HomeHome Metamath Proof Explorer
Theorem List (p. 85 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulasspi 8401 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 8402 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 8403 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 8404 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 8405 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 8406* 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 8407 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 8408 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 8409 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pi 8410 No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  A  <N  1o
 
Theoremindpi 8411* 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 8412* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8418) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 8415). (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
 ) ) >. )
 
Definitiondf-mpq 8413* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
 |- 
 .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |-> 
 <. ( ( 1st `  x )  .N  ( 1st `  y
 ) ) ,  (
 ( 2nd `  x )  .N  ( 2nd `  y
 ) ) >. )
 
Definitiondf-ltpq 8414* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
 |- 
 <pQ  =  { <. x ,  y >.  |  ( ( x  e.  ( N. 
 X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( ( 1st `  x )  .N  ( 2nd `  y
 ) )  <N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) }
 
Definitiondf-enq 8415* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |- 
 ~Q  =  { <. x ,  y >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X. 
 N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .N  u )  =  ( w  .N  v ) ) ) }
 
Definitiondf-nq 8416* Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) (New usage is discouraged.)
 |- 
 Q.  =  { x  e.  ( N.  X.  N. )  |  A. y  e.  ( N.  X.  N. ) ( x  ~Q  y  ->  -.  ( 2nd `  y )  <N  ( 2nd `  x ) ) }
 
Definitiondf-erq 8417 Define a convenience function that "reduces" a fraction to lowest terms. Note that in this form, it is not obviously a function; we prove this in nqerf 8434. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |- 
 /Q  =  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. )
 )
 
Definitiondf-plq 8418 Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)
 |- 
 +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) )
 
Definitiondf-mq 8419 Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.) (New usage is discouraged.)
 |- 
 .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) )
 
Definitiondf-1nq 8420 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
 |- 
 1Q  =  <. 1o ,  1o >.
 
Definitiondf-rq 8421 Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by NM, 6-Mar-1996.) (New usage is discouraged.)
 |- 
 *Q  =  ( `' 
 .Q  " { 1Q }
 )
 
Definitiondf-ltnq 8422 Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) (New usage is discouraged.)
 |- 
 <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
 
Theoremenqbreq 8423 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  ~Q  <. C ,  D >.  <-> 
 ( A  .N  D )  =  ( B  .N  C ) ) )
 
Theoremenqbreq2 8424 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B ) )  =  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
 
Theoremenqer 8425 The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
 |- 
 ~Q  Er  ( N.  X. 
 N. )
 
Theoremenqex 8426 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |- 
 ~Q  e.  _V
 
Theoremnqex 8427 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 8428 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  Q.
 
Theoremelpqn 8429 Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  A  e.  ( N. 
 X.  N. ) )
 
Theoremltrelnq 8430 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 <Q  C_  ( Q.  X.  Q. )
 
Theorempinq 8431 The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  N.  -> 
 <. A ,  1o >.  e. 
 Q. )
 
Theorem1nq 8432 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |- 
 1Q  e.  Q.
 
Theoremnqereu 8433* There is a unique element of  Q. equivalent to each element of  N.  X.  N.. (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  E! x  e.  Q.  x  ~Q  A )
 
Theoremnqerf 8434 Corollary of nqereu 8433: the function  /Q is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |- 
 /Q : ( N. 
 X.  N. ) --> Q.
 
Theoremnqercl 8435 Corollary of nqereu 8433: closure of  /Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q `  A )  e.  Q. )
 
Theoremnqerrel 8436 Any member of  ( N.  X.  N. ) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
 
Theoremnqerid 8437 Corollary of nqereu 8433: the function  /Q acts as the identity on members of  Q.. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
 
Theoremenqeq 8438 Corollary of nqereu 8433: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  A  ~Q  B ) 
 ->  A  =  B )
 
Theoremnqereq 8439 The function  /Q acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  ~Q  B  <->  ( /Q `  A )  =  ( /Q `  B ) ) )
 
Theoremaddpipq2 8440 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  +pQ  B )  =  <. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  (
 ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) >. )
 
Theoremaddpipq 8441 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  +pQ  <. C ,  D >. )  =  <. ( ( A  .N  D )  +N  ( C  .N  B ) ) ,  ( B  .N  D ) >. )
 
Theoremaddpqnq 8442 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( /Q `  ( A  +pQ  B ) ) )
 
Theoremmulpipq2 8443 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  .pQ  B )  =  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
 >. )
 
Theoremmulpipq 8444 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )
 
Theoremmulpqnq 8445 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
 
Theoremordpipq 8446 Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( <. A ,  B >. 
 <pQ  <. C ,  D >.  <-> 
 ( A  .N  D )  <N  ( C  .N  B ) )
 
Theoremordpinq 8447 Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( ( 1st `  A )  .N  ( 2nd `  B ) )  <N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
 
Theoremaddpqf 8448 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |- 
 +pQ  : ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
 
Theoremaddclnq 8449 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulpqf 8450 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |- 
 .pQ  : ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
 
Theoremmulclnq 8451 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremaddnqf 8452 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 +Q  : ( Q. 
 X.  Q. ) --> Q.
 
