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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcanap5rd 8601 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  x.  C )  /  ( B  x.  C ) )  =  ( A  /  B ) )
 
Theoremdivcanap7d 8602 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  /  ( B  /  C ) )  =  ( A  /  B ) )
 
Theoremdmdcanapd 8603 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( B  /  C )  x.  ( A  /  B ) )  =  ( A  /  C ) )
 
Theoremdmdcanap2d 8604 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( B  /  C ) )  =  ( A  /  C ) )
 
Theoremdivdivap1d 8605 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  /  C )  =  ( A  /  ( B  x.  C ) ) )
 
Theoremdivdivap2d 8606 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  ( A  /  ( B  /  C ) )  =  ( ( A  x.  C )  /  B ) )
 
Theoremdivmulap2d 8607 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( C  x.  B ) ) )
 
Theoremdivmulap3d 8608 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( B  x.  C ) ) )
 
Theoremdivassapd 8609 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv12apd 8610 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  ( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdiv23apd 8611 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( ( A  /  C )  x.  B ) )
 
Theoremdivdirapd 8612 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivsubdirapd 8613 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  -  B )  /  C )  =  ( ( A  /  C )  -  ( B  /  C ) ) )
 
Theoremdiv11apd 8614 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   &    |-  ( ph  ->  ( A  /  C )  =  ( B  /  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuldivapd 8615 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  D #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( C  /  D ) )  =  ( ( A  x.  C )  /  ( B  x.  D ) ) )
 
Theoremrerecclapd 8616 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremredivclapd 8617 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR )
 
Theoremdiveqap1bd 8618 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8488. Converse of diveqap1d 8581. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  /  B )  =  1 )
 
Theoremdiv2subap 8619 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C #  D ) )  ->  ( ( A  -  B ) 
 /  ( C  -  D ) )  =  ( ( B  -  A )  /  ( D  -  C ) ) )
 
Theoremdiv2subapd 8620 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8619. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  C #  D )   =>    |-  ( ph  ->  (
 ( A  -  B )  /  ( C  -  D ) )  =  ( ( B  -  A )  /  ( D  -  C ) ) )
 
Theoremsubrecap 8621 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  /  A )  -  (
 1  /  B )
 )  =  ( ( B  -  A ) 
 /  ( A  x.  B ) ) )
 
Theoremsubrecapi 8622 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( ( 1  /  A )  -  (
 1  /  B )
 )  =  ( ( B  -  A ) 
 /  ( A  x.  B ) )
 
Theoremsubrecapd 8623 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( 1 
 /  A )  -  ( 1  /  B ) )  =  (
 ( B  -  A )  /  ( A  x.  B ) ) )
 
Theoremmvllmulapd 8624 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  ( A  x.  B )  =  C )   =>    |-  ( ph  ->  B  =  ( C  /  A ) )
 
4.3.9  Ordering on reals (cont.)
 
Theoremltp1 8625 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 8626 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1 8627 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <  A )
 
Theoremlem1 8628 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <_  A )
 
Theoremletrp1 8629 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  ( B  +  1 ) )
 
Theoremp1le 8630 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  1 )  <_  B )  ->  A  <_  B )
 
Theoremrecgt0 8631 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  0  <  (
 1  /  A )
 )
 
Theoremprodgt0gt0 8632 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8633 for the same theorem with  0  < 
A replaced by the weaker condition 
0  <_  A. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  B )
 
Theoremprodgt0 8633 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  B )
 
Theoremprodgt02 8634 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  B  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  A )
 
Theoremprodge0 8635 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.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <_  ( A  x.  B ) ) ) 
 ->  0  <_  B )
 
Theoremprodge02 8636 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  B  /\  0  <_  ( A  x.  B ) ) ) 
 ->  0  <_  A )
 
Theoremltmul2 8637 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 ) ) )
 
Theoremlemul2 8638 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 8639 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.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2a 8640 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 8641 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 8642 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 8643 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 8644 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 8645 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 8646 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 8647 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 8648 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 8649 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 8650 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 8651 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 8652 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 8653 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 8654 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 8655 '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 8656 '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 8657 '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 8658 '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 8659 '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 8660 '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 8661 '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 8662 '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 8663 '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 8664 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 8665 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 8666 Lemma for lt2msq 8667. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B ) )
 
Theoremlt2msq 8667 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 8668 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 8669 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 8670 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 8671 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 8672 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 8673 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 8674 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 8675 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 8676 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 8677 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 8678 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 8679 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 8680 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 8681 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 8682 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 8683 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 8684 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 8685* 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 8686 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  RR   =>    |-  A  <  ( A  +  1 )
 
Theoremrecgt0i 8687 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 8688 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 8689 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 8690 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 8691 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 8692 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 8693 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 8694 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 8695 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 8696 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 8697 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 8698 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 8699 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 8700 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 )
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