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Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgruf 8401 A Grothendieck's universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  U  /\  F : A --> U ) 
 ->  F  e.  U )
 
Theoremgruen 8402 A Grothendieck's universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  C_  U  /\  ( B  e.  U  /\  B  ~~  A ) )  ->  A  e.  U )
 
Theoremgruwun 8403 A nonempty Grothendieck's universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  U  e. WUni )
 
Theoremintgru 8404 The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |-  ( ( A  C_  Univ  /\  A  =/=  (/) )  ->  |^| A  e.  Univ )
 
Theoremingru 8405* The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( Tr  A  /\  A. x  e.  A  ( ~P x  e.  A  /\  A. y  e.  A  { x ,  y }  e.  A  /\  A. y
 ( y : x --> A  ->  U. ran  y  e.  A ) ) ) 
 ->  ( U  e.  Univ  ->  ( U  i^i  A )  e.  Univ ) )
 
Theoremwfgru 8406 The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( U  e.  Univ  ->  ( U  i^i  U. ( R1 " On ) )  e.  Univ )
 
Theoremgrudomon 8407 Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( U  e.  Univ  /\  A  e.  On  /\  ( B  e.  U  /\  A  ~<_  B ) ) 
 ->  A  e.  U )
 
Theoremgruina 8408 If a Grothendieck's universe  U is nonempty, then the height of the ordinals in  U is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  A  =  ( U  i^i  On )   =>    |-  ( ( U  e.  Univ  /\  U  =/=  (/) )  ->  A  e.  Inacc
 )
 
Theoremgrur1a 8409 A characterization of Grothendieck's universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  A  =  ( U  i^i  On )   =>    |-  ( U  e.  Univ 
 ->  ( R1 `  A )  C_  U )
 
Theoremgrur1 8410 A characterization of Grothendieck's universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  A  =  ( U  i^i  On )   =>    |-  ( ( U  e.  Univ  /\  U  e.  U. ( R1 " On ) )  ->  U  =  ( R1 `  A ) )
 
Theoremgrutsk1 8411 Grothendieck's universes are the same as transitive Tarski's classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 8373.) (Contributed by Mario Carneiro, 17-Jun-2013.)
 |-  ( ( T  e.  Tarski  /\ 
 Tr  T )  ->  T  e.  Univ )
 
Theoremgrutsk 8412 Grothendieck's universes are the same as transitive Tarski's classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.)
 |- 
 Univ  =  { x  e.  Tarski  |  Tr  x }
 
4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
 
4.2.1  Introduce the Tarksi-Grothendieck Axiom
 
Axiomax-groth 8413* The Tarksi-Grothendieck Axiom. For every set  x there is an inaccessible cardinal  y such that  y is not in  x. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 8424. An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( A. w ( w  C_  z  ->  w  e.  y
 )  /\  E. w  e.  y  A. v ( v  C_  z  ->  v  e.  w ) ) 
 /\  A. z ( z 
 C_  y  ->  (
 z  ~~  y  \/  z  e.  y )
 ) )
 
Theoremaxgroth5 8414* The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
 ( z  ~~  y  \/  z  e.  y
 ) )
 
Theoremaxgroth2 8415* Alternate version of the Tarksi-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( A. w ( w  C_  z  ->  w  e.  y
 )  /\  E. w  e.  y  A. v ( v  C_  z  ->  v  e.  w ) ) 
 /\  A. z ( z 
 C_  y  ->  (
 y  ~<_  z  \/  z  e.  y ) ) )
 
4.2.2  Derive the Power Set, Infinity and Choice Axioms
 
Theoremgrothpw 8416* Derive the Axiom of Power Sets ax-pow 4160 from the Tarksi-Grothendieck axiom ax-groth 8413. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html. Note that ax-pow 4160 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.)
 |- 
 E. y A. z
 ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
 
Theoremgrothpwex 8417 Derive the Axiom of Power Sets from the Tarksi-Grothendieck axiom ax-groth 8413. Note that ax-pow 4160 is not used by the proof. Use axpweq 4159 to obtain ax-pow 4160. (Contributed by Gérard Lang, 22-Jun-2009.)
 |- 
 ~P x  e.  _V
 
Theoremaxgroth6 8418* The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set  x, there exists a set  y containing  x, the subsets of the members of  y, the power sets of the members of  y, and the subsets of  y of cardinality less than that of  y. (Contributed by NM, 21-Jun-2009.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  ~P z  e.  y ) 
 /\  A. z  e.  ~P  y ( z  ~<  y 
 ->  z  e.  y
 ) )
 
Theoremgrothomex 8419 The Tarksi-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 7312). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.)
 |- 
 om  e.  _V
 
Theoremgrothac 8420 The Tarksi-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8064). This can be put in a more conventional form via ween 7630 and dfac8 7729. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.)
 |- 
 dom  card  =  _V
 
