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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.4.21  Finite recursion
 
Theoremfrfnom 6401 The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( rec ( F ,  A )  |`  om )  Fn  om
 
Theoremfr0g 6402 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
 |-  ( A  e.  B  ->  ( ( rec ( F ,  A )  |` 
 om ) `  (/) )  =  A )
 
Theoremfrsuc 6403 The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( B  e.  om  ->  ( ( rec ( F ,  A )  |` 
 om ) `  suc  B )  =  ( F `
  ( ( rec ( F ,  A )  |`  om ) `  B ) ) )
 
Theoremfrsucmpt 6404 The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  ( x  =  ( F `  B )  ->  C  =  D )   =>    |-  ( ( B  e.  om 
 /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
 
Theoremfrsucmptn 6405 The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 
D is a proper class). This is a technical lemma that can be used together with frsucmpt 6404 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  ( x  =  ( F `  B )  ->  C  =  D )   =>    |-  ( -.  D  e.  _V 
 ->  ( F `  suc  B )  =  (/) )
 
Theoremfrsucmpt2 6406* The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  (
 y  =  x  ->  E  =  C )   &    |-  (
 y  =  ( F `
  B )  ->  E  =  D )   =>    |-  (
 ( B  e.  om  /\  D  e.  V ) 
 ->  ( F `  suc  B )  =  D )
 
Theoremtz7.48lem 6407* A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( ( A 
 C_  On  /\  A. x  e.  A  A. y  e.  x  -.  ( F `
  x )  =  ( F `  y
 ) )  ->  Fun  `' ( F  |`  A ) )
 
Theoremtz7.48-2 6408* Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F )
 
Theoremtz7.48-1 6409* Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A )
 
Theoremtz7.48-3 6410* Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
 
Theoremtz7.49 6411* Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
 |-  F  Fn  On   &    |-  ( ph 
 <-> 
 A. x  e.  On  ( ( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )   =>    |-  ( ( A  e.  B  /\  ph )  ->  E. x  e.  On  ( A. y  e.  x  ( A  \  ( F " y
 ) )  =/=  (/)  /\  ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
 
Theoremtz7.49c 6412* Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
 |-  F  Fn  On   =>    |-  ( ( A  e.  B  /\  A. x  e.  On  (
 ( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
 
Syntaxcseqom 6413 Extend class notation to include index-aware recursive definitions.
 class seq𝜔 ( F ,  I )
 
Definitiondf-seqom 6414* Index-aware recursive definitions over  om. A mashup of df-rdg 6377 and df-seq 10999, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |- seq𝜔 ( F ,  I )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. ) " om )
 
Theoremseqomlem0 6415* Lemma for seq𝜔. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |- 
 rec ( ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. )  =  rec ( ( c  e. 
 om ,  d  e. 
 _V  |->  <. suc  c ,  ( c F d ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. )
 
Theoremseqomlem1 6416* Lemma for seq𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( A  e.  om 
 ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `  A ) ) >. )
 
Theoremseqomlem2 6417* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( Q " om )  Fn  om
 
Theoremseqomlem3 6418* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( ( Q
 " om ) `  (/) )  =  (  _I  `  I )
 
Theoremseqomlem4 6419* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( A  e.  om 
 ->  ( ( Q " om ) `  suc  A )  =  ( A F ( ( Q
 " om ) `  A ) ) )
 
Theoremseqomeq12 6420 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D ) )
 
Theoremfnseqom 6421 An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  G  Fn  om
 
Theoremseqom0g 6422 Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  ( I  e.  _V  ->  ( G `  (/) )  =  I )
 
Theoremseqomsuc 6423 Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( A F ( G `  A ) ) )
 
2.4.22  Abian's "most fundamental" fixed point theorem
 
Theoremabianfplem 6424* Lemma for abianfp 6425. We prove by transfinite induction that if  F has a fixed point  x, then its iterates also equal  x. This lemma is used for the "trivial" direction of the main theorem. (Contributed by NM, 4-Sep-2004.)
 |-  A  e.  _V   &    |-  G  =  rec ( ( z  e.  _V  |->  ( F `
  z ) ) ,  x )   =>    |-  ( v  e. 
 On  ->  ( ( F `
  x )  =  x  ->  ( G `  v )  =  x ) )
 
Theoremabianfp 6425* "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let  G `  0  =  x,  G `  1  =  F `  x,  G `  2  =  F `  ( F `
 x ),... be the iterates of  F. The theorem reads (using our variable names): "Let  F be a mapping from a set  A into itself. Then  F has a fixed point if and only if: There exists an element  x of  A such that for every ordinal  v,  G `  v is an element of  A, and if  G `  v is not a fixed point of  F then the  G `  u's are all distinct for every ordinal  u  e.  v." See df-rdg 6377 for the  rec operation. The proof's key idea is to assume that  F does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 5671 to derive that the class of all ordinal numbers exists, contradicting onprc 4534. Our version of this theorem does not require the hypothesis that  F be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem. (Contributed by NM, 5-Sep-2004.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  A  e.  _V   &    |-  G  =  rec ( ( z  e.  _V  |->  ( F `
  z ) ) ,  x )   =>    |-  ( E. x  e.  A  ( F `  x )  =  x  <->  E. x  e.  A  A. v  e.  On  (
 ( G `  v
 )  e.  A  /\  ( -.  ( F `  ( G `  v ) )  =  ( G `
  v )  ->  A. u  e.  v  -.  ( G `  v
 )  =  ( G `
  u ) ) ) )
 
2.4.23  Ordinal arithmetic
 
Syntaxc1o 6426 Extend the definition of a class to include the ordinal number 1.
 class  1o
 
Syntaxc2o 6427 Extend the definition of a class to include the ordinal number 2.
 class  2o
 
Syntaxc3o 6428 Extend the definition of a class to include the ordinal number 3.
 class  3o
 
Syntaxc4o 6429 Extend the definition of a class to include the ordinal number 4.
 class  4o
 
Syntaxcoa 6430 Extend the definition of a class to include the ordinal addition operation.
 class  +o
 
Syntaxcomu 6431 Extend the definition of a class to include the ordinal multiplication operation.
 class  .o
 
Syntaxcoe 6432 Extend the definition of a class to include the ordinal exponentiation operation.
 class  ^o
 
Definitiondf-1o 6433 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  =  suc  (/)
 
Definitiondf-2o 6434 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
 |- 
 2o  =  suc  1o
 
Definitiondf-3o 6435 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 3o  =  suc  2o
 
Definitiondf-4o 6436 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 4o  =  suc  3o
 
Definitiondf-oadd 6437* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
 |- 
 +o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  suc  z ) ,  x ) `  y ) )
 
Definitiondf-omul 6438* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
 |- 
 .o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  ( z  +o  x ) ) ,  (/) ) `  y ) )
 
Definitiondf-oexp 6439* Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
 |- 
 ^o  =  ( x  e.  On ,  y  e.  On  |->  if ( x  =  (/) ,  ( 1o  \  y ) ,  ( rec ( ( z  e. 
 _V  |->  ( z  .o  x ) ) ,  1o ) `  y
 ) ) )
 
Theorem1on 6440 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  On
 
Theorem2on 6441 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |- 
 2o  e.  On
 
Theorem2on0 6442 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
 |- 
 2o  =/=  (/)
 
Theorem3on 6443 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  On
 
Theorem4on 6444 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  On
 
Theoremdf1o2 6445 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
 |- 
 1o  =  { (/) }
 
Theoremdf2o3 6446 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 2o  =  { (/) ,  1o }
 
Theoremdf2o2 6447 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
 |- 
 2o  =  { (/) ,  { (/)
 } }
 
Theorem1n0 6448 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
 |- 
 1o  =/=  (/)
 
Theoremxp01disj 6449 Cross 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 6450 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
 
Theoremordge1n0 6451 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
 
Theoremel1o 6452 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  ( A  e.  1o  <->  A  =  (/) )
 
Theoremdif1o 6453 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 6454 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 6455 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 6456 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
 
Theorem0lt1o 6457 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  (/)  e.  1o
 
Theoremdif20el 6458 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  ( A  e.  ( On  \  2o )  ->  (/) 
 e.  A )
 
Theorem0we1 6459 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  (/)  We  1o
 
Theorembrwitnlem 6460 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 6461 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 6462 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 6463 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
 |- 
 ^o  Fn  ( On  X. 
 On )
 
Theoremoav 6464* 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 6465* 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 6466 A helper lemma for oe0 6475 and others. (Contributed by NM, 6-Jan-2005.)
 |-  ( ( ph  /\  A  =  (/) )  ->  ps )   &    |-  (
 ( ( A  e.  On  /\  ph )  /\  (/)  e.  A )  ->  ps )   =>    |-  ( ( A  e.  On  /\  ph )  ->  ps )
 
Theoremoev 6467* 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 6468* 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 6469 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 6470 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 6471 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 6472 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 6470, 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 6473 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
 |-  ( (/)  ^o  (/) )  =  1o
 
Theoremoe0m1 6474 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 6475 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 6476* Alternate value of ordinal exponentiation. Compare oev 6467. (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 6477 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 6478* Lemma for oesuc 6480. (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 6479 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 6480 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 6481 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 6477 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 6482 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 6483 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 6484 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 6485* 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 6486* 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 6487* 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 6488 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 6489 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 6490 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 6491 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
 |-  ( A  e.  On  ->  ( (/)  +o  A )  =  A )
 
Theoremom0r 6492 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
 |-  ( A  e.  On  ->  ( (/)  .o  A )  =  (/) )
 
Theoremo1p1e2 6493 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 |-  ( 1o  +o  1o )  =  2o
 
Theoremom1 6494 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 6495 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 6496 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
 |-  ( A  e.  On  ->  ( A  ^o  1o )  =  A )
 
Theoremoe1m 6497 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 6498 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 6499 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 6500 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 ) )
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