have the basic tools to
move class substitution in and out of any wff, since all other wff
connectives are defined in terms of implication, negation, and
universal quantification.
(3-Feb-04) rabid is trivial but it can simplify some proofs
such as nnwos. sbcn gets a basic class substitution property out of the
way for future use.
(2-Feb-04) hbf is needed for fopab2. fvopab2 is a more general form
of an earlier version that had separate hypotheses. fopab2 is a
generalization of an earlier version that only proved the forward
direction of the conclusion; it is expected to be a nice tool for some
future equinumerosity theorems.
(1-Feb-04) dmopabss is a simple result that previously was proved
explicitly in a couple of other proofs, and it simplifies those proofs.
dmopab2 is a powerful equivalence that is expected to be useful in the
future and has already been used to simplify some proofs; previously
only the forward direction was proved.
(31-Jan-04) Got a bunch of bound-variable hypothesis builders out of
the way. Actually they all chain together to help prove fnopabg, which
replaces an earlier version with a more powerful bidirectional
conclusion. dmsn is kind of curious - it holds with *all*
of its class existence hypotheses eliminated, although the proof of this
is somewhat tedious; but it shortens proofs that use it, over an
earlier version that required that the classes exist.
(30-Jan-04) Eliminated the requirement that C be a set in elmap.
fnopabfv is now a bidirectional result, making it more powerful. dmsnsn0 is
a cute technical result that will be used in the future. Curiously dom
0 = dom {0} = dom{{0}} = 0, but dom {{{0}}} and beyond are not 0.
(29-Jan-04) sb7 should make some logicians happy who don't like
the x both free and bound in df-sb. However I still think df-sb is
neat because it doesn't have any distinct variable requirements.
(28-Jan-04) Self-explanatory.
(27-Jan-04) I think sbidm is nice but I don't know if it will ever
be useful. I simplified Raph's proof of nfunv, but later he wrote about
my version, "Neat. I still kinda like my proof, because it's portable
to second-order arithmetic with Cantor pairing (in which the universe is
a relation, the complete relation to be specific)." (His Ghilbert HOL
database has his version.)
(26-Jan-04) cleq2tr is on the one hand trivial, on the other hand to
me it seemed non-intuitive at first. As an exercise, try to convince
yourself of it informally.
(25-Jan-04) The very basic cleqtr had never been proved in the
database before, and suprisingly it seems to have little use. Only one
proof was shortened by it. alephnbtwn, on the other hand, is a pretty
deep and basic "must have" fact about alephs that will be of much use in
the future.
(24-Jan-04) opnz is an easy theorem I thought should be there although
I don't know if it will ever be useful. iunrab gets a basic fact of
indexed unions out of the way for future use.
Copyright terms for this file: Public domain (see
http://us.metamath.org/copyright.html#pd).