Theorem List for Intuitionistic Logic Explorer - 8701-8800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | div0api 8701 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|
#  
 |
|
Theorem | divclapzi 8702 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
 # 
   |
|
Theorem | divcanap1zi 8703 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #   

  |
|
Theorem | divcanap2zi 8704 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 # 
     |
|
Theorem | divrecapzi 8705 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
 # 
       |
|
Theorem | divcanap3zi 8706 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #   
   |
|
Theorem | divcanap4zi 8707 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #   
   |
|
Theorem | rec11api 8708 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
|
  # #    

    |
|
Theorem | divclapi 8709 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#  
 |
|
Theorem | divcanap2i 8710 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#  
   |
|
Theorem | divcanap1i 8711 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#      |
|
Theorem | divrecapi 8712 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
#  
     |
|
Theorem | divcanap3i 8713 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#    
 |
|
Theorem | divcanap4i 8714 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#    
 |
|
Theorem | divap0i 8715 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
|
# #   #  |
|
Theorem | rec11apii 8716 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# #   

   |
|
Theorem | divassapzi 8717 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
 #
          |
|
Theorem | divmulapzi 8718 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
 #
    
   |
|
Theorem | divdirapzi 8719 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
 #
      
     |
|
Theorem | divdiv23apzi 8720 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
  # #        
   |
|
Theorem | divmulapi 8721 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
#   
    |
|
Theorem | divdiv32api 8722 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
# #   
      |
|
Theorem | divassapi 8723 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
  
   |
|
Theorem | divdirapi 8724 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
    
   |
|
Theorem | div23api 8725 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
|
#   
      |
|
Theorem | div11api 8726 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
 
  |
|
Theorem | divmuldivapi 8727 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# #   
      
   |
|
Theorem | divmul13api 8728 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# #   
          |
|
Theorem | divadddivapi 8729 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# #   
          
   |
|
Theorem | divdivdivapi 8730 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# # #   
      
   |
|
Theorem | rerecclapzi 8731 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
 #  
  |
|
Theorem | rerecclapi 8732 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#  
 |
|
Theorem | redivclapzi 8733 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
 # 
   |
|
Theorem | redivclapi 8734 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#  
 |
|
Theorem | div1d 8735 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
       |
|
Theorem | recclapd 8736 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #   
   |
|
Theorem | recap0d 8737 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
|
   #   
 #   |
|
Theorem | recidapd 8738 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
   #         |
|
Theorem | recidap2d 8739 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
   #         |
|
Theorem | recrecapd 8740 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
   #   
     |
|
Theorem | dividapd 8741 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #       |
|
Theorem | div0apd 8742 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #   
   |
|
Theorem | apmul1 8743 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
|
   #    #   #
     |
|
Theorem | apmul2 8744 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
|
   #    #   #
     |
|
Theorem | divclapd 8745 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
   
  |
|
Theorem | divcanap1d 8746 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
        |
|
Theorem | divcanap2d 8747 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
   
    |
|
Theorem | divrecapd 8748 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
     #
   
      |
|
Theorem | divrecap2d 8749 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
     #
   
      |
|
Theorem | divcanap3d 8750 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
     
  |
|
Theorem | divcanap4d 8751 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
     
  |
|
Theorem | diveqap0d 8752 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
     #
   
    |
|
Theorem | diveqap1d 8753 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #
   
    |
|
Theorem | diveqap1ad 8754 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8660. Generalization of diveqap1d 8753.
(Contributed by Jim Kingdon, 19-Mar-2020.)
|
     #
    
   |
|
Theorem | diveqap0ad 8755 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8637. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
     #
    
   |
|
Theorem | divap1d 8756 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
|
     #
  #
    #
  |
|
Theorem | divap0bd 8757 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
     #
   #   #    |
|
Theorem | divnegapd 8758 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #
    
     |
|
Theorem | divneg2apd 8759 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #
    
