HomeHome Metamath Proof Explorer
Theorem List (p. 97 of 315)
< 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
(1-21493)
  Hilbert Space Explorer  Hilbert Space Explorer
(21494-23016)
  Users' Mathboxes  Users' Mathboxes
(23017-31457)
 

Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltmul1 9601 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltmul2 9602 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1 9603 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2 9604 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremlemul1a 9605 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2a 9606 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( C  x.  A )  <_  ( C  x.  B ) )
 
Theoremltmul12a 9607 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 9608 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 9609 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 9610 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 9611 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 9612 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 9613 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 9614 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 9615 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 9616 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 9617 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 9618 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 9619 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 9620 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 9621 '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 9622 '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 9623 '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 9624 '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 ) ) )
 
TheoremledivmulOLD 9625 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  (
 ( A  /  B )  <_  C  <->  A  <_  ( B  x.  C ) ) )
 
Theoremltdivmul2 9626 '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 9627 '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 9628 '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 ) ) )
 
Theoremledivmul2OLD 9629 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  (
 ( A  /  B )  <_  C  <->  A  <_  ( C  x.  B ) ) )
 
Theoremlemuldiv 9630 '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 9631 '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 9632 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 9633 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 9634 Lemma for lt2msq 9635. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B ) )
 
Theoremlt2msq 9635 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 9636 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 ) ) )
 
Theoremltdiv2OLD 9637 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( 0  <  A  /\  0  <  B  /\  0  <  C ) )  ->  ( A  <  B  <->  ( C  /  B )  <  ( C 
 /  A ) ) )
 
Theoremltrec1 9638 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 9639 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 9640 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 9641 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 9642 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 9643 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 9644 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 9645 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 9646 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 9647 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 9648 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 9649 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 9650 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 9651 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 9652 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 9653 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 9654* 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 9655 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  RR   =>    |-  A  <  ( A  +  1 )
 
Theoremrecgt0i 9656 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 9657 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 9658 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  ( A  x.  B ) ) 
 ->  0  <  B )
 
Theoremprodge0i 9659 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <_  ( A  x.  B ) ) 
 ->  0  <_  B )
 
Theoremdivgt0i 9660 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 9661 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 9662 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 9663 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 9664 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 9665 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 9666 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 9667 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 9668 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 9669 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 9670 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 9671 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 9672 '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 9673 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 9674 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 9675 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 9676 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 ) )
 
Theoremledivp1i 9677 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1
 ) ) )  ->  ( A  x.  C )  <_  B )
 
Theoremltdivp1i 9678 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1
 ) ) )  ->  ( A  x.  C )  <  B )
 
Theoremltdiv23ii 9679 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 9680 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 9681 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 9682 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 9683 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 9684 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 9685 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 9686 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 9687 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 9688 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 9689 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 9690 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 9691 The square of a nonnegative number is a one-to-one function. (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 9692 The square of a nonnegative number is a one-to-one function. (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 9693 The square of a nonnegative number is a one-to-one function. (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 9694 The square of a nonnegative number is a one-to-one function. (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 9695 The square of a nonnegative number is a one-to-one function. (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 ) )
 
5.3.8  Completeness Axiom and Suprema
 
Theoremfimaxre 9696* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  y 
 <_  x )
 
Theoremfimaxre2 9697* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. x  e.  RR  A. y  e.  A  y 
 <_  x )
 
Theoremfimaxre3 9698* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  ( ( A  e.  Fin  /\  A. y  e.  A  B  e.  RR )  ->  E. x  e.  RR  A. y  e.  A  B  <_  x )
 
Theoremlbreu 9699* 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 9700* 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 )
    < 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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31457
  Copyright terms: Public domain < Previous  Next >