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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltmul12a 7901 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
 |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B ) )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  < 
 ( B  x.  D ) )
 
Theoremlemul12b 7902 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <_  D )
 ) )  ->  (
 ( A  <_  B  /\  C  <_  D )  ->  ( A  x.  C )  <_  ( B  x.  D ) ) )
 
Theoremlemul12a 7903 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR )  /\  ( ( C  e.  RR  /\  0  <_  C )  /\  D  e.  RR ) )  ->  ( ( A  <_  B  /\  C  <_  D )  ->  ( A  x.  C )  <_  ( B  x.  D ) ) )
 
Theoremmulgt1 7904 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 1  <  A  /\  1  <  B ) ) 
 ->  1  <  ( A  x.  B ) )
 
Theoremltmulgt11 7905 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A ) 
 ->  ( 1  <  B  <->  A  <  ( A  x.  B ) ) )
 
Theoremltmulgt12 7906 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A ) 
 ->  ( 1  <  B  <->  A  <  ( B  x.  A ) ) )
 
Theoremlemulge11 7907 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  1  <_  B ) ) 
 ->  A  <_  ( A  x.  B ) )
 
Theoremlemulge12 7908 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  1  <_  B ) ) 
 ->  A  <_  ( B  x.  A ) )
 
Theoremltdiv1 7909 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
Theoremlediv1 7910 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( A  /  C )  <_  ( B 
 /  C ) ) )
 
Theoremgt0div 7911 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B ) 
 ->  ( 0  <  A  <->  0  <  ( A  /  B ) ) )
 
Theoremge0div 7912 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  B ) 
 ->  ( 0  <_  A  <->  0 
 <_  ( A  /  B ) ) )
 
Theoremdivgt0 7913 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
 0  <  ( A  /  B ) )
 
Theoremdivge0 7914 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
 0  <_  ( A  /  B ) )
 
Theoremltmuldiv 7915 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  x.  C )  <  B 
 <->  A  <  ( B 
 /  C ) ) )
 
Theoremltmuldiv2 7916 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  A )  <  B 
 <->  A  <  ( B 
 /  C ) ) )
 
Theoremltdivmul 7917 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  C )  <  B 
 <->  A  <  ( C  x.  B ) ) )
 
Theoremledivmul 7918 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  C )  <_  B 
 <->  A  <_  ( C  x.  B ) ) )
 
Theoremltdivmul2 7919 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  C )  <  B 
 <->  A  <  ( B  x.  C ) ) )
 
Theoremlt2mul2div 7920 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  (
 ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C 
 /  B ) ) )
 
Theoremledivmul2 7921 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  C )  <_  B 
 <->  A  <_  ( B  x.  C ) ) )
 
Theoremlemuldiv 7922 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  x.  C )  <_  B 
 <->  A  <_  ( B  /  C ) ) )
 
Theoremlemuldiv2 7923 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  A )  <_  B 
 <->  A  <_  ( B  /  C ) ) )
 
Theoremltrec 7924 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <  B  <->  ( 1  /  B )  <  ( 1 
 /  A ) ) )
 
Theoremlerec 7925 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.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <_  B  <->  ( 1  /  B )  <_  ( 1 
 /  A ) ) )
 
Theoremlt2msq1 7926 Lemma for lt2msq 7927. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B ) )
 
Theoremlt2msq 7927 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <  B  <->  ( A  x.  A )  <  ( B  x.  B ) ) )
 
Theoremltdiv2 7928 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( C  /  B )  <  ( C 
 /  A ) ) )
 
Theoremltrec1 7929 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( 1  /  A )  <  B  <->  ( 1  /  B )  <  A ) )
 
Theoremlerec2 7930 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <_  ( 1 
 /  B )  <->  B  <_  ( 1 
 /  A ) ) )
 
Theoremledivdiv 7931 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
 |-  ( ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( ( C  e.  RR  /\  0  <  C )  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  (
 ( A  /  B )  <_  ( C  /  D )  <->  ( D  /  C )  <_  ( B 
 /  A ) ) )
 
Theoremlediv2 7932 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
 |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( C  /  B )  <_  ( C 
 /  A ) ) )
 
Theoremltdiv23 7933 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
 |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
 
Theoremlediv23 7934 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
 |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( A  /  B ) 
 <_  C  <->  ( A  /  C )  <_  B ) )
 
Theoremlediv12a 7935 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
 |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <_  B )
 )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  (
 0  <  C  /\  C  <_  D ) ) )  ->  ( A  /  D )  <_  ( B  /  C ) )
 
Theoremlediv2a 7936 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B ) 
 /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( C  /  B )  <_  ( C  /  A ) )
 
Theoremreclt1 7937 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  ( A  <  1  <-> 
 1  <  ( 1  /  A ) ) )
 
Theoremrecgt1 7938 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  ( 1  <  A 
 <->  ( 1  /  A )  <  1 ) )
 
Theoremrecgt1i 7939 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  ( 0  < 
 ( 1  /  A )  /\  ( 1  /  A )  <  1 ) )
 
