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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoa0 6401 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
 
Theoremom0 6402 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
 
Theoremoe0m 6403 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  On  ->  ( (/)  ^o  A )  =  ( 1o  \  A ) )
 
Theoremom0x 6404 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6402, this version works whether or not  A 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.)
 |-  ( A  .o  (/) )  =  (/)
 
Theoremoe0m0 6405 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
 |-  ( (/)  ^o  (/) )  =  1o
 
Theoremoe0m1 6406 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.)
 |-  ( A  e.  On  ->  ( (/)  e.  A  <->  ( (/)  ^o  A )  =  (/) ) )
 
Theoremoe0 6407 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.)
 |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
 
Theoremoev2 6408* Alternate value of ordinal exponentiation. Compare oev 6399. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B )  =  (
 ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  ( ( _V  \  |^| A )  u.  |^| B ) ) )
 
Theoremoasuc 6409 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremoesuclem 6410* Lemma for oesuc 6412. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 Lim  X   &    |-  ( B  e.  X  ->  ( rec (
 ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B )  =  ( ( x  e. 
 _V  |->  ( x  .o  A ) ) `  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) ) )   =>    |-  ( ( A  e.  On  /\  B  e.  X )  ->  ( A  ^o  suc 
 B )  =  ( ( A  ^o  B )  .o  A ) )
 
Theoremomsuc 6411 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremoesuc 6412 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  suc 
 B )  =  ( ( A  ^o  B )  .o  A ) )
 
Theoremonasuc 6413 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6409 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremonmsuc 6414 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremonesuc 6415 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ^o  suc 
 B )  =  ( ( A  ^o  B )  .o  A ) )
 
Theoremoa1suc 6416 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.)
 |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )
 
Theoremoalim 6417* 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.)
 |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B )
 )  ->  ( A  +o  B )  =  U_ x  e.  B  ( A  +o  x ) )
 
Theoremomlim 6418* 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.)
 |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B )
 )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
 
Theoremoelim 6419* 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.)
 |-  ( ( ( A  e.  On  /\  ( B  e.  C  /\  Lim 
 B ) )  /\  (/) 
 e.  A )  ->  ( A  ^o  B )  =  U_ x  e.  B  ( A  ^o  x ) )
 
Theoremoacl 6420 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  e.  On )
 
Theoremomcl 6421 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B )  e.  On )
 
Theoremoecl 6422 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B )  e.  On )
 
Theoremoa0r 6423 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
 |-  ( A  e.  On  ->  ( (/)  +o  A )  =  A )
 
Theoremom0r 6424 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
 |-  ( A  e.  On  ->  ( (/)  .o  A )  =  (/) )
 
Theoremo1p1e2 6425 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 |-  ( 1o  +o  1o )  =  2o
 
Theoremom1 6426 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)
 |-  ( A  e.  On  ->  ( A  .o  1o )  =  A )
 
Theoremom1r 6427 Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
 |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )
 
Theoremoe1 6428 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
 |-  ( A  e.  On  ->  ( A  ^o  1o )  =  A )
 
Theoremoe1m 6429 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
 |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
 
Theoremoaordi 6430 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremoaord 6431 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremoacan 6432 Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  +o  B )  =  ( A  +o  C )  <->  B  =  C ) )
 
Theoremoaword 6433 Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( C  +o  A ) 
 C_  ( C  +o  B ) ) )
 
Theoremoawordri 6434 Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( A  +o  C )  C_  ( B  +o  C ) ) )
 
Theoremoaord1 6435 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  A  e.  ( A  +o  B ) ) )
 
Theoremoaword1 6436 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 6435.) (Contributed by NM, 6-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
 
Theoremoaword2 6437 An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( B  +o  A ) )
 
Theoremoawordeulem 6438* Lemma for oawordex 6441. (Contributed by NM, 11-Dec-2004.)
 |-  A  e.  On   &    |-  B  e.  On   &    |-  S  =  {
 y  e.  On  |  B  C_  ( A  +o  y ) }   =>    |-  ( A  C_  B  ->  E! x  e. 
 On  ( A  +o  x )  =  B )
 
Theoremoawordeu 6439* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B )  ->  E! x  e.  On  ( A  +o  x )  =  B )
 
Theoremoawordexr 6440* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)
 |-  ( ( A  e.  On  /\  E. x  e. 
 On  ( A  +o  x )  =  B )  ->  A  C_  B )
 
