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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlediv12a 7601 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
 RR  RR  0  <_  <_  C  RR  D  RR  0  <  C  C  <_  D  D  <_  C
 
Theoremlediv2a 7602 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 RR  0  <  RR  0  <  C  RR  0  <_  C  <_  C  <_  C
 
Theoremreclt1 7603 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
 RR  0  <  <  1  1  <  1
 
Theoremrecgt1 7604 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
 RR  0  <  1  <  1  <  1
 
Theoremrecgt1i 7605 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
 RR  1  <  0  <  1  1  <  1
 
Theoremrecp1lt1 7606 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
 RR  0  <_  1  +  <  1
 
Theoremrecreclt 7607 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)
 RR  0  <  1  1  + 
 1  <  1  1  1  +  1  <
 
Theoremle2msq 7608 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  0  <_  <_  x.  <_  x.
 
Theoremmsq11 7609 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  0  <_  x.  x.
 
Theoremledivp1 7610 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
 RR  0  <_  RR  0  <_  +  1  x.  <_
 
Theoremsqueeze0 7611* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
 RR  0  <_  RR  0  < 
 <  0
 
Theoremltp1i 7612 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 RR   =>     <  +  1
 
Theoremrecgt0i 7613 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 RR   =>     0  <  0  <  1
 
Theoremrecgt0ii 7614 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 RR   &     0  <    =>     0  <  1
 
Theoremprodgt0i 7615 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
 RR   &     RR   =>     0  <_  0  <  x.  0  <
 
Theoremprodge0i 7616 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 RR   &     RR   =>     0  <  0  <_  x.  0  <_
 
Theoremdivgt0i 7617 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 RR   &     RR   =>     0  <  0  <  0  <
 
Theoremdivge0i 7618 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <  0  <_
 
Theoremltreci 7619 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 RR   &     RR   =>     0  <  0  <  <  1 
 <  1
 
Theoremlereci 7620 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
 RR   &     RR   =>     0  <  0  <  <_  1 
 <_  1
 
Theoremlt2msqi 7621 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <_  <  x.  <  x.
 
Theoremle2msqi 7622 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <_  <_  x.  <_  x.
 
Theoremmsq11i 7623 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
 RR   &     RR   =>     0  <_  0  <_  x.  x.
 
Theoremdivgt0i2i 7624 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     0  <    =>     0  <  0  <
 
Theoremltrecii 7625 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     <  1 
 <  1
 
Theoremdivgt0ii 7626 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <
 
Theoremltmul1i 7627 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  <  x.  C  <  x.  C
 
Theoremltdiv1i 7628 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  <  C 
 <  C
 
Theoremltmuldivi 7629 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  x.  C  <  <  C
 
Theoremltmul2i 7630 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  <  C  x.  <  C  x.
 
Theoremlemul1i 7631 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  <_  x.  C  <_  x.  C
 
Theoremlemul2i 7632 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  <_  C  x.  <_  C  x.
 
Theoremltdiv23i 7633 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 RR   &     RR   &     C  RR   =>     0  <  0  <  C  <  C  C 
 <
 
Theoremltdiv23ii 7634 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 RR   &     RR   &     C  RR   &     0  <    &     0  <  C   =>     <  C  C  <
 
Theoremltmul1ii 7635 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
 RR   &     RR   &     C  RR   &     0  <  C   =>     <  x.  C  <  x.  C
 
Theoremltdiv1ii 7636 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     C  RR   &     0  <  C   =>     <  C 
 <  C
 
Theoremltp1d 7637 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     <  +  1
 
Theoremlep1d 7638 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     <_  +  1
 
Theoremltm1d 7639 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     -  1  <
 
Theoremlem1d 7640 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     -  1  <_
 
Theoremrecgt0d 7641 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     0  <    =>     0  <  1
 
Theoremdivgt0d 7642 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <
 
Theoremmulgt1d 7643 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     1  <    &     1  <    =>     1  <  x.
 
Theoremlemulge11d 7644 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     1  <_    =>     <_  x.
 
Theoremlemulge12d 7645 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     1  <_    =>     <_  x.
 
Theoremlemul1ad 7646 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR   &     0  <_  C   &     <_    =>     x.  C  <_  x.  C
 
Theoremlemul2ad 7647 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR   &     0  <_  C   &     <_    =>     C  x.  <_  C  x.
 
