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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoawordri 7201 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 7202 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 7203 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 7202.) (Contributed by NM, 6-Dec-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
 
Theoremoaword2 7204 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 7205* Lemma for oawordex 7208. (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 7206* 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 7207* 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 7208* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7206 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 7209* 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 7210 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 7211 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 7212 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 7213* 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 7214* 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 7215* Lemma for oacomf1o 7216. (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 7216* 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 8071). (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 7217 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 7218 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 7219 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 7220 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 7221 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 7222 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 7223 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 7224 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 7225 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 7226 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 7227 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 7228* 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 7229 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 7230 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 7231 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 7232 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 7233* Lemma for omeu 7236: 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 7234 Lemma for omeu 7236: 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 7235 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 7236* 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 7237 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 7238 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 7239 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 7240 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 7241 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 7242 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 7243 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 7244 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 7245 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 7246* 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 7247 Lemma for oeoa 7248. (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 7248 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 7249 Lemma for oeoe 7250. (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 7250 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 7251 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 7252* Lemma for oeeu 7254. (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 7253* The division algorithm for ordinal exponentiation. (This version of oeeu 7254 gives an explicit expression for the unique solution of the equation, in terms of the solution  P to omeu 7236.) (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 7254* 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 ) )
 
2.4.18  Natural number arithmetic
 
Theoremnna0 7255 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 7256 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 7257 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 7258 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 7259 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 7260 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 7190) so that we can avoid ax-rep 4548, 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 7261 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 7262 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 7263 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 7264 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 7265  om is closed under addition. Inference form of nnacl 7262. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  +o  B )  e.  om
 
Theoremnnmcli 7266  om is closed under multiplication. Inference form of nnmcl 7263. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  .o  B )  e.  om
 
Theoremnnarcl 7267 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 )
 ) )
 
Theoremnnacom 7268 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  =  ( B  +o  A ) )
 
Theoremnnaordi 7269 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.)
 |-  ( ( B  e.  om 
 /\  C  e.  om )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremnnaord 7270 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )
 
Theoremnnaordr 7271 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  e.  B  <->  ( A  +o  C )  e.  ( B  +o  C ) ) )
 
Theoremnnawordi 7272 Adding to both sides of an inequality in  om. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( A  +o  C )  C_  ( B  +o  C ) ) )
 
Theoremnnaass 7273 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) )
 
Theoremnndi 7274 Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  .o  ( B  +o  C ) )  =  ( ( A  .o  B )  +o  ( A  .o  C ) ) )
 
Theoremnnmass 7275 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  .o  B )  .o  C )  =  ( A  .o  ( B  .o  C ) ) )
 
Theoremnnmsucr 7276 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  .o  B )  =  ( ( A  .o  B )  +o  B ) )
 
Theoremnnmcom 7277 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  B )  =  ( B  .o  A ) )
 
Theoremnnaword 7278 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  <->  ( C  +o  A ) 
 C_  ( C  +o  B ) ) )
 
Theoremnnacan 7279 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  +o  B )  =  ( A  +o  C )  <->  B  =  C ) )
 
Theoremnnaword1 7280 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  A  C_  ( A  +o  B ) )
 
Theoremnnaword2 7281 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  A  C_  ( B  +o  A ) )
 
Theoremnnmordi 7282 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.)
 |-  ( ( ( B  e.  om  /\  C  e.  om )  /\  (/)  e.  C )  ->  ( A  e.  B  ->  ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremnnmord 7283 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  e.  B  /\  (/)  e.  C )  <-> 
 ( C  .o  A )  e.  ( C  .o  B ) ) )
 
Theoremnnmword 7284 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  /\  (/)  e.  C )  ->  ( A  C_  B 
 <->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremnnmcan 7285 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  /\  (/)  e.  A )  ->  ( ( A  .o  B )  =  ( A  .o  C ) 
 <->  B  =  C ) )
 
Theoremnnmwordi 7286 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( C  .o  A )  C_  ( C  .o  B ) ) )
 
Theoremnnmwordri 7287 Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( A  C_  B  ->  ( A  .o  C )  C_  ( B  .o  C ) ) )
 
Theoremnnawordex 7288* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  C_  B 
 <-> 
 E. x  e.  om  ( A  +o  x )  =  B )
 )
 
Theoremnnaordex 7289* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <-> 
 E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x )  =  B )
 ) )
 
Theorem1onn 7290 One is a natural number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  om
 
Theorem2onn 7291 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
 |- 
 2o  e.  om
 
Theorem3onn 7292 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  om
 
Theorem4onn 7293 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  om
 
Theoremoaabslem 7294 Lemma for oaabs 7295. (Contributed by NM, 9-Dec-2004.)
 |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
 
Theoremoaabs 7295 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.)
 |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B )  ->  ( A  +o  B )  =  B )
 
Theoremoaabs2 7296 The absorption law oaabs 7295 is also a property of higher powers of  om. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ( ( A  e.  ( om  ^o  C )  /\  B  e.  On )  /\  ( om  ^o  C )  C_  B )  ->  ( A  +o  B )  =  B )
 
Theoremomabslem 7297 Lemma for omabs 7298. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( om  e.  On  /\  A  e.  om  /\  (/)  e.  A )  ->  ( A  .o  om )  =  om )
 
Theoremomabs 7298 Ordinal multiplication is also absorbed by powers of  om. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ( ( A  e.  om  /\  (/)  e.  A )  /\  ( B  e.  On  /\  (/)  e.  B ) )  ->  ( A  .o  ( om  ^o  B ) )  =  ( om  ^o  B ) )
 
Theoremnnm1 7299 Multiply an element of  om by  1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  1o )  =  A )
 
Theoremnnm2 7300 Multiply an element of  om by  2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  om  ->  ( A  .o  2o )  =  ( A  +o  A ) )
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