Theoremoawordex 6441* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 6439 for uniqueness. (Contributed by NM, 12-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <-> 
 E. x  e.  On  ( A  +o  x )  =  B )
 )
 
Theoremoaordex 6442* 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <-> 
 E. x  e.  On  ( (/)  e.  x  /\  ( A  +o  x )  =  B )
 ) )
 
Theoremoa00 6443 An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  =  (/) 
 <->  ( A  =  (/)  /\  B  =  (/) ) ) )
 
Theoremoalimcl 6444 The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.)
 |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B )
 )  ->  Lim  ( A  +o  B ) )
 
Theoremoaass 6445 Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) )
 
Theoremoarec 6446* Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  =  ( A  u.  ran  (  x  e.  B  |->  ( A  +o  x ) ) ) )
 
Theoremoaf1o 6447* Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( A  e.  On  ->  ( x  e.  On  |->  ( A  +o  x ) ) : On -1-1-onto-> ( On  \  A ) )
 
Theoremoacomf1olem 6448* Lemma for oacomf1o 6449. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  A  |->  ( B  +o  x ) )   =>    |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F : A
 -1-1-onto-> ran  F  /\  ( ran 
 F  i^i  B )  =  (/) ) )
 
Theoremoacomf1o 6449* Define a bijection from  A  +o  B to  B  +o  A. Thus the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 7236). (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( ( x  e.  A  |->  ( B  +o  x ) )  u.  `' ( x  e.  B  |->  ( A  +o  x ) ) )   =>    |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : ( A  +o  B ) -1-1-onto-> ( B  +o  A ) )
 
Theoremomordi 6450 Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
 |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremomord2 6451 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  e.  B 
 <->  ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremomord 6452 Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  e.  B  /\  (/)  e.  C )  <-> 
 ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremomcan 6453 Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C ) 
 <->  B  =  C ) )
 
Theoremomword 6454 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B 
 <->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremomwordi 6455 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremomwordri 6456 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( A  .o  C )  C_  ( B  .o  C ) ) )
 
Theoremomword1 6457 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( A  .o  B ) )
 
Theoremomword2 6458 An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  A  C_  ( B  .o  A ) )
 
Theoremom00 6459 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/) 
 <->  ( A  =  (/)  \/  B  =  (/) ) ) )
 
Theoremom00el 6460 The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( (/)  e.  A  /\  (/)  e.  B ) ) )
 
Theoremomordlim 6461* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)
 |-  ( ( ( A  e.  On  /\  ( B  e.  D  /\  Lim 
 B ) )  /\  C  e.  ( A  .o  B ) )  ->  E. x  e.  B  C  e.  ( A  .o  x ) )
 
Theoremomlimcl 6462 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.)
 |-  ( ( ( A  e.  On  /\  ( B  e.  C  /\  Lim 
 B ) )  /\  (/) 
 e.  A )  ->  Lim  ( A  .o  B ) )
 
Theoremodi 6463 Distributive law for ordinal arithmetic. Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  .o  ( B  +o  C ) )  =  ( ( A  .o  B )  +o  ( A  .o  C ) ) )
 
Theoremomass 6464 Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  .o  B )  .o  C )  =  ( A  .o  ( B  .o  C ) ) )
 
Theoremoneo 6465 If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  =  ( 2o 
 .o  A ) ) 
 ->  -.  suc  C  =  ( 2o  .o  B ) )
 
Theoremomeulem1 6466* Lemma for omeu 6469: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
 
Theoremomeulem2 6467 Lemma for omeu 6469: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)
 |-  ( ( ( A  e.  On  /\  A  =/= 
 (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  ( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D )  +o  E ) ) )
 
Theoremomopth2 6468 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ( ( A  e.  On  /\  A  =/= 
 (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  ->  ( ( ( A  .o  B )  +o  C )  =  (
 ( A  .o  D )  +o  E )  <->  ( B  =  D  /\  C  =  E ) ) )
 
Theoremomeu 6469* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E! z E. x  e. 
 On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
 ( A  .o  x )  +o  y )  =  B ) )
 
Theoremoen0 6470 Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  (/)  e.  ( A 
 ^o  B ) )
 
Theoremoeordi 6471 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.)
 |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) ) 
 ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
 
Theoremoeord 6472 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.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
 
Theoremoecan 6473 Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  =  ( A  ^o  C )  <->  B  =  C ) )
 