Theoremltmul12ad 7648 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     0  <_    &     <    &     0  <_  C   &     C  <  D   =>     x.  C  <  x.  D
 
Theoremlemul12ad 7649 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     0  <_    &     0  <_  C   &     <_    &     C  <_  D   =>     x.  C  <_  x.  D
 
Theoremlemul12bd 7650 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     0  <_    &     0  <_  D   &     <_    &     C  <_  D   =>     x.  C  <_  x.  D
 
3.3.10  Imaginary and complex number properties
 
Theoremcrap0 7651 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
 RR  RR #  0 #  0  +  _i  x. #  0
 
Theoremcreur 7652* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  RR  RR  +  _i  x.
 
Theoremcreui 7653* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  RR  RR  +  _i  x.
 
Theoremcju 7654* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
 CC  CC  +  RR  _i  x.  -  RR
 
3.4  Integer sets
 
3.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 7655 Extend class notation to include the class of positive integers.

 NN
 
Definitiondf-inn 7656* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7657 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)

 NN  |^| {  |  1  +  1  }
 
Theoremdfnn2 7657* Definition of the set of positive integers. Another name for df-inn 7656. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

 NN  |^| {  |  1  +  1  }
 
Theorempeano5nni 7658* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 1  +  1  NN  C_
 
Theoremnnssre 7659 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

 NN  C_  RR
 
Theoremnnsscn 7660 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

 NN  C_  CC
 
Theoremnnex 7661 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

 NN  _V
 
Theoremnnre 7662 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 NN  RR
 
Theoremnncn 7663 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 NN  CC
 
Theoremnnrei 7664 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 NN   =>     RR
 
Theoremnncni 7665 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 NN   =>     CC
 
Theorem1nn 7666 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
 1  NN
 
Theorempeano2nn 7667 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 NN  +  1  NN
 
Theoremnnred 7668 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     RR
 
Theoremnncnd 7669 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     CC
 
Theorempeano2nnd 7670 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     +  1 
 NN
 
3.4.2  Principle of mathematical induction
 
Theoremnnind 7671* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 7675 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 1    &       &     +  1    &       &       &     NN    =>     NN
 
TheoremnnindALT 7672* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 7671 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

 NN    &       &     1    &       &     +  1    &       =>     NN
 
Theoremnn1m1nn 7673 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 NN  1  -  1  NN
 
Theoremnn1suc 7674* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 1    &     +  1    &       &       &     NN    =>     NN
 
Theoremnnaddcl 7675 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
 NN  NN  +  NN
 
Theoremnnmulcl 7676 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
 NN  NN  x.  NN
 
Theoremnnmulcli 7677 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN   &     NN   =>     x.  NN
 
Theoremnnge1 7678 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
 NN  1  <_
 
Theoremnnle1eq1 7679 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
 NN  <_  1  1
 
Theoremnngt0 7680 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
 NN  0  <
 
Theoremnnnlt1 7681 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 NN  <  1
 
Theorem0nnn 7682 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
 0  NN
 
Theoremnnne0 7683 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
 NN  =/=  0
 
Theoremnnap0 7684 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
 NN #  0
 
Theoremnngt0i 7685 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 NN   =>     0  <
 
Theoremnnne0i 7686 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 NN   =>     =/=  0
 
Theoremnn2ge 7687* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
 NN  NN  NN  <_  <_
 
Theoremnn1gt1 7688 A positive integer is either one or greater than one. This is for  NN; 0elnn 4283 is a similar theorem for  om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
 NN  1  1  <
 
Theoremnngt1ne1 7689 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
 NN  1  <  =/=  1
 
Theoremnndivre 7690 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
 RR  N  NN  N  RR
 
Theoremnnrecre 7691 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
 N  NN  1  N  RR
 
Theoremnnrecgt0 7692 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
 NN  0  <  1
 
Theoremnnsub 7693 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
 NN  NN  <  -  NN
 
Theoremnnsubi 7694 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
 NN   &     NN   =>     <  -  NN
 
Theoremnndiv 7695* Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 NN  NN  NN  x.  NN
 
Theoremnndivtr 7696 Transitive property of divisibility: if divides and divides  C, then divides  C. Typically,  C would be an integer, although the theorem holds for complex  C. (Contributed by NM, 3-May-2005.)
 NN  NN  C  CC  NN  C 
 NN  C  NN
 
Theoremnnge1d 7697 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     1  <_
 
Theoremnngt0d 7698 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     0  <
 
Theoremnnne0d 7699 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     =/=  0
 
Theoremnnrecred 7700 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>    
 1  RR
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