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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-tpos 5801* | Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposss 5802 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeq 5803 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposeqd 5804 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
tpos tpos | ||
Theorem | tposssxp 5805 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
tpos | ||
Theorem | reltpos 5806 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos2 5807 | Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtpos0 5808 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | reldmtpos 5809 | Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | brtposg 5810 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
tpos | ||
Theorem | ottposg 5811 | The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.) |
tpos | ||
Theorem | dmtpos 5812 | The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | rntpos 5813 | The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposexg 5814 | The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | ovtposg 5815 | 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 5816 | The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos2 5817* | Alternate definition of tpos when has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos3 5818* | Alternate definition of tpos when has relational domain. Compare df-cnv 4296. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | dftpos4 5819* | Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tpostpos 5820 | Value of the double transposition for a general class . (Contributed by Mario Carneiro, 16-Sep-2015.) |
tpos tpos | ||
Theorem | tpostpos2 5821 | Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
tpos tpos | ||
Theorem | tposfn2 5822 | The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfo2 5823 | Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf2 5824 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf12 5825 | Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf1o2 5826 | Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfo 5827 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposf 5828 | The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposfn 5829 | Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tpos0 5830 | Transposition of the empty set. (Contributed by NM, 10-Sep-2015.) |
tpos | ||
Theorem | tposco 5831 | Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos tpos | ||
Theorem | tpossym 5832* | Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
tpos | ||
Theorem | tposeqi 5833 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos tpos | ||
Theorem | tposex 5834 | A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | nftpos 5835 | Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposoprab 5836* | Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | tposmpt2 5837* | Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.) |
tpos | ||
Theorem | pwuninel2 5838 | 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 5839 | 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 5840* | 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 5841 | Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals. |
Definition | df-smo 5842* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | dfsmo2 5843* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Theorem | issmo 5844* | Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | issmo2 5845* | Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smoeq 5846 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smodm 5847 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smores 5848 | 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 5849 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Theorem | smores2 5850 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Theorem | smodm2 5851 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smofvon2dm 5852 | The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | iordsmo 5853 | The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smo0 5854 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smofvon 5855 | 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 5856 | 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 5857* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiso 5858 | 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 5859 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Syntax | crecs 5860 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 5861* |
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 5897 for more details on why
this definition is desirable. Unlike df-irdg 5897 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 5890 and
tfri2d 5891 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 5862 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 5863 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 5864* | 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 5865* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3a 5866* | 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 5867* | 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 5868* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.) |
Theorem | tfrlem3-2d 5869* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 5870* | 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 5871* | 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 5872* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 5873* | 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 5874* | 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 5875* | 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 5876* | 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 5877 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0 5878 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 5879* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 5880* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 5881* | Each element of is an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 5882* | The union of is defined on all ordinals. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 5883* | The union of is a function defined on . Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 5884* | The set exists. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 5885* | The union of satisfies the recursion rule (lemma for tfrlemi1 5887). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 5886* | Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 5887* |
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 5888* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 5889* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 5890* |
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 5891* | 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 5892). 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 5892* |
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 5893* | 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 5892). 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 5894* | 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 5892). 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 5895* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 5896 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 5897* |
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 5861
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 5918 and for suitable characteristic functions df-frec 5918 yields the same result as restricted to , as seen at frecrdg 5931. 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 5898 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 5899 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 5900 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
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