Theoremrecp1lt1 7940 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A  /  ( 1  +  A ) )  <  1 )
 
Theoremrecreclt 7941 Given a positive number  A, construct a new positive number less than both  A and 1. (Contributed by NM, 28-Dec-2005.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  ( ( 1 
 /  ( 1  +  ( 1  /  A ) ) )  < 
 1  /\  ( 1  /  ( 1  +  (
 1  /  A )
 ) )  <  A ) )
 
Theoremle2msq 7942 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <_  B  <->  ( A  x.  A )  <_  ( B  x.  B ) ) )
 
Theoremmsq11 7943 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.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  x.  A )  =  ( B  x.  B )  <->  A  =  B ) )
 
Theoremledivp1 7944 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.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  /  ( B  +  1
 ) )  x.  B )  <_  A )
 
Theoremsqueeze0 7945* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  ( 0  <  x  ->  A  <  x ) )  ->  A  =  0 )
 
Theoremltp1i 7946 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  RR   =>    |-  A  <  ( A  +  1 )
 
Theoremrecgt0i 7947 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  0  <  (
 1  /  A )
 )
 
Theoremrecgt0ii 7948 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  0  <  ( 1 
 /  A )
 
Theoremprodgt0i 7949 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  ( A  x.  B ) ) 
 ->  0  <  B )
 
Theoremprodge0i 7950 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <_  ( A  x.  B ) ) 
 ->  0  <_  B )
 
Theoremdivgt0i 7951 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  0  <  ( A  /  B ) )
 
Theoremdivge0i 7952 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  B )  ->  0  <_  ( A  /  B ) )
 
Theoremltreci 7953 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  ( A  <  B  <-> 
 ( 1  /  B )  <  ( 1  /  A ) ) )
 
Theoremlereci 7954 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  ( A  <_  B  <-> 
 ( 1  /  B )  <_  ( 1  /  A ) ) )
 
Theoremlt2msqi 7955 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <  B  <-> 
 ( A  x.  A )  <  ( B  x.  B ) ) )
 
Theoremle2msqi 7956 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <_  B  <-> 
 ( A  x.  A )  <_  ( B  x.  B ) ) )
 
Theoremmsq11i 7957 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( A  x.  A )  =  ( B  x.  B ) 
 <->  A  =  B ) )
 
Theoremdivgt0i2i 7958 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  B   =>    |-  (
 0  <  A  ->  0  <  ( A  /  B ) )
 
Theoremltrecii 7959 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  ( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) )
 
Theoremdivgt0ii 7960 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  0  <  ( A 
 /  B )
 
Theoremltmul1i 7961 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltdiv1i 7962 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
Theoremltmuldivi 7963 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( ( A  x.  C )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremltmul2i 7964 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1i 7965 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2i 7966 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 0  <  C  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremltdiv23i 7967 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <  B  /\  0  <  C ) 
 ->  ( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
 
Theoremltdiv23ii 7968 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  B   &    |-  0  <  C   =>    |-  (
 ( A  /  B )  <  C  <->  ( A  /  C )  <  B )
 
Theoremltmul1ii 7969 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.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  C   =>    |-  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) )
 
Theoremltdiv1ii 7970 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  0  <  C   =>    |-  ( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) )
 
Theoremltp1d 7971 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 7972 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1d 7973 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 7974 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <_  A )
 
Theoremrecgt0d 7975 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  0  <  ( 1  /  A ) )
 
Theoremdivgt0d 7976 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  /  B ) )
 
Theoremmulgt1d 7977 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  1  <  B )   =>    |-  ( ph  ->  1  <  ( A  x.  B ) )
 
Theoremlemulge11d 7978 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( A  x.  B ) )
 
Theoremlemulge12d 7979 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( B  x.  A ) )
 
Theoremlemul1ad 7980 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2ad 7981 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( C  x.  A )  <_  ( C  x.  B ) )
 
Theoremltmul12ad 7982 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  ->  ( A  x.  C )  < 
 ( B  x.  D ) )
 
Theoremlemul12ad 7983 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremlemul12bd 7984 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  D )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremmulle0r 7985 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  0  /\  0  <_  B ) )  ->  ( A  x.  B )  <_  0 )
 
3.3.10  Imaginary and complex number properties
 
Theoremcrap0 7986 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A #  0  \/  B #  0
 ) 
 <->  ( A  +  ( _i  x.  B ) ) #  0 ) )
 
Theoremcreur 7987* 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.)
 |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcreui 7988* 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.)
 |-  ( A  e.  CC  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcju 7989* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  E! x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )
 
3.4  Integer sets
 
3.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 7990 Extend class notation to include the class of positive integers.
 class  NN
 
Definitiondf-inn 7991* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7992 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremdfnn2 7992* Definition of the set of positive integers. Another name for df-inn 7991. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theorempeano5nni 7993* 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  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  A )
 
Theoremnnssre 7994 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 7995 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 NN  C_  CC
 
Theoremnnex 7996 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 NN  e.  _V
 
Theoremnnre 7997 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  RR )
 
Theoremnncn 7998 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  CC )
 
Theoremnnrei 7999 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 8000 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
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