Theoremaxgroth3 8421* Alternate version of the Tarksi-Grothendieck Axiom. ax-cc 8029 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  ( A. w ( w  C_  z  ->  w  e.  y
 )  /\  E. w  e.  y  A. v ( v  C_  z  ->  v  e.  w ) ) 
 /\  A. z ( z 
 C_  y  ->  (
 ( y  \  z
 )  ~<_  z  \/  z  e.  y ) ) )
 
Theoremaxgroth4 8422* Alternate version of the Tarksi-Grothendieck Axiom. ax-ac 8053 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z  e.  y  E. v  e.  y  A. w ( w  C_  z  ->  w  e.  (
 y  i^i  v )
 )  /\  A. z ( z  C_  y  ->  ( ( y  \  z
 )  ~<_  z  \/  z  e.  y ) ) )
 
Theoremgrothprimlem 8423* Lemma for grothprim 8424. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
 |-  ( { u ,  v }  e.  w  <->  E. g ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v )
 ) ) )
 
Theoremgrothprim 8424* The Tarksi-Grothendieck Axiom ax-groth 8413 expanded into set theory primitives using 163 symbols (allowing the defined symbols  /\,  \/,  <->, and  E.). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
 |- 
 E. y ( x  e.  y  /\  A. z ( ( z  e.  y  ->  E. v
 ( v  e.  y  /\  A. w ( A. u ( u  e.  w  ->  u  e.  z )  ->  ( w  e.  y  /\  w  e.  v ) ) ) )  /\  E. w ( ( w  e.  z  ->  w  e.  y )  ->  ( A. v ( ( v  e.  z  ->  E. t A. u ( E. g
 ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  v  \/  h  =  u ) ) ) 
 ->  u  =  t
 ) )  /\  (
 v  e.  y  ->  ( v  e.  z  \/  E. u ( u  e.  z  /\  E. g ( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v )
 ) ) ) ) ) )  \/  z  e.  y ) ) ) )
 
Theoremgrothtsk 8425 The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
 |- 
 U. Tarski  =  _V
 
Theoreminaprc 8426 An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)
 |- 
 Inacc  e/  _V
 
4.2.3  Tarski map function
 
Syntaxctskm 8427 Extend class definition to include the map whose value is the smallest Tarski's class.
 class  tarskiMap
 
Definitiondf-tskm 8428* A function that maps a set  x to the smallest Tarski's class that contains the set. (Contributed by FL, 30-Dec-2010.)
 |-  tarskiMap 
 =  ( x  e. 
 _V  |->  |^| { y  e.  Tarski  |  x  e.  y } )
 
Theoremtskmval 8429* Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
 
Theoremtskmid 8430 The set  A is an element of the smallest Tarski's class that contains  A. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A ) )
 
Theoremtskmcl 8431 A Tarski's class that contains  A is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( tarskiMap `  A )  e.  Tarski
 
Theoremsstskm 8432* Being a part of  ( tarskiMap `  A
). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A )  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) ) )
 
Theoremeltskm 8433* Belonging to  ( tarskiMap `  A
). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
 |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A )  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )
 ) )
 
PART 5  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers. We first construct and axiomitize real and complex numbers (e.g., ax-resscn 8762). After that we derive their basic properties, various operations like addition (df-plus 8716) and sine (df-sin 12313), and subsets such as the integers (df-z 9992) and natural numbers (df-n 9715).

 
5.1  Construction and axiomatization of real and complex numbers
 
5.1.1  Dedekind-cut construction of real and complex numbers
 
Syntaxcnpi 8434 The set of positive integers, which is the set of natural numbers  om with 0 removed.

Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 8734. The actual set of Dedekind cuts is defined by df-np 8573.

 class  N.
 
Syntaxcpli 8435 Positive integer addition.
 class  +N
 
Syntaxcmi 8436 Positive integer multiplication.
 class  .N
 
Syntaxclti 8437 Positive integer ordering relation.
 class  <N
 
Syntaxcplpq 8438 Positive pre-fraction addition.
 class  +pQ
 
Syntaxcmpq 8439 Positive pre-fraction multiplication.
 class  .pQ
 
Syntaxcltpq 8440 Positive pre-fraction ordering relation.
 class  <pQ
 
Syntaxceq 8441 Equivalence class used to construct positive fractions.
 class  ~Q
 
Syntaxcnq 8442 Set of positive fractions.
 class  Q.
 
Syntaxc1q 8443 The positive fraction constant 1.
 class  1Q
 
Syntaxcerq 8444 Positive fraction equivalence class.
 class  /Q
 
Syntaxcplq 8445 Positive fraction addition.
 class  +Q
 
Syntaxcmq 8446 Positive fraction multiplication.
 class  .Q
 
Syntaxcrq 8447 Positive fraction reciprocal operation.
 class  *Q
 
Syntaxcltq 8448 Positive fraction ordering relation.
 class  <Q
 
Syntaxcnp 8449 Set of positive reals.
 class  P.
 
Syntaxc1p 8450 Positive real constant 1.
 class  1P
 
Syntaxcpp 8451 Positive real addition.
 class  +P.
 
Syntaxcmp 8452 Positive real multiplication.
 class  .P.
 
Syntaxcltp 8453 Positive real ordering relation.
 class  <P
 
Syntaxcplpr 8454 Signed real pre-addition.
 class  +pR
 
Syntaxcmpr 8455 Signed real pre-multiplication.
 class  .pR
 
Syntaxcer 8456 Equivalence class used to construct signed reals.
 class  ~R
 
Syntaxcnr 8457 Set of signed reals.
 class  R.
 
Syntaxc0r 8458 The signed real constant 0.
 class  0R
 
Syntaxc1r 8459 The signed real constant 1.
 class  1R
 
Syntaxcm1r 8460 The signed real constant -1.
 class  -1R
 
Syntaxcplr 8461 Signed real addition.
 class  +R
 
Syntaxcmr 8462 Signed real multiplication.
 class  .R
 
Syntaxcltr 8463 Signed real ordering relation.
 class  <R
 
Definitiondf-ni 8464 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
 |- 
 N.  =  ( om  \  { (/) } )
 
Definitiondf-pli 8465 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
 |- 
 +N  =  (  +o  |`  ( N.  X.  N. ) )
 
Definitiondf-mi 8466 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
 |- 
 .N  =  (  .o  |`  ( N.  X.  N. ) )
 
Definitiondf-lti 8467 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.)
 |- 
 <N  =  (  _E  i^i  ( N.  X.  N. ) )
 
Theoremelni 8468 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/= 
 (/) ) )
 
Theoremelni2 8469 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A ) )
 
Theorempinn 8470 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  A  e.  om )
 
Theorempion 8471 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  A  e.  On )
 
Theorempiord 8472 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  Ord  A )
 
Theoremniex 8473 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
 |- 
 N.  e.  _V
 
Theorem0npi 8474 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  N.
 
Theorem1pi 8475 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
 |- 
 1o  e.  N.
 
Theoremaddpiord 8476 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
 
Theoremmulpiord 8477 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  =  ( A  .o  B ) )
 
Theoremmulidpi 8478 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  ( A  .N  1o )  =  A )
 
Theoremltpiord 8479 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B )
 )
 
Theoremltsopi 8480 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 <N  Or  N.
 
Theoremltrelpi 8481 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)
 |- 
 <N  C_  ( N.  X.  N. )
 
Theoremdmaddpi 8482 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  +N  =  ( N.  X.  N. )
 
Theoremdmmulpi 8483 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  .N  =  ( N.  X.  N. )
 
Theoremaddclpi 8484 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  e.  N. )
 
Theoremmulclpi 8485 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  e.  N. )
 
Theoremaddcompi 8486 Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( A  +N  B )  =  ( B  +N  A )
 
Theoremaddasspi 8487 Addition of positive integers is associative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  +N  B )  +N  C )  =  ( A  +N  ( B  +N  C ) )
 
Theoremmulcompi 8488 Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
 |-  ( A  .N  B )  =  ( B  .N  A )
 
Theoremmulasspi 8489 Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  .N  B )  .N  C )  =  ( A  .N  ( B  .N  C ) )
 
Theoremdistrpi 8490 Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
 |-  ( A  .N  ( B  +N  C ) )  =  ( ( A  .N  B )  +N  ( A  .N  C ) )
 
Theoremaddcanpi 8491 Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C ) 
 <->  B  =  C ) )
 
Theoremmulcanpi 8492 Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( ( A  .N  B )  =  ( A  .N  C ) 
 <->  B  =  C ) )
 
Theoremaddnidpi 8493 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  -.  ( A  +N  B )  =  A )
 
Theoremltexpi 8494* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  E. x  e.  N.  ( A  +N  x )  =  B )
 )
 
Theoremltapi 8495 Ordering property of addition for positive integers. (Contributed by NM, 7-Mar-1996.) (New usage is discouraged.)
 |-  ( C  e.  N.  ->  ( A  <N  B  <->  ( C  +N  A )  <N  ( C  +N  B ) ) )
 
Theoremltmpi 8496 Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)
 |-  ( C  e.  N.  ->  ( A  <N  B  <->  ( C  .N  A )  <N  ( C  .N  B ) ) )
 
Theorem1lt2pi 8497 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pi 8498 No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  A  <N  1o
 
Theoremindpi 8499* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
 |-  ( x  =  1o  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +N  1o )  ->  ( ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  N.  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  N.  ->  ta )
 
Definitiondf-plpq 8500* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition  +Q (df-plq 8506) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 8503). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
 |- 
 +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |-> 
 <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
 ) ) >. )
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