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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremseqomlem3 6701* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Theoremseqomlem4 6702* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Theoremseqomeq12 6703 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔 seq𝜔

Theoremfnseqom 6704 An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremseqom0g 6705 Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

Theoremseqomsuc 6706 Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔

2.4.25  Abian's "most fundamental" fixed point theorem

Theoremabianfplem 6707* Lemma for abianfp 6708. 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 6708* "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 6660 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 5963 to derive that the class of all ordinal numbers exists, contradicting onprc 4757. 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.)

2.4.26  Ordinal arithmetic

Syntaxc1o 6709 Extend the definition of a class to include the ordinal number 1.

Syntaxc2o 6710 Extend the definition of a class to include the ordinal number 2.

Syntaxc3o 6711 Extend the definition of a class to include the ordinal number 3.

Syntaxc4o 6712 Extend the definition of a class to include the ordinal number 4.

Syntaxcoa 6713 Extend the definition of a class to include the ordinal addition operation.

Syntaxcomu 6714 Extend the definition of a class to include the ordinal multiplication operation.

Syntaxcoe 6715 Extend the definition of a class to include the ordinal exponentiation operation.

Definitiondf-1o 6716 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)

Definitiondf-2o 6717 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)

Definitiondf-3o 6718 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-4o 6719 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-omul 6721* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)

Definitiondf-oexp 6722* Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)

Theorem1on 6723 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)

Theorem2on 6724 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theorem2on0 6725 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)

Theorem3on 6726 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4on 6727 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremdf1o2 6728 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)

Theoremdf2o3 6729 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremdf2o2 6730 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)

Theorem1n0 6731 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)

Theoremxp01disj 6732 Cross products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)

Theoremordgt0ge1 6733 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)

Theoremordge1n0 6734 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)

Theoremel1o 6735 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Theoremdif1o 6736 Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.)

Theoremondif1 6737 Two ways to say that is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)

Theoremondif2 6738 Two ways to say that is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)

Theorem2oconcl 6739 Closure of the pair swapping function on . (Contributed by Mario Carneiro, 27-Sep-2015.)

Theorem0lt1o 6740 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)

Theoremdif20el 6741 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)

Theorem0we1 6742 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theorembrwitnlem 6743 Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremfnoa 6744 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnom 6745 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnoe 6746 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoav 6747* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomv 6748* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoe0lem 6749 A helper lemma for oe0 6758 and others. (Contributed by NM, 6-Jan-2005.)

Theoremoev 6750* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoevn0 6751* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoa0 6752 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0 6753 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoe0m 6754 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0x 6755 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6753, this version works whether or not is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.)

Theoremoe0m0 6756 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)

Theoremoe0m1 6757 Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)

Theoremoe0 6758 Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoev2 6759* Alternate value of ordinal exponentiation. Compare oev 6750. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoasuc 6760 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuclem 6761* Lemma for oesuc 6763. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsuc 6762 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuc 6763 Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremonasuc 6764 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6760 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremonmsuc 6765 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremonesuc 6766 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremoa1suc 6767 Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremoalim 6768* Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomlim 6769* Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoelim 6770* Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoacl 6771 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremomcl 6772 Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoecl 6773 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoa0r 6774 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremom0r 6775 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremo1p1e2 6776 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)

Theoremom1 6777 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)

Theoremom1r 6778 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremoe1 6779 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)

Theoremoe1m 6780 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Theoremoaordi 6781 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaord 6782 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)

Theoremoacan 6783 Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaword 6784 Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoawordri 6785 Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)

Theoremoaord1 6786 An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of [Suppes] p. 209 and its converse. (Contributed by NM, 6-Dec-2004.)

Theoremoaword1 6787 An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (For the other part see oaord1 6786.) (Contributed by NM, 6-Dec-2004.)

Theoremoaword2 6788 An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.)

Theoremoawordeulem 6789* Lemma for oawordex 6792. (Contributed by NM, 11-Dec-2004.)

Theoremoawordeu 6790* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)

Theoremoawordexr 6791* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)

Theoremoawordex 6792* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6790 for uniqueness. (Contributed by NM, 12-Dec-2004.)

Theoremoaordex 6793* Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse. (Contributed by NM, 12-Dec-2004.)

Theoremoa00 6794 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)

Theoremoalimcl 6795 The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)

Theoremoaass 6796 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)

Theoremoarec 6797* Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)

Theoremoaf1o 6798* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1olem 6799* Lemma for oacomf1o 6800. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1o 6800* Define a bijection from to . Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 7596). (Contributed by Mario Carneiro, 30-May-2015.)

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