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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtfrlem9 6401* 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

Theoremtfrlem9a 6402* Lemma for transfinite recursion. Without using ax-rep 4131, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
recs recs

Theoremtfrlem10 6403* Lemma for transfinite recursion. We define class by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, . Using this assumption we will prove facts about that will lead to a contradiction in tfrlem14 6407, thus showing the domain of recs does in fact equal . Here we show (under the false assumption) that is a function extending the domain of recs by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem11 6404* Lemma for transfinite recursion. Compute the value of . (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem12 6405* Lemma for transfinite recursion. Show is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs

Theoremtfrlem13 6406* Lemma for transfinite recursion. If recs is a set function, then is acceptable, and thus a subset of recs, but is bigger than recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
recs

Theoremtfrlem14 6407* Lemma for transfinite recursion. Assuming ax-rep 4131, recs recs , so since recs is an ordinal, it must be equal to . (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem15 6408* Lemma for transfinite recursion. Without assuming ax-rep 4131, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
recs recs

Theoremtfrlem16 6409* Lemma for finite recursion. Without assuming ax-rep 4131, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
recs

Theoremtfr1a 6410 A weak version of tfr1 6413 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr2a 6411 A weak version of tfr2 6414 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr2b 6412 Without assuming ax-rep 4131, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr1 6413 Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class , normally a function, and 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 NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
recs

Theoremtfr2 6414 Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function has the property that for any function whatsoever, the "next" value of is recursively applied to all "previous" values of . (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfr3 6415* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. 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 NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremrecsfnon 6416 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremrecsval 6417 Strong transfinite recursion in terms of all previous values. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremtz7.44lem1 6418* is a function. Lemma for tz7.44-1 6419, tz7.44-2 6420, and tz7.44-3 6421. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtz7.44-1 6419* The value of at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremtz7.44-2 6420* The value of at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremtz7.44-3 6421* The value of at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

2.4.22  Recursive definition generator

Syntaxcrdg 6422 Extend class notation with the recursive definition generator, with characteristic function and initial value .

Definitiondf-rdg 6423* Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . This combines functions in tfr1 6413 and in tz7.44-1 6419 into one definition. 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. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 6510, from which we prove the recursive textbook definition as theorems oa0 6515, oasuc 6523, and oalim 6531 (with the help of theorems rdg0 6434, rdgsuc 6437, and rdglim2a 6446). We can also restrict the operation to define otherwise recursive functions on the natural numbers ; see fr0g 6448 and frsuc 6449. Our operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the operations (see df-if 3566) select cases based on whether the domain of is zero, a successor, or a limit ordinal.

An important use of this definition is in the recursive sequence generator df-seq 11047 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 11289 and integer powers df-exp 11105.

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 NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

recs

Theoremrdgeq1 6424 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgeq2 6425 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgeq12 6426 Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremnfrdg 6427 Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdglem1 6428* Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)

Theoremrdgfun 6429 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremrdgdmlim 6430 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)

Theoremrdgfnon 6431 The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgvalg 6432* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdgval 6433* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdg0 6434 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdgseg 6435 The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremrdgsucg 6436 The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)

Theoremrdgsuc 6437 The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdglimg 6438 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)

Theoremrdglim 6439 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdg0g 6440 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)

Theoremrdgsucmptf 6441 The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrdgsucmptnf 6442 The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class is a proper class). This is a technical lemma that can be used together with rdgsucmptf 6441 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrdgsucmpt2 6443* This version of rdgsucmpt 6444 avoids the not-free hypothesis of rdgsucmptf 6441 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremrdgsucmpt 6444* The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)

Theoremrdglim2 6445* The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)

Theoremrdglim2a 6446* The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)

2.4.23  Finite recursion

Theoremfrfnom 6447 The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremfr0g 6448 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)

Theoremfrsuc 6449 The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremfrsucmpt 6450 The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)

Theoremfrsucmptn 6451 The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class is a proper class). This is a technical lemma that can be used together with frsucmpt 6450 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfrsucmpt2 6452* The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremtz7.48lem 6453* A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)

Theoremtz7.48-2 6454* Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)

Theoremtz7.48-1 6455* Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

Theoremtz7.48-3 6456* Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

Theoremtz7.49 6457* Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)

Theoremtz7.49c 6458* Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)

Syntaxcseqom 6459 Extend class notation to include index-aware recursive definitions.
seq𝜔

Definitiondf-seqom 6460* Index-aware recursive definitions over . A mashup of df-rdg 6423 and df-seq 11047, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremseqomlem0 6461* Lemma for seq𝜔. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremseqomlem1 6462* Lemma for seq𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Theoremseqomlem2 6463* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Theoremseqomlem3 6464* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremseqomlem4 6465* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Theoremseqomeq12 6466 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔 seq𝜔

Theoremfnseqom 6467 An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremseqom0g 6468 Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremseqomsuc 6469 Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

2.4.24  Abian's "most fundamental" fixed point theorem

Theoremabianfplem 6470* Lemma for abianfp 6471. We prove by transfinite induction that if has a fixed point , then its iterates also equal . This lemma is used for the "trivial" direction of the main theorem. (Contributed by NM, 4-Sep-2004.)

Theoremabianfp 6471* "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let , , ,... be the iterates of . The theorem reads (using our variable names): "Let be a mapping from a set into itself. Then has a fixed point if and only if: There exists an element of such that for every ordinal , is an element of , and if is not a fixed point of then the 's are all distinct for every ordinal ." See df-rdg 6423 for the operation. The proof's key idea is to assume that does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 5751 to derive that the class of all ordinal numbers exists, contradicting onprc 4576. Our version of this theorem does not require the hypothesis that be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem. (Contributed by NM, 5-Sep-2004.) (Revised by David Abernethy, 19-Jun-2012.)

2.4.25  Ordinal arithmetic

Syntaxc1o 6472 Extend the definition of a class to include the ordinal number 1.

Syntaxc2o 6473 Extend the definition of a class to include the ordinal number 2.

Syntaxc3o 6474 Extend the definition of a class to include the ordinal number 3.

Syntaxc4o 6475 Extend the definition of a class to include the ordinal number 4.

Syntaxcoa 6476 Extend the definition of a class to include the ordinal addition operation.

Syntaxcomu 6477 Extend the definition of a class to include the ordinal multiplication operation.

Syntaxcoe 6478 Extend the definition of a class to include the ordinal exponentiation operation.

Definitiondf-1o 6479 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)

Definitiondf-2o 6480 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)

Definitiondf-3o 6481 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-4o 6482 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-omul 6484* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)

Definitiondf-oexp 6485* Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)

Theorem1on 6486 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)

Theorem2on 6487 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theorem2on0 6488 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)

Theorem3on 6489 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4on 6490 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremdf1o2 6491 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)

Theoremdf2o3 6492 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremdf2o2 6493 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)

Theorem1n0 6494 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)

Theoremxp01disj 6495 Cross products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)

Theoremordgt0ge1 6496 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)

Theoremordge1n0 6497 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)

Theoremel1o 6498 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Theoremdif1o 6499 Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.)

Theoremondif1 6500 Two ways to say that is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)

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