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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodge0 7601 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR 
 0  <  0  <_  x.  0  <_
 
Theoremprodge02 7602 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 RR  RR 
 0  <  0  <_  x.  0  <_
 
Theoremltmul2 7603 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
 RR  RR  C  RR  0  <  C 
 <  C  x.  <  C  x.
 
Theoremlemul2 7604 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
 RR  RR  C  RR  0  <  C 
 <_  C  x.  <_  C  x.
 
Theoremlemul1a 7605 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)
 RR  RR  C  RR  0  <_  C  <_  x.  C  <_  x.  C
 
Theoremlemul2a 7606 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 RR  RR  C  RR  0  <_  C  <_  C  x.  <_  C  x.
 
Theoremltmul12a 7607 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
 RR  RR  0  <_  <  C  RR  D  RR  0  <_  C  C  <  D  x.  C  <  x.  D
 
Theoremlemul12b 7608 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 RR  0  <_  RR  C  RR  D  RR  0  <_  D  <_  C  <_  D  x.  C  <_  x.  D
 
Theoremlemul12a 7609 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 RR  0  <_  RR  C  RR  0  <_  C  D  RR 
 <_  C  <_  D  x.  C  <_  x.  D
 
Theoremmulgt1 7610 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
 RR  RR 
 1  <  1  <  1  <  x.
 
Theoremltmulgt11 7611 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 RR  RR  0  <  1  <  <  x.
 
Theoremltmulgt12 7612 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 RR  RR  0  <  1  <  <  x.
 
Theoremlemulge11 7613 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
 RR  RR 
 0  <_  1  <_  <_  x.
 
Theoremlemulge12 7614 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
 RR  RR 
 0  <_  1  <_  <_  x.
 
Theoremltdiv1 7615 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  0  <  C 
 <  C  <  C
 
Theoremlediv1 7616 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C 
 <_  C  <_  C
 
Theoremgt0div 7617 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
 RR  RR  0  <  0  <  0  <
 
Theoremge0div 7618 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
 RR  RR  0  <  0  <_  0  <_
 
Theoremdivgt0 7619 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
 RR  0  <  RR  0  <  0  <
 
Theoremdivge0 7620 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
 RR  0  <_  RR  0  <  0  <_
 
Theoremltmuldiv 7621 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  0  <  C  x.  C 
 <  <  C
 
Theoremltmuldiv2 7622 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C  C  x. 
 <  <  C
 
Theoremltdivmul 7623 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 RR  RR  C  RR  0  <  C  C 
 <  <  C  x.
 
Theoremledivmul 7624 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 RR  RR  C  RR  0  <  C  C 
 <_  <_  C  x.
 
Theoremltdivmul2 7625 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
 RR  RR  C  RR  0  <  C  C 
 <  <  x.  C
 
Theoremlt2mul2div 7626 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
 RR  RR  0  <  C  RR  D  RR  0  <  D  x.  <  C  x.  D  D  <  C
 
Theoremledivmul2 7627 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 RR  RR  C  RR  0  <  C  C 
 <_  <_  x.  C
 
Theoremlemuldiv 7628 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 RR  RR  C  RR  0  <  C  x.  C 
 <_  <_  C
 
Theoremlemuldiv2 7629 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 RR  RR  C  RR  0  <  C  C  x. 
 <_  <_  C
 
Theoremltrec 7630 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <  RR  0  <  <  1  < 
 1
 
Theoremlerec 7631 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  0  <  RR  0  <  <_  1  <_ 
 1
 
Theoremlt2msq1 7632 Lemma for lt2msq 7633. (Contributed by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  <  x.  <  x.
 
Theoremlt2msq 7633 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <_  RR  0  <_  <  x.  <  x.
 
Theoremltdiv2 7634 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
 RR  0  <  RR  0  <  C  RR  0  <  C  <  C  <  C
 
Theoremltrec1 7635 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
 RR  0  <  RR  0  <  1  <  1  <
 
Theoremlerec2 7636 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
 RR  0  <  RR  0  <  <_  1  <_  1
 
Theoremledivdiv 7637 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
 RR  0  <  RR  0  <  C  RR  0  <  C  D  RR  0  <  D 
 <_  C  D  D  C  <_
 
Theoremlediv2 7638 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
 RR  0  <  RR  0  <  C  RR  0  <  C  <_  C  <_  C
 
Theoremltdiv23 7639 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
 RR  RR  0  <  C  RR  0  <  C 
 <  C  C  <
 
Theoremlediv23 7640 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
 RR  RR  0  <  C  RR  0  <  C 
 <_  C  C  <_
 
Theoremlediv12a 7641 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 7642 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 7643 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 7644 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 7645 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 7646 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
 RR  0  <_  1  +  <  1
 
Theoremrecreclt 7647 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 7648 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 7649 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 7650 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 7651* 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 7652 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 RR   =>     <  +  1
 
Theoremrecgt0i 7653 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 7654 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 7655 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 7656 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 7657 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 RR   &     RR   =>     0  <  0  <  0  <
 
Theoremdivge0i 7658 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <  0  <_
 
Theoremltreci 7659 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 RR   &     RR   =>     0  <  0  <  <  1 
 <  1
 
Theoremlereci 7660 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
 RR   &     RR   =>     0  <  0  <  <_  1 
 <_  1
 
Theoremlt2msqi 7661 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <_  <  x.  <  x.
 
Theoremle2msqi 7662 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
 RR   &     RR   =>     0  <_  0  <_  <_  x.  <_  x.
 
Theoremmsq11i 7663 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 7664 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 RR   &     RR   &     0  <    =>     0  <  0  <
 
Theoremltrecii 7665 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     <  1 
 <  1
 
Theoremdivgt0ii 7666 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <
 
Theoremltmul1i 7667 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 7668 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 7669 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
 RR   &     RR   &     C  RR   =>     0  <  C  x.  C  <  <  C
 
Theoremltmul2i 7670 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 7671 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 7672 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 7673 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 RR   &     RR   &     C  RR   =>     0  <  0  <  C  <  C  C 
 <
 
Theoremltdiv23ii 7674 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 RR   &     RR   &     C  RR   &     0  <    &     0  <  C   =>     <  C  C  <
 
Theoremltmul1ii 7675 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 7676 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 7677 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     <  +  1
 
Theoremlep1d 7678 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     <_  +  1
 
Theoremltm1d 7679 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     -  1  <
 
Theoremlem1d 7680 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     -  1  <_
 
Theoremrecgt0d 7681 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 7682 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <
 
Theoremmulgt1d 7683 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 7684 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     1  <_    =>     <_  x.
 
Theoremlemulge12d 7685 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     1  <_    =>     <_  x.
 
Theoremlemul1ad 7686 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 7687 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 7688 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 7689 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 7690 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 7691 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 7692* 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 7693* 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 7694* 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 7695 Extend class notation to include the class of positive integers.

 NN
 
Definitiondf-inn 7696* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7697 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)

 NN  |^| {  |  1  +  1  }
 
Theoremdfnn2 7697* Definition of the set of positive integers. Another name for df-inn 7696. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

 NN  |^| {  |  1  +  1  }
 
Theorempeano5nni 7698* 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 7699 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 7700 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

 NN  C_  CC
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