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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4on 7201 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  On
 
Theoremdf1o2 7202 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
 |- 
 1o  =  { (/) }
 
Theoremdf2o3 7203 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 2o  =  { (/) ,  1o }
 
Theoremdf2o2 7204 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
 |- 
 2o  =  { (/) ,  { (/)
 } }
 
Theorem1n0 7205 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
 |- 
 1o  =/=  (/)
 
Theoremxp01disj 7206 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
 |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
 ) )  =  (/)
 
Theoremordgt0ge1 7207 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
 
Theoremordge1n0 7208 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
 
Theoremel1o 7209 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  ( A  e.  1o  <->  A  =  (/) )
 
Theoremdif1o 7210 Two ways to say that  A is a nonzero number of the set  B. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( B  \  1o )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
 
Theoremondif1 7211 Two ways to say that  A is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  (/)  e.  A ) )
 
Theoremondif2 7212 Two ways to say that  A is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
 
Theorem2oconcl 7213 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
 
Theorem0lt1o 7214 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  (/)  e.  1o
 
Theoremdif20el 7215 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  ( A  e.  ( On  \  2o )  ->  (/) 
 e.  A )
 
Theorem0we1 7216 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  (/)  We  1o
 
Theorembrwitnlem 7217 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.)
 |-  R  =  ( `' O " ( _V  \  1o ) )   &    |-  O  Fn  X   =>    |-  ( A R B  <->  ( A O B )  =/=  (/) )
 
Theoremfnoa 7218 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 +o  Fn  ( On  X. 
 On )
 
Theoremfnom 7219 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 .o  Fn  ( On  X. 
 On )
 
Theoremfnoe 7220 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
 |- 
 ^o  Fn  ( On  X. 
 On )
 
Theoremoav 7221* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  =  ( rec ( ( x  e. 
 _V  |->  suc  x ) ,  A ) `  B ) )
 
Theoremomv 7222* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B )  =  ( rec ( ( x  e. 
 _V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
 
Theoremoe0lem 7223 A helper lemma for oe0 7232 and others. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( ph  /\  A  =  (/) )  ->  ps )   &    |-  (
 ( ( A  e.  On  /\  ph )  /\  (/)  e.  A )  ->  ps )   =>    |-  ( ( A  e.  On  /\  ph )  ->  ps )
 
Theoremoev 7224* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B )  =  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) ) )
 
Theoremoevn0 7225* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e. 
 _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
 
Theoremoa0 7226 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 7227 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 7228 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 7229 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 7227, 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 7230 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
 |-  ( (/)  ^o  (/) )  =  1o
 
Theoremoe0m1 7231 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 7232 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 7233* Alternate value of ordinal exponentiation. Compare oev 7224. (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 7234 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 7235* Lemma for oesuc 7237. (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 7236 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 7237 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 7238 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7234 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 7239 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 7240 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 7241 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 7242* 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 7243* 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 7244* 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 7245 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 7246 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 7247 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 7248 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
 |-  ( A  e.  On  ->  ( (/)  +o  A )  =  A )
 
Theoremom0r 7249 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
 |-  ( A  e.  On  ->  ( (/)  .o  A )  =  (/) )
 
Theoremo1p1e2 7250 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 |-  ( 1o  +o  1o )  =  2o
 
Theoremom1 7251 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 7252 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 7253 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
 |-  ( A  e.  On  ->  ( A  ^o  1o )  =  A )
 
Theoremoe1m 7254 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 7255 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 7256 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 7257 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 7258 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 7259 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 7260 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 7261 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 7260.) (Contributed by NM, 6-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
 
Theoremoaword2 7262 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 7263* Lemma for oawordex 7266. (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 7264* 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 7265* 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 7266* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7264 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 7267* 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 7268 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 7269 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 7270 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 7271* 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 7272* 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 7273* Lemma for oacomf1o 7274. (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 7274* 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 8156). (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 7275 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 7276 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 7277 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 7278 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 7279 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 7280 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 7281 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 7282 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 7283 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 7284 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 7285 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 7286* 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 7287 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 7288 Distributive law for ordinal arithmetic (left-distributivity). 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 7289 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 7290 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 7291* Lemma for omeu 7294: 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 7292 Lemma for omeu 7294: 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 7293 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 7294* 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 7295 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 7296 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 7297 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 7298 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 7299 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 7300 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 ) ) )
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