Theorem List for Intuitionistic Logic Explorer - 8701-8800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | divdivdivap 8701 |
Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by
Jim Kingdon, 25-Feb-2020.)
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#     #   #            
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Theorem | divcanap5 8702 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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   #   #  
   
 
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Theorem | divmul13ap 8703 |
Swap the denominators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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      #  
#   
     
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Theorem | divmul24ap 8704 |
Swap the numerators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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      #  
#   
     
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Theorem | divmuleqap 8705 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
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      #  
#   
      
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Theorem | recdivap 8706 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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   # 
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Theorem | divcanap6 8707 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
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   # 
 #     
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Theorem | divdiv32ap 8708 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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   #   #  
   
      |
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Theorem | divcanap7 8709 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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   #   #  
   
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Theorem | dmdcanap 8710 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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   # 
 # 

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Theorem | divdivap1 8711 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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   #   #  
   
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Theorem | divdivap2 8712 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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   #   #  
   
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Theorem | recdivap2 8713 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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   # 
 #     
  
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Theorem | ddcanap 8714 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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   # 
 #   
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Theorem | divadddivap 8715 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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      #  
#   
   
     
   
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Theorem | divsubdivap 8716 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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      #  
#   
   
         
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Theorem | conjmulap 8717 |
Two numbers whose reciprocals sum to 1 are called "conjugates" and
satisfy
this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
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   # 
 #         
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Theorem | rerecclap 8718 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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  #   
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Theorem | redivclap 8719 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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  #  
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Theorem | eqneg 8720 |
A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | eqnegd 8721 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8720. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 8722 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8720. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 8723 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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  #     
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Theorem | divneg2ap 8724 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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  #    
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Theorem | recclapzi 8725 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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 #  
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Theorem | recap0apzi 8726 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
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 #   #   |
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Theorem | recidapzi 8727 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | div1i 8728 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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Theorem | eqnegi 8729 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
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Theorem | recclapi 8730 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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#  
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Theorem | recidapi 8731 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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#  
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Theorem | recrecapi 8732 |
A number is equal to the reciprocal of its reciprocal. Theorem I.10
of [Apostol] p. 18. (Contributed by
NM, 9-Feb-1995.)
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#  
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Theorem | dividapi 8733 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
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#  
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Theorem | div0api 8734 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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#  
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Theorem | divclapzi 8735 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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 # 
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Theorem | divcanap1zi 8736 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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 #   

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Theorem | divcanap2zi 8737 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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 # 
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Theorem | divrecapzi 8738 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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 # 
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Theorem | divcanap3zi 8739 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap4zi 8740 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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 #   
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Theorem | rec11api 8741 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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  # #    

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Theorem | divclapi 8742 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#  
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Theorem | divcanap2i 8743 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#  
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Theorem | divcanap1i 8744 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#      |
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Theorem | divrecapi 8745 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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#  
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Theorem | divcanap3i 8746 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#    
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Theorem | divcanap4i 8747 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#    
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Theorem | divap0i 8748 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
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# #   #  |
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Theorem | rec11apii 8749 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# #   

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Theorem | divassapzi 8750 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapzi 8751 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divdirapzi 8752 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
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 #
      
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Theorem | divdiv23apzi 8753 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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  # #        
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Theorem | divmulapi 8754 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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#   
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Theorem | divdiv32api 8755 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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# #   
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Theorem | divassapi 8756 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#   
  
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Theorem | divdirapi 8757 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#   
    
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Theorem | div23api 8758 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
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#   
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Theorem | div11api 8759 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#   
 
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Theorem | divmuldivapi 8760 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #   
      
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Theorem | divmul13api 8761 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #   
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Theorem | divadddivapi 8762 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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# #   
          
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Theorem | divdivdivapi 8763 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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# # #   
      
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Theorem | rerecclapzi 8764 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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 #  
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Theorem | rerecclapi 8765 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#  
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Theorem | redivclapzi 8766 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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 # 
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Theorem | redivclapi 8767 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#  
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Theorem | div1d 8768 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | recclapd 8769 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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   #   
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Theorem | recap0d 8770 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
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   #   
 #   |
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Theorem | recidapd 8771 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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   #         |
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Theorem | recidap2d 8772 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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   #         |
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Theorem | recrecapd 8773 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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   #   
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Theorem | dividapd 8774 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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   #       |
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Theorem | div0apd 8775 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
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   #   
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Theorem | apmul1 8776 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
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   #    #   #
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Theorem | apmul2 8777 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
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   #    #   #
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Theorem | divclapd 8778 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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     #
   
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Theorem | divcanap1d 8779 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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     #
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Theorem | divcanap2d 8780 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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     #
   
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Theorem | divrecapd 8781 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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     #
   
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Theorem | divrecap2d 8782 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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     #
   
      |
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Theorem | divcanap3d 8783 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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     #
     
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Theorem | divcanap4d 8784 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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     #
     
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Theorem | diveqap0d 8785 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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     #
   
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Theorem | diveqap1d 8786 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
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     #
   
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Theorem | diveqap1ad 8787 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8693. Generalization of diveqap1d 8786.
(Contributed by Jim Kingdon, 19-Mar-2020.)
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     #
    
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Theorem | diveqap0ad 8788 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8670. (Contributed by Jim Kingdon, 19-Mar-2020.)
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     #
    
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Theorem | divap1d 8789 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
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     #
  #
    #
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Theorem | divap0bd 8790 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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     #
   #   #    |
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Theorem | divnegapd 8791 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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     #
    
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Theorem | divneg2apd 8792 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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     #
    
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Theorem | div2negapd 8793 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
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     #
          |
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Theorem | divap0d 8794 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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     #
  #
    #
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Theorem | recdivapd 8795 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
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     #
  #
   
      |
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Theorem | recdivap2d 8796 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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     #
  #
     

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Theorem | divcanap6d 8797 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
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     #
  #
       
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Theorem | ddcanapd 8798 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
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     #
  #
   
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Theorem | rec11apd 8799 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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     #
  #
   

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Theorem | divmulapd 8800 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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       #     
 
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