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

Theorembrwitnlem 6701 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 6702 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnom 6703 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfnoe 6704 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoav 6705* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremomv 6706* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoe0lem 6707 A helper lemma for oe0 6716 and others. (Contributed by NM, 6-Jan-2005.)

Theoremoev 6708* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremoevn0 6709* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoa0 6710 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0 6711 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 6712 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremom0x 6713 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6711, 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 6714 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)

Theoremoe0m1 6715 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 6716 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 6717* Alternate value of ordinal exponentiation. Compare oev 6708. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoasuc 6718 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuclem 6719* Lemma for oesuc 6721. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremomsuc 6720 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremoesuc 6721 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 6722 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6718 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremonmsuc 6723 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremonesuc 6724 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremoa1suc 6725 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 6726* 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 6727* 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 6728* 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 6729 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremomcl 6730 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 6731 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoa0r 6732 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)

Theoremom0r 6733 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremo1p1e2 6734 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)

Theoremom1 6735 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)

Theoremom1r 6736 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)

Theoremoe1 6737 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)

Theoremoe1m 6738 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Theoremoaordi 6739 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaord 6740 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)

Theoremoacan 6741 Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)

Theoremoaword 6742 Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoawordri 6743 Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)

Theoremoaord1 6744 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 6745 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 6744.) (Contributed by NM, 6-Dec-2004.)

Theoremoaword2 6746 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 6747* Lemma for oawordex 6750. (Contributed by NM, 11-Dec-2004.)

Theoremoawordeu 6748* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)

Theoremoawordexr 6749* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)

Theoremoawordex 6750* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6748 for uniqueness. (Contributed by NM, 12-Dec-2004.)

Theoremoaordex 6751* 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 6752 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)

Theoremoalimcl 6753 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 6754 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)

Theoremoarec 6755* 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 6756* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1olem 6757* Lemma for oacomf1o 6758. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremoacomf1o 6758* Define a bijection from to . Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 7553). (Contributed by Mario Carneiro, 30-May-2015.)

Theoremomordi 6759 Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomord2 6760 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)

Theoremomord 6761 Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Theoremomcan 6762 Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)

Theoremomword 6763 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordi 6764 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Theoremomwordri 6765 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)

Theoremomword1 6766 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremomword2 6767 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)

Theoremom00 6768 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 6769 The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)

Theoremomordlim 6770* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)

Theoremomlimcl 6771 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 6772 Distributive law for ordinal arithmetic. Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)

Theoremomass 6773 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)

Theoremoneo 6774 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)

Theoremomeulem1 6775* Lemma for omeu 6778: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremomeulem2 6776 Lemma for omeu 6778: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Theoremomopth2 6777 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremomeu 6778* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoen0 6779 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)

Theoremoeordi 6780 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 6781 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 6782 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoeword 6783 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Theoremoewordi 6784 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Theoremoewordri 6785 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)

Theoremoeworde 6786 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 6787 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)

Theoremoelim2 6788* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)

Theoremoeoalem 6789 Lemma for oeoa 6790. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoa 6790 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 6791 Lemma for oeoe 6792. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremoeoe 6792 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 6793 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremoeeulem 6794* Lemma for oeeu 6796. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremoeeui 6795* The division algorithm for ordinal exponentiation. (This version of oeeu 6796 gives an explicit expression for the unique solution of the equation, in terms of the solution to omeu 6778.) (Contributed by Mario Carneiro, 25-May-2015.)

Theoremoeeu 6796* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)

2.4.27  Natural number arithmetic

Theoremnna0 6797 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)

Theoremnnm0 6798 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)

Theoremnnasuc 6799 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremnnmsuc 6800 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

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