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

Theoremoawordexr 6801* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)

Theoremoawordex 6802* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6800 for uniqueness. (Contributed by NM, 12-Dec-2004.)

Theoremoaordex 6803* 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 6804 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)

Theoremoalimcl 6805 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 6806 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)

Theoremoarec 6807* 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 6808* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1olem 6809* Lemma for oacomf1o 6810. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1o 6810* Define a bijection from to . Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 7608). (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomordi 6811 Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomord2 6812 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)

Theoremomord 6813 Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomcan 6814 Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)

Theoremomword 6815 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordi 6816 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordri 6817 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)

Theoremomword1 6818 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremomword2 6819 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremom00 6820 The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.)

Theoremom00el 6821 The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)

Theoremomordlim 6822* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Theoremomlimcl 6823 The product of any nonzero ordinal with a limit ordinal is a limit ordinal. Proposition 8.24 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Theoremodi 6824 Distributive law for ordinal arithmetic. Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)

Theoremomass 6825 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)

Theoremoneo 6826 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)

Theoremomeulem1 6827* Lemma for omeu 6830: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremomeulem2 6828 Lemma for omeu 6830: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Theoremomopth2 6829 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomeu 6830* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoen0 6831 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)

Theoremoeordi 6832 Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeord 6833 Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoecan 6834 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeword 6835 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoewordi 6836 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Theoremoewordri 6837 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)

Theoremoeworde 6838 Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeordsuc 6839 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)

Theoremoelim2 6840* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)

Theoremoeoalem 6841 Lemma for oeoa 6842. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoa 6842 Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoelem 6843 Lemma for oeoe 6844. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoe 6844 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 6845 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoeeulem 6846* Lemma for oeeu 6848. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoeeui 6847* The division algorithm for ordinal exponentiation. (This version of oeeu 6848 gives an explicit expression for the unique solution of the equation, in terms of the solution to omeu 6830.) (Contributed by Mario Carneiro, 25-May-2015.)

Theoremoeeu 6848* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)

2.4.27  Natural number arithmetic

Theoremnna0 6849 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)

Theoremnnm0 6850 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)

Theoremnnasuc 6851 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnmsuc 6852 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnesuc 6853 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremnna0r 6854 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 6784) so that we can avoid ax-rep 4322, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnm0r 6855 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacl 6856 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 6857 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 6858 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 6859 is closed under addition. Inference form of nnacl 6856. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnmcli 6860 is closed under multiplication. Inference form of nnmcl 6857. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnarcl 6861 Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)

Theoremnnacom 6862 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 6863 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 6864 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 6865 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)

Theoremnnawordi 6866 Adding to both sides of an inequality in (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Theoremnnaass 6867 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 6868 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 6869 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 6870 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnnmcom 6871 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 6872 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnacan 6873 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword1 6874 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaword2 6875 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)

Theoremnnmordi 6876 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 6877 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 6878 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmcan 6879 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnmwordi 6880 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnmwordri 6881 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 6882* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnaordex 6883* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorem1onn 6884 One is a natural number. (Contributed by NM, 29-Oct-1995.)

Theorem2onn 6885 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)

Theorem3onn 6886 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorem4onn 6887 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremoaabslem 6888 Lemma for oaabs 6889. (Contributed by NM, 9-Dec-2004.)

Theoremoaabs 6889 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 6890 The absorption law oaabs 6889 is also a property of higher powers of . (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomabslem 6891 Lemma for omabs 6892. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomabs 6892 Ordinal multiplication is also absorbed by powers of . (Contributed by Mario Carneiro, 30-May-2015.)

Theoremnnm1 6893 Multiply an element of by . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremnnm2 6894 Multiply an element of by (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnn2m 6895 Multiply an element of by (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnneo 6896 If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremnneob 6897* A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsmolem 6898* Lemma for omsmo 6899. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)

Theoremomsmo 6899* 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 6900 Lemma for omopthi 6902. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

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