Theoremoeword 6474 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C 
 ^o  B ) ) )
 
Theoremoewordi 6475 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C 
 ^o  B ) ) )
 
Theoremoewordri 6476 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( A  ^o  C )  C_  ( B 
 ^o  C ) ) )
 
Theoremoeworde 6477 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.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  On )  ->  B  C_  ( A  ^o  B ) )
 
Theoremoeordsuc 6478 Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( A  ^o  suc 
 C )  e.  ( B  ^o  suc  C )
 ) )
 
Theoremoelim2 6479* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B )
 )  ->  ( A  ^o  B )  =  U_ x  e.  ( B  \  1o ) ( A 
 ^o  x ) )
 
Theoremoeoalem 6480 Lemma for oeoa 6481. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  A  e.  On   &    |-  (/)  e.  A   &    |-  B  e.  On   =>    |-  ( C  e.  On  ->  ( A  ^o  ( B  +o  C ) )  =  ( ( A 
 ^o  B )  .o  ( A  ^o  C ) ) )
 
Theoremoeoa 6481 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.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  ( B  +o  C ) )  =  ( ( A 
 ^o  B )  .o  ( A  ^o  C ) ) )
 
Theoremoeoelem 6482 Lemma for oeoe 6483. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  A  e.  On   &    |-  (/)  e.  A   =>    |-  (
 ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
 
Theoremoeoe 6483 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.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
 
Theoremoelimcl 6484 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  ( B  e.  C  /\  Lim  B ) ) 
 ->  Lim  ( A  ^o  B ) )
 
Theoremoeeulem 6485* Lemma for oeeu 6487. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  X  =  U. |^| { x  e.  On  |  B  e.  ( A  ^o  x ) }   =>    |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o )
 )  ->  ( X  e.  On  /\  ( A 
 ^o  X )  C_  B  /\  B  e.  ( A  ^o  suc  X )
 ) )
 
Theoremoeeui 6486* The division algorithm for ordinal exponentiation. (This version of oeeu 6487 gives an explicit expression for the unique solution of the equation, in terms of the solution  P to omeu 6469.) (Contributed by Mario Carneiro, 25-May-2015.)
 |-  X  =  U. |^| { x  e.  On  |  B  e.  ( A  ^o  x ) }   &    |-  P  =  ( iota w E. y  e.  On  E. z  e.  ( A  ^o  X ) ( w  = 
 <. y ,  z >.  /\  ( ( ( A 
 ^o  X )  .o  y )  +o  z
 )  =  B ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   =>    |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o ) )  ->  ( ( ( C  e.  On  /\  D  e.  ( A  \  1o )  /\  E  e.  ( A  ^o  C ) ) 
 /\  ( ( ( A  ^o  C )  .o  D )  +o  E )  =  B ) 
 <->  ( C  =  X  /\  D  =  Y  /\  E  =  Z )
 ) )
 
Theoremoeeu 6487* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  ( ( A  e.  ( On  \  2o )  /\  B  e.  ( On  \  1o ) )  ->  E! w E. x  e. 
 On  E. y  e.  ( A  \  1o ) E. z  e.  ( A  ^o  x ) ( w  =  <. x ,  y ,  z >.  /\  ( ( ( A  ^o  x )  .o  y )  +o  z )  =  B ) )
 
2.4.24  Natural number arithmetic
 
Theoremnna0 6488 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 6489 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 6490 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremnnmsuc 6491 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremnnesuc 6492 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ^o  suc 
 B )  =  ( ( A  ^o  B )  .o  A ) )
 
Theoremnna0r 6493 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 6423) so that we can avoid ax-rep 4028, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( A  e.  om  ->  ( (/)  +o  A )  =  A )
 
Theoremnnm0r 6494 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( A  e.  om  ->  ( (/)  .o  A )  =  (/) )
 
Theoremnnacl 6495 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  e.  om )
 
Theoremnnmcl 6496 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  B )  e.  om )
 
Theoremnnecl 6497 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ^o  B )  e.  om )
 
Theoremnnacli 6498  om is closed under addition. Inference form of nnacl 6495. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  +o  B )  e.  om
 
Theoremnnmcli 6499  om is closed under multiplication. Inference form of nnmcl 6496. (Contributed by Scott Fenton, 20-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  .o  B )  e.  om
 
Theoremnnarcl 6500 Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e. 
 om 
 <->  ( A  e.  om  /\  B  e.  om )
 ) )
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