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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ottposg 5901 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
tpos | ||
Theorem | dmtpos 5902 | The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | rntpos 5903 | The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposexg 5904 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | ovtposg 5905 | The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from ( 1 ... m ) ( 1 ... n ) to the reals or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposfun 5906 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos2 5907* | Alternate definition of tpos when has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos3 5908* | Alternate definition of tpos when has relational domain. Compare df-cnv 4381. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos4 5909* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tpostpos 5910 | Value of the double transposition for a general class . (Contributed by Mario Carneiro, 16-Sep-2015.) |
tpos tpos | ||
Theorem | tpostpos2 5911 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
tpos tpos | ||
Theorem | tposfn2 5912 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfo2 5913 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf2 5914 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf12 5915 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf1o2 5916 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfo 5917 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf 5918 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfn 5919 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tpos0 5920 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposco 5921 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos tpos | ||
Theorem | tpossym 5922* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tposeqi 5923 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposex 5924 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | nftpos 5925 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposoprab 5926* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposmpt2 5927* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | pwuninel2 5928 | The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | 2pwuninelg 5929 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) |
Theorem | iunon 5930* | The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.) |
Syntax | wsmo 5931 | Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals. |
Definition | df-smo 5932* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | dfsmo2 5933* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Theorem | issmo 5934* | Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | issmo2 5935* | Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smoeq 5936 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smodm 5937 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smores 5938 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Theorem | smores3 5939 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Theorem | smores2 5940 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Theorem | smodm2 5941 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smofvon2dm 5942 | The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | iordsmo 5943 | The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smo0 5944 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smofvon 5945 | If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smoel 5946 | If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiun 5947* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiso 5948 | If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.) |
Theorem | smoel2 5949 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Syntax | crecs 5950 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 5951* |
Define a function recs on , the class of ordinal
numbers, by transfinite recursion given a rule which sets the next
value given all values so far. See df-irdg 5988 for more details on why
this definition is desirable. Unlike df-irdg 5988 which restricts the
update rule to use only the previous value, this version allows the
update rule to use all previous values, which is why it is
described
as "strong", although it is actually more primitive. See tfri1d 5980 and
tfri2d 5981 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 5952 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 5953 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 5954* | A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlem3ag 5955* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3a 5956* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3 5957* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3-2 5958* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.) |
Theorem | tfrlem3-2d 5959* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 5960* | Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem5 5961* | Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | recsfval 5962* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 5963* | Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem7 5964* | Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfrlem8 5965* | Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
recs | ||
Theorem | tfrlem9 5966* | Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.) |
recs recs recs | ||
Theorem | tfr2a 5967 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0 5968 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 5969* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 5970* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 5971* | Each element of is an acceptable function. Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 5972* | The union of is defined on all ordinals. Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 5973* | The union of is a function defined on . Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 5974* | The set exists. Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 5975* | The union of satisfies the recursion rule (lemma for tfrlemi1 5977). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 5976* | Lemma for tfrlemi1 5977. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 5977* |
We can define an acceptable function on any ordinal.
As with many of the transfinite recursion theorems, we have a hypothesis that states that is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi14d 5978* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 5979* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 5980* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2d 5981* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5982). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri1 5982* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2 5983* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5982). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri3 5984* | Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5982). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfrex 5985* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfrfun 5986 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
recs | ||
Syntax | crdg 5987 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 5988* |
Define a recursive definition generator on (the class of ordinal
numbers) with characteristic function and initial value .
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our operation (especially when df-recs 5951
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple. In classical logic it would be easier to divide this definition
into cases based on whether the domain of is zero, a successor, or
a limit ordinal. Cases do not (in general) work that way in
intuitionistic logic, so instead we choose a definition which takes the
union of all the results of the characteristic function for ordinals in
the domain of .
This means that this definition has the expected
properties for increasing and continuous ordinal functions, which
include ordinal addition and multiplication.
For finite recursion we also define df-frec 6009 and for suitable characteristic functions df-frec 6009 yields the same result as restricted to , as seen at frecrdg 6023. Note: We introduce with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.) |
recs | ||
Theorem | rdgeq1 5989 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 5990 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 5991 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 5992* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 5993* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 5994* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 5995 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 5996 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 5997 | The recursive definition generator is a function on ordinal numbers. The condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6004; in cases like df-oadd 6036 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 5998* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 5999* | Value of the recursive definition generator. Lemma for rdgival 6000 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 6000* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
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