     |
|
Theorem | div2negapd 8760 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #
          |
|
Theorem | divap0d 8761 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
     #
  #
    #
  |
|
Theorem | recdivapd 8762 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #
  #
   
      |
|
Theorem | recdivap2d 8763 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #
  #
     

     |
|
Theorem | divcanap6d 8764 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #
  #
       
  |
|
Theorem | ddcanapd 8765 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #
  #
   
    |
|
Theorem | rec11apd 8766 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #
  #
   

     |
|
Theorem | divmulapd 8767 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #     
 
   |
|
Theorem | apdivmuld 8768 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 26-Dec-2022.)
|
       #      #   #
   |
|
Theorem | div32apd 8769 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #             |
|
Theorem | div13apd 8770 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #         
   |
|
Theorem | divdiv32apd 8771 |
Swap denominators in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
       #   #         
   |
|
Theorem | divcanap5d 8772 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #             |
|
Theorem | divcanap5rd 8773 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #             |
|
Theorem | divcanap7d 8774 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
       #   #             |
|
Theorem | dmdcanapd 8775 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #             |
|
Theorem | dmdcanap2d 8776 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #             |
|
Theorem | divdivap1d 8777 |
Division into a fraction. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
       #   #             |
|
Theorem | divdivap2d 8778 |
Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
|
       #   #         
   |
|
Theorem | divmulap2d 8779 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #     
     |
|
Theorem | divmulap3d 8780 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #     
     |
|
Theorem | divassapd 8781 |
An associative law for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #             |
|
Theorem | div12apd 8782 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #             |
|
Theorem | div23apd 8783 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #         
   |
|
Theorem | divdirapd 8784 |
Distribution of division over addition. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #         
     |
|
Theorem | divsubdirapd 8785 |
Distribution of division over subtraction. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #         
     |
|
Theorem | div11apd 8786 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #           |
|
Theorem | divmuldivapd 8787 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
30-Jul-2021.)
|
         #   #           
     |
|
Theorem | divmuleqapd 8788 |
Cross-multiply in an equality of ratios. (Contributed by Mario
Carneiro, 27-May-2016.)
|
         #   #     
   
     |
|
Theorem | rerecclapd 8789 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
   #   
   |
|
Theorem | redivclapd 8790 |
Closure law for division of reals. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #
   
  |
|
Theorem | diveqap1bd 8791 |
If two complex numbers are equal, their quotient is one. One-way
deduction form of diveqap1 8660. Converse of diveqap1d 8753. (Contributed
by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
|
   #         |
|
Theorem | div2subap 8792 |
Swap the order of subtraction in a division. (Contributed by Scott
Fenton, 24-Jun-2013.)
|
    
#  
        
     |
|
Theorem | div2subapd 8793 |
Swap subtrahend and minuend inside the numerator and denominator of a
fraction. Deduction form of div2subap 8792. (Contributed by David Moews,
28-Feb-2017.)
|
         #           
     |
|
Theorem | subrecap 8794 |
Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
|
   # 
 #     

          |
|
Theorem | subrecapi 8795 |
Subtraction of reciprocals. (Contributed by Scott Fenton,
9-Jan-2017.)
|
# #   

         |
|
Theorem | subrecapd 8796 |
Subtraction of reciprocals. (Contributed by Scott Fenton,
9-Jan-2017.)
|
     #
  #
       
        |
|
Theorem | mvllmulapd 8797 |
Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
10-Jun-2020.)
|
     #
   
      |
|
Theorem | rerecapb 8798* |
A real number has a multiplicative inverse if and only if it is apart
from zero. Theorem 11.2.4 of [HoTT], p.
(varies). (Contributed by Jim
Kingdon, 18-Jan-2025.)
|
  #  

   |
|
4.3.9 Ordering on reals (cont.)
|
|
Theorem | ltp1 8799 |
A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|
     |
|
Theorem | lep1 8800 |
A number is less than or equal to itself plus 1. (Contributed by NM,
5-Jan-2006.)
|

    |