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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecidapi 7501 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     x. 
 1  1
 
Theoremrecrecapi 7502 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     1 
 1
 
Theoremdividapi 7503 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 CC   &    #  0   =>     1
 
Theoremdiv0api 7504 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 CC   &    #  0   =>     0  0
 
Theoremdivclapzi 7505 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  CC
 
Theoremdivcanap1zi 7506 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivcanap2zi 7507 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivrecapzi 7508 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.  1
 
Theoremdivcanap3zi 7509 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremdivcanap4zi 7510 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 CC   &     CC   =>    #  0  x.
 
Theoremrec11api 7511 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   =>    #  0 #  0  1  1
 
Theoremdivclapi 7512 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     CC
 
Theoremdivcanap2i 7513 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap1i 7514 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivrecapi 7515 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.  1
 
Theoremdivcanap3i 7516 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap4i 7517 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivap0i 7518 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    #  0
 
Theoremrec11apii 7519 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     1 
 1
 
Theoremdivassapzi 7520 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>     C #  0  x.  C  x.  C
 
Theoremdivmulapzi 7521 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>    #  0  C  x.  C
 
Theoremdivdirapzi 7522 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>     C #  0  +  C  C  +  C
 
Theoremdivdiv23apzi 7523 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 CC   &     CC   &     C  CC   =>    #  0  C #  0  C  C
 
Theoremdivmulapi 7524 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     C  x.  C
 
Theoremdivdiv32api 7525 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdivassapi 7526 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdivdirapi 7527 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     +  C  C  +  C
 
Theoremdiv23api 7528 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  C  x.
 
Theoremdiv11api 7529 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  C
 
Theoremdivmuldivapi 7530 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     x.  C  D  x.  C  x.  D
 
Theoremdivmul13api 7531 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     x.  C  D  C  x.  D
 
Theoremdivadddivapi 7532 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   =>     +  C  D  x.  D  +  C  x.  x.  D
 
Theoremdivdivdivapi 7533 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 CC   &     CC   &     C  CC   &     D  CC   &    #  0   &     D #  0   &     C #  0   =>     C  D  x.  D  x.  C
 
Theoremrerecclapzi 7534 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   =>    #  0  1  RR
 
Theoremrerecclapi 7535 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &    #  0   =>     1  RR
 
Theoremredivclapzi 7536 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &     RR   =>    #  0  RR
 
Theoremredivclapi 7537 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 RR   &     RR   &    #  0   =>     RR
 
Theoremdiv1d 7538 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     1
 
Theoremrecclapd 7539 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1  CC
 
Theoremrecap0d 7540 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1 #  0
 
Theoremrecidapd 7541 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     x.  1  1
 
Theoremrecidap2d 7542 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     1  x.  1
 
Theoremrecrecapd 7543 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 1  1
 
Theoremdividapd 7544 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>     1
 
Theoremdiv0apd 7545 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &    #  0   =>    
 0  0
 
Theoremapmul1 7546 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
 CC  CC  C  CC  C #  0 #  x.  C #  x.  C
 
Theoremdivclapd 7547 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>    
 CC
 
Theoremdivcanap1d 7548 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap2d 7549 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivrecapd 7550 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x. 
 1
 
Theoremdivrecap2d 7551 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     1  x.
 
Theoremdivcanap3d 7552 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdivcanap4d 7553 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 CC   &     CC   &    #  0   =>     x.
 
Theoremdiveqap0d 7554 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   &     0   =>     0
 
Theoremdiveqap1d 7555 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   &     1   =>   
 
Theoremdiveqap1ad 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.)
 CC   &     CC   &    #  0   =>     1
 
Theoremdiveqap0ad 7557 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7443. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     0  0
 
Theoremdivap1d 7558 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
 CC   &     CC   &    #  0   &    #    =>    #  1
 
Theoremdivap0bd 7559 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>    #  0 #  0
 
Theoremdivnegapd 7560 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdivneg2apd 7561 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdiv2negapd 7562 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
 CC   &     CC   &    #  0   =>     -u  -u
 
Theoremdivap0d 7563 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    #  0
 
Theoremrecdivapd 7564 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>    
 1
 
Theoremrecdivap2d 7565 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     1  1  x.
 
Theoremdivcanap6d 7566 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>     x.  1
 
Theoremddcanapd 7567 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   =>   
 
Theoremrec11apd 7568 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 CC   &     CC   &    #  0   &    #  0   &    
 1  1    =>   
 
Theoremdivmulapd 7569 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     C  x.  C
 
Theoremdiv32apd 7570 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     x.  C  x.  C
 
Theoremdiv13apd 7571 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   =>     x.  C  C  x.
 
Theoremdivdiv32apd 7572 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdivcanap5d 7573 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C  x.
 
Theoremdivcanap5rd 7574 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     x.  C  x.  C
 
Theoremdivcanap7d 7575 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  C
 
Theoremdmdcanapd 7576 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdmdcanap2d 7577 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     x.  C  C
 
Theoremdivdivap1d 7578 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdivdivap2d 7579 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 CC   &     CC   &     C  CC   &    #  0   &     C #  0   =>     C  x.  C
 
Theoremdivmulap2d 7580 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  C  x.
 
Theoremdivmulap3d 7581 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     C  x.  C
 
Theoremdivassapd 7582 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdiv12apd 7583 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  x.  C
 
Theoremdiv23apd 7584 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     x.  C  C  x.
 
Theoremdivdirapd 7585 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     +  C  C  +  C
 
Theoremdivsubdirapd 7586 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   =>     -  C  C  -  C
 
Theoremdiv11apd 7587 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 CC   &     CC   &     C  CC   &     C #  0   &     C  C   =>   
 
Theoremrerecclapd 7588 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR   &    #  0   =>    
 1  RR
 
Theoremredivclapd 7589 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR   &     RR   &    #  0   =>    
 RR
 
Theoremmvllmulapd 7590 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
 CC   &     CC   &    #  0   &     x.  C   =>     C
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7591 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 RR  <  +  1
 
Theoremlep1 7592 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 RR  <_  +  1
 
Theoremltm1 7593 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 RR  -  1  <
 
Theoremlem1 7594 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
 RR  -  1  <_
 
Theoremletrp1 7595 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
 RR  RR  <_  <_  +  1
 
Theoremp1le 7596 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
 RR  RR  +  1  <_  <_
 
Theoremrecgt0 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.)
 RR  0  <  0  < 
 1
 
Theoremprodgt0gt0 7598 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7599 for the same theorem with  0  < replaced by the weaker condition 
0  <_ . (Contributed by Jim Kingdon, 29-Feb-2020.)
 RR  RR 
 0  <  0  <  x.  0  <
 
Theoremprodgt0 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.)
 RR  RR 
 0  <_  0  <  x.  0  <
 
Theoremprodgt02 7600 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
 RR  RR 
 0  <_  0  <  x.  0  <
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