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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoeoe 6801 Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoelimcl 6802 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoeeulem 6803* Lemma for oeeu 6805. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoeeui 6804* The division algorithm for ordinal exponentiation. (This version of oeeu 6805 gives an explicit expression for the unique solution of the equation, in terms of the solution to omeu 6787.) (Contributed by Mario Carneiro, 25-May-2015.)

Theoremoeeu 6805* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)

2.4.27  Natural number arithmetic

Theoremnna0 6806 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)

Theoremnnm0 6807 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)

Theoremnnasuc 6808 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnmsuc 6809 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnesuc 6810 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremnna0r 6811 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 6741) so that we can avoid ax-rep 4280, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnm0r 6812 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacl 6813 Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnmcl 6814 Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnecl 6815 Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnacli 6816 is closed under addition. Inference form of nnacl 6813. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnmcli 6817 is closed under multiplication. Inference form of nnmcl 6814. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnarcl 6818 Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)

Theoremnnacom 6819 Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaordi 6820 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaord 6821 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaordr 6822 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)

Theoremnnawordi 6823 Adding to both sides of an inequality in (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnaass 6824 Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnndi 6825 Distributive law for natural numbers. Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmass 6826 Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmsucr 6827 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnmcom 6828 Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnaword 6829 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacan 6830 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword1 6831 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword2 6832 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)

Theoremnnmordi 6833 Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmord 6834 Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmword 6835 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmcan 6836 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmwordi 6837 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmwordri 6838 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnawordex 6839* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaordex 6840* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorem1onn 6841 One is a natural number. (Contributed by NM, 29-Oct-1995.)

Theorem2onn 6842 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)

Theorem3onn 6843 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4onn 6844 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremoaabslem 6845 Lemma for oaabs 6846. (Contributed by NM, 9-Dec-2004.)

Theoremoaabs 6846 Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.)

Theoremoaabs2 6847 The absorption law oaabs 6846 is also a property of higher powers of . (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomabslem 6848 Lemma for omabs 6849. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomabs 6849 Ordinal multiplication is also absorbed by powers of . (Contributed by Mario Carneiro, 30-May-2015.)

Theoremnnm1 6850 Multiply an element of by . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnm2 6851 Multiply an element of by (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnn2m 6852 Multiply an element of by (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnneo 6853 If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremnneob 6854* A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsmolem 6855* Lemma for omsmo 6856. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)

Theoremomsmo 6856* A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)

Theoremomopthlem1 6857 Lemma for omopthi 6859. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopthlem2 6858 Lemma for omopthi 6859. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopthi 6859 An ordered pair theorem for . Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 11518. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremomopth 6860 An ordered pair theorem for finite integers. Analagous to nn0opthi 11518. (Contributed by Scott Fenton, 1-May-2012.) (Revised by Mario Carneiro, 12-May-2012.)

2.4.28  Equivalence relations and classes

Syntaxwer 6861 Extend the definition of a wff to include the equivalence predicate.

Syntaxcec 6862 Extend the definition of a class to include equivalence class.

Syntaxcqs 6863 Extend the definition of a class to include quotient set.

Definitiondf-er 6864 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6865 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6884, ersymb 6878, and ertr 6879. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)

Theoremdfer2 6865* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)

Definitiondf-ec 6866 Define the -coset of . Exercise 35 of [Enderton] p. 61. This is called the equivalence class of modulo when is an equivalence relation (i.e. when ; see dfer2 6865). In this case, is a representative (member) of the equivalence class , which contains all sets that are equivalent to . Definition of [Enderton] p. 57 uses the notation (subscript) , although we simply follow the brackets by since we don't have subscripted expressions. For an alternate definition, see dfec2 6867. (Contributed by NM, 23-Jul-1995.)

Theoremdfec2 6867* Alternate definition of -coset of . Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theoremecexg 6868 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)

Theoremecexr 6869 A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Definitiondf-qs 6870* Define quotient set. is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)

Theoremereq1 6871 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremereq2 6872 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrel 6873 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerdm 6874 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremercl 6875 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersym 6876 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercl2 6877 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersymb 6878 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertr 6879 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertrd 6880 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr2d 6881 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr3d 6882 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr4d 6883 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremerref 6884 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercnv 6885 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrn 6886 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerssxp 6887 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerex 6888 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremerexb 6889 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremiserd 6890* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theorembrdifun 6891 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremswoer 6892* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremswoord1 6893* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoord2 6894* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoso 6895* If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremeqerlem 6896* Lemma for eqer 6897. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremeqer 6897* Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremider 6898 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theorem0er 6899 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremeceq1 6900 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

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