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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltmuldiv2 8601 '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 8602 '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 8603 '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 8604 '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 8605 '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 8606 '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 8607 '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 8608 '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 8609 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 8610 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 8611 Lemma for lt2msq 8612. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B ) )
 
Theoremlt2msq 8612 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 8613 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 8614 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 8615 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 8616 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 8617 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 8618 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 8619 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 8620 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 8621 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 8622 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 8623 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 8624 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 8625 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 8626 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 8627 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 8628 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 8629 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 8630* 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 8631 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  RR   =>    |-  A  <  ( A  +  1 )
 
Theoremrecgt0i 8632 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 8633 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 8634 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 8635 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 8636 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 8637 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 8638 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 8639 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 8640 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 8641 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 8642 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 8643 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 8644 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 8645 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 8646 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 8647 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 8648 '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 8649 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 8650 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 8651 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 8652 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 8653 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 8654 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 8655 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 8656 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 8657 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 8658 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 8659 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 8660 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 8661 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 8662 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 8663 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 8664 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 8665 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 8666 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 8667 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 8668 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 8669 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 8670 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 )
 
4.3.10  Suprema
 
Theoremlbreu 8671* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  E! x  e.  S  A. y  e.  S  x  <_  y
 )
 
Theoremlbcl 8672* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y
 )  e.  S )
 
Theoremlble 8673* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y )  <_  A )
 
Theoremlbinf 8674* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  =  ( iota_ x  e.  S  A. y  e.  S  x  <_  y ) )
 
Theoremlbinfcl 8675* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  e.  S )
 
Theoremlbinfle 8676* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 -> inf ( S ,  RR ,  <  )  <_  A )
 
Theoremsuprubex 8677* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprlubex 8678* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
 
Theoremsuprnubex 8679* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( -.  B  <  sup ( A ,  RR ,  <  )  <->  A. z  e.  A  -.  B  <  z ) )
 
Theoremsuprleubex 8680* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  <_  B  <->  A. z  e.  A  z 
 <_  B ) )
 
Theoremnegiso 8681 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  F  =  ( x  e.  RR  |->  -u x )   =>    |-  ( F  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
 
Theoremdfinfre 8682* The infimum of a set of reals  A. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( A  C_  RR  -> inf ( A ,  RR ,  <  )  =  U. { x  e.  RR  |  ( A. y  e.  A  x  <_  y  /\  A. y  e.  RR  ( x  <  y  ->  E. z  e.  A  z  <  y
 ) ) } )
 
Theoremsup3exmid 8683* If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
 |-  ( ( u  C_  RR  /\  E. w  w  e.  u  /\  E. x  e.  RR  A. y  e.  u  y  <_  x )  ->  E. x  e.  RR  ( A. y  e.  u  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
 x  ->  E. z  e.  u  y  <  z ) ) )   =>    |- DECID  ph
 
4.3.11  Imaginary and complex number properties
 
Theoremcrap0 8684 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 8685* 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 8686* 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 8687* 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 ) )
 
4.4  Integer sets
 
4.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 8688 Extend class notation to include the class of positive integers.
 class  NN
 
Definitiondf-inn 8689* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8690 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 8690* Definition of the set of positive integers. Another name for df-inn 8689. (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 8691* 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 8692 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 8693 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 NN  C_  CC
 
Theoremnnex 8694 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 NN  e.  _V
 
Theoremnnre 8695 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  RR )
 
Theoremnncn 8696 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  CC )
 
Theoremnnrei 8697 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 8698 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 8699 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
 |-  1  e.  NN
 
Theorempeano2nn 8700 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.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
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