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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivdirap 8601 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( A  +  B )  /  C )  =  ( ( A 
 /  C )  +  ( B  /  C ) ) )
 
Theoremdivcanap3 8602 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcanap4 8603 