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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcan6d 10401 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( B  /  A ) )  =  1 )
 
Theoremddcand 10402 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A  /  ( A  /  B ) )  =  B )
 
Theoremrec11d 10403 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  ( 1  /  A )  =  ( 1  /  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuld 10404 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremdiv32d 10405 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( A  x.  ( C  /  B ) ) )
 
Theoremdiv13d 10406 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( ( C  /  B )  x.  A ) )
 
Theoremdivdiv32d 10407 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  /  B )  /  C )  =  ( ( A  /  C )  /  B ) )
 
Theoremdivcan5d 10408 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( C  x.  A )  /  ( C  x.  B ) )  =  ( A  /  B ) )
 
Theoremdivcan5rd 10409 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( 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 ) )
 
Theoremdivcan7d 10410 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) )
 
Theoremdmdcand 10411 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) )
 
Theoremdmdcan2d 10412 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( 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 ) )
 
Theoremdivdiv1d 10413 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) ) )
 
Theoremdivdiv2d 10414 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) )
 
Theoremdivmul2d 10415 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( C  x.  B ) ) )
 
Theoremdivmul3d 10416 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( B  x.  C ) ) )
 
Theoremdivassd 10417 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv12d 10418 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdiv23d 10419 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( ( A  /  C )  x.  B ) )
 
Theoremdivdird 10420 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivsubdird 10421 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  -  B )  /  C )  =  ( ( A  /  C )  -  ( B  /  C ) ) )
 
Theoremdiv11d 10422 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( A  /  C )  =  ( B  /  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuldivd 10423 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) ) )
 
Theoremdivmul13d 10424 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 ) )  =  ( ( C  /  B )  x.  ( A  /  D ) ) )
 
Theoremdivmul24d 10425 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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  /  D )  x.  ( C  /  B ) ) )
 
Theoremdivadddivd 10426 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 )  +  ( C  /  D ) )  =  ( ( ( A  x.  D )  +  ( C  x.  B ) )  /  ( B  x.  D ) ) )
 
Theoremdivsubdivd 10427 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 )  -  ( C  /  D ) )  =  ( ( ( A  x.  D )  -  ( C  x.  B ) )  /  ( B  x.  D ) ) )
 
Theoremdivmuleqd 10428 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( 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 )  =  ( C  /  D )  <->  ( A  x.  D )  =  ( C  x.  B ) ) )
 
Theoremdivdivdivd 10429 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C ) ) )
 
Theoremdiveq1bd 10430 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 10300. Converse of diveq1d 10390. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  /  B )  =  1 )
 
Theoremdiv2sub 10431 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 ) ) )
 
Theoremdiv2subd 10432 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 10431. (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 ) ) )
 
Theoremrereccld 10433 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  RR )
 
Theoremredivcld 10434 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 RR )
 
Theoremsubrec 10435 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 ) ) )
 
Theoremsubreci 10436 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 ) )
 
Theoremsubrecd 10437 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 ) ) )
 
Theoremmvllmuld 10438 Move LHS left multiplication to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  ( A  x.  B )  =  C )   =>    |-  ( ph  ->  B  =  ( C  /  A ) )
 
Theoremmvllmuli 10439 Move LHS left multiplication to RHS. Uses divcan4i 10353. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A  =/=  0   &    |-  ( A  x.  B )  =  C   =>    |-  B  =  ( C 
 /  A )
 
5.3.7  Ordering on reals (cont.)
 
Theoremelimgt0 10440 Hypothesis for weak deduction theorem to eliminate  0  <  A. (Contributed by NM, 15-May-1999.)
 |-  0  <  if (
 0  <  A ,  A ,  1 )
 
Theoremelimge0 10441 Hypothesis for weak deduction theorem to eliminate  0  <_  A. (Contributed by NM, 30-Jul-1999.)
 |-  0  <_  if (
 0  <_  A ,  A ,  0 )
 
Theoremltp1 10442 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 10443 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1 10444 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <  A )
 
Theoremlem1 10445 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 10446 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 10447 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 10448 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 )
 )
 
Theoremprodgt0 10449 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 10450 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 10451 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 10452 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 )
 
Theoremltmul1a 10453 Lemma for ltmul1 10454. Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (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 ) )
 
Theoremltmul1 10454 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 10455 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 10456 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 10457 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 10458 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 10459 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 10460 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 10461 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 10462 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 10463 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 10464 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 10465 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 10466 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 10467 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 10468 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 10469 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 10470 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 10471 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 10472 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 10473 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 ) )
 
Theoremmulge0b 10474 A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  x.  B ) 
 <->  ( ( A  <_  0 
 /\  B  <_  0
 )  \/  ( 0 
 <_  A  /\  0  <_  B ) ) ) )
 
Theoremmulle0b 10475 A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <_ 
 0 
 <->  ( ( A  <_  0 
 /\  0  <_  B )  \/  ( 0  <_  A  /\  B  <_  0
 ) ) ) )
 
Theoremmulsuble0b 10476 A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( ( A  -  B )  x.  ( C  -  B ) )  <_  0  <->  ( ( A 
 <_  B  /\  B  <_  C )  \/  ( C 
 <_  B  /\  B  <_  A ) ) ) )
 
Theoremltmuldiv 10477 '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 10478 '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 10479 '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 10480 '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 10481 '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 10482 '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 10483 '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 10484 '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 10485 '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 10486 '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 10487 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 10488 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 10489 Lemma for lt2msq 10490. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR  /\  A  <  B )  ->  ( A  x.  A )  <  ( B  x.  B ) )
 
Theoremlt2msq 10490 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 10491 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 10492 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 10493 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 10494 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 10495 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 10496 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 10497 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 10498 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 10499 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 10500 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 ) )
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