Theorem List for Intuitionistic Logic Explorer - 7501-7600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | recidapi 7501 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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#  
 
 |
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Theorem | recrecapi 7502 |
A number is equal to the reciprocal of its reciprocal. Theorem I.10
of [Apostol] p. 18. (Contributed by
NM, 9-Feb-1995.)
|
#  
 
 |
|
Theorem | dividapi 7503 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
|
#  
 |
|
Theorem | div0api 7504 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|
#  
 |
|
Theorem | divclapzi 7505 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
 # 
   |
|
Theorem | divcanap1zi 7506 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #       |
|
Theorem | divcanap2zi 7507 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 # 
     |
|
Theorem | divrecapzi 7508 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
 # 
  
    |
|
Theorem | divcanap3zi 7509 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #       |
|
Theorem | divcanap4zi 7510 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
 #       |
|
Theorem | rec11api 7511 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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  # #    
     |
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Theorem | divclapi 7512 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#    |
|
Theorem | divcanap2i 7513 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#      |
|
Theorem | divcanap1i 7514 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#   

 |
|
Theorem | divrecapi 7515 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
#  
     |
|
Theorem | divcanap3i 7516 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#   
  |
|
Theorem | divcanap4i 7517 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
#   
  |
|
Theorem | divap0i 7518 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
|
# #   #  |
|
Theorem | rec11apii 7519 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# #   


  |
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Theorem | divassapzi 7520 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
 #           |
|
Theorem | divmulapzi 7521 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
 #   
     |
|
Theorem | divdirapzi 7522 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
 #       
     |
|
Theorem | divdiv23apzi 7523 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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  # #        
   |
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Theorem | divmulapi 7524 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
#   


  |
|
Theorem | divdiv32api 7525 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
# #   
      |
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Theorem | divassapi 7526 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
  
   |
|
Theorem | divdirapi 7527 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
        |
|
Theorem | div23api 7528 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
|
#   
      |
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Theorem | div11api 7529 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#   
    |
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Theorem | divmuldivapi 7530 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# #   
      
   |
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Theorem | divmul13api 7531 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# #   
          |
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Theorem | divadddivapi 7532 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# #   
              |
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Theorem | divdivdivapi 7533 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# # #   
     
    |
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Theorem | rerecclapzi 7534 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
 # 
   |
|
Theorem | rerecclapi 7535 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#    |
|
Theorem | redivclapzi 7536 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
 # 
   |
|
Theorem | redivclapi 7537 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
#    |
|
Theorem | div1d 7538 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
       |
|
Theorem | recclapd 7539 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #   
   |
|
Theorem | recap0d 7540 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
|
   #   
 #   |
|
Theorem | recidapd 7541 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
   #         |
|
Theorem | recidap2d 7542 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
   #    
    |
|
Theorem | recrecapd 7543 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
   #   
     |
|
Theorem | dividapd 7544 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #       |
|
Theorem | div0apd 7545 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
   #   
   |
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Theorem | apmul1 7546 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
|
   #    #   #      |
|
Theorem | divclapd 7547 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #       |
|
Theorem | divcanap1d 7548 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #    
    |
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Theorem | divcanap2d 7549 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #         |
|
Theorem | divrecapd 7550 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
     #      
    |
|
Theorem | divrecap2d 7551 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
     #       
   |
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Theorem | divcanap3d 7552 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #    
    |
|
Theorem | divcanap4d 7553 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #    
    |
|
Theorem | diveqap0d 7554 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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     #         |
|
Theorem | diveqap1d 7555 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #         |
|
Theorem | diveqap1ad 7556 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 7464. Generalization of diveqap1d 7555.
(Contributed by Jim Kingdon, 19-Mar-2020.)
|
     #    
    |
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Theorem | diveqap0ad 7557 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 7443. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
     #    
    |
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Theorem | divap1d 7558 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
|
     #   #     #   |
|
Theorem | divap0bd 7559 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
     #    #
  #    |
|
Theorem | divnegapd 7560 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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     #           |
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Theorem | divneg2apd 7561 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #           |
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Theorem | div2negapd 7562 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
     #      
    |
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Theorem | divap0d 7563 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
     #   #     #   |
|
Theorem | recdivapd 7564 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #   #   
       |
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Theorem | recdivap2d 7565 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #   #    
        |
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Theorem | divcanap6d 7566 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #   #    
      |
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Theorem | ddcanapd 7567 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #   #         |
|
Theorem | rec11apd 7568 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
     #   #   
       |
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Theorem | divmulapd 7569 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #    
      |
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Theorem | div32apd 7570 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #    
        |
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Theorem | div13apd 7571 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #    
    
   |
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Theorem | divdiv32apd 7572 |
Swap denominators in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
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       #   #    
        |
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Theorem | divcanap5d 7573 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #    
        |
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Theorem | divcanap5rd 7574 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #    
        |
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Theorem | divcanap7d 7575 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
       #   #    
   
    |
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Theorem | dmdcanapd 7576 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #    
        |
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Theorem | dmdcanap2d 7577 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
       #   #    
        |
|
Theorem | divdivap1d 7578 |
Division into a fraction. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
       #   #    
        |
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Theorem | divdivap2d 7579 |
Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
|
       #   #             |
|
Theorem | divmulap2d 7580 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #    
 
    |
|
Theorem | divmulap3d 7581 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #    
 
    |
|
Theorem | divassapd 7582 |
An associative law for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #    
        |
|
Theorem | div12apd 7583 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #             |
|
Theorem | div23apd 7584 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #    
    
   |
|
Theorem | divdirapd 7585 |
Distribution of division over addition. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #    
    
     |
|
Theorem | divsubdirapd 7586 |
Distribution of division over subtraction. (Contributed by Jim
Kingdon, 2-Mar-2020.)
|
       #    
    
     |
|
Theorem | div11apd 7587 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
|
       #           |
|
Theorem | rerecclapd 7588 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
   #   
   |
|
Theorem | redivclapd 7589 |
Closure law for division of reals. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
     #       |
|
Theorem | mvllmulapd 7590 |
Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
10-Jun-2020.)
|
     #           |
|
3.3.9 Ordering on reals (cont.)
|
|
Theorem | ltp1 7591 |
A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
|
     |
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Theorem | lep1 7592 |
A number is less than or equal to itself plus 1. (Contributed by NM,
5-Jan-2006.)
|

    |
|
Theorem | ltm1 7593 |
A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
|
     |
|
Theorem | lem1 7594 |
A number minus 1 is less than or equal to itself. (Contributed by Mario
Carneiro, 2-Oct-2015.)
|
     |
|
Theorem | letrp1 7595 |
A transitive property of 'less than or equal' and plus 1. (Contributed by
NM, 5-Aug-2005.)
|
       |
|
Theorem | p1le 7596 |
A transitive property of plus 1 and 'less than or equal'. (Contributed by
NM, 16-Aug-2005.)
|
   

  |
|
Theorem | recgt0 7597 |
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.)
|
   
   |
|
Theorem | prodgt0gt0 7598 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 7599 for the same theorem with
replaced by the weaker condition
. (Contributed by
Jim
Kingdon, 29-Feb-2020.)
|
    
      |
|
Theorem | prodgt0 7599 |
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.)
|
    
      |
|
Theorem | prodgt02 7600 |
Infer that a multiplier is positive from a nonnegative multiplicand and
positive product. (Contributed by NM, 24-Apr-2005.)
|
    
      |