Theoremmulnqf 8453 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 .Q  : ( Q. 
 X.  Q. ) --> Q.
 
Theoremaddcompq 8454 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  +pQ  B )  =  ( B  +pQ  A )
 
Theoremaddcomnq 8455 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  +Q  B )  =  ( B  +Q  A )
 
Theoremmulcompq 8456 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .pQ  B )  =  ( B  .pQ  A )
 
Theoremmulcomnq 8457 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .Q  B )  =  ( B  .Q  A )
 
Theoremadderpqlem 8458 Lemma for adderpq 8460. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A 
 ~Q  B  <->  ( A  +pQ  C )  ~Q  ( B 
 +pQ  C ) ) )
 
Theoremmulerpqlem 8459 Lemma for mulerpq 8461. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A 
 ~Q  B  <->  ( A  .pQ  C )  ~Q  ( B 
 .pQ  C ) ) )
 
Theoremadderpq 8460 Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( /Q `  A )  +Q  ( /Q `  B ) )  =  ( /Q `  ( A  +pQ  B ) )
 
Theoremmulerpq 8461 Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( /Q `  A )  .Q  ( /Q `  B ) )  =  ( /Q `  ( A  .pQ  B ) )
 
Theoremaddassnq 8462 Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) )
 
Theoremmulassnq 8463 Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )
 
Theoremmulcanenq 8464 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  -> 
 <. ( A  .N  B ) ,  ( A  .N  C ) >.  ~Q  <. B ,  C >. )
 
Theoremdistrnq 8465 Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) )
 
Theorem1nqenq 8466 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  1Q  ~Q  <. A ,  A >. )
 
Theoremmulidnq 8467 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecmulnq 8468 Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( ( *Q `  A )  =  B  <->  ( A  .Q  B )  =  1Q ) )
 
Theoremrecidnq 8469 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
 
Theoremrecclnq 8470 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( *Q `  A )  e.  Q. )
 
Theoremrecrecnq 8471 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )
 
Theoremdmrecnq 8472 Domain of reciprocal on positive fractions. (Contributed by Mario Carneiro, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 dom  *Q  =  Q.
 
Theoremltsonq 8473 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
 |- 
 <Q  Or  Q.
 
Theoremlterpq 8474 Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
 |-  ( A  <pQ  B  <->  ( /Q `  A )  <Q  ( /Q
 `  B ) )
 
Theoremltanq 8475 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
 
Theoremltmnq 8476 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
 
Theorem1lt2nq 8477 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |- 
 1Q  <Q  ( 1Q  +Q  1Q )
 
Theoremltaddnq 8478 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
 
Theoremltexnq 8479* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x ( A  +Q  x )  =  B ) )
 
Theoremhalfnq 8480* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x ( x  +Q  x )  =  A )
 
Theoremnsmallnq 8481* The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
 
Theoremltbtwnnq 8482* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  E. x ( A 
 <Q  x  /\  x  <Q  B ) )
 
Theoremltrnq 8483 Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  ( *Q `  B )  <Q  ( *Q `  A ) )
 
Theoremarchnq 8484* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >.
 )
 
Definitiondf-np 8485* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  =  { x  |  ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
 <Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
 
Definitiondf-1p 8486 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |- 
 1P  =  { x  |  x  <Q  1Q }
 
Definitiondf-plp 8487* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u ) } )
 
Definitiondf-mp 8488* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
 
Definitiondf-ltp 8489* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
 
Theoremnpex 8490 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  e.  _V
 
Theoremelnp 8491* Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  (
 A. y ( y 
 <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremelnpi 8492* Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  _V  /\  (/)  C.  A  /\  A  C. 
 Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x 
 ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremprn0 8493 A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  =/=  (/) )
 
Theoremprpssnq 8494 A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  C.  Q. )
 
Theoremelprnq 8495 A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  B  e.  Q. )
 
Theorem0npr 8496 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnq 8497 A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  ( C  <Q  B 
 ->  C  e.  A ) )
 
Theoremprub 8498 A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  B 
 <Q  C ) )
 
Theoremprnmax 8499* A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
 
Theoremnpomex 8500 A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of  P. is an infinite set, the negation of Infinity implies that  P., and hence 
RR, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 8497 and nsmallnq 8481). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  om  e.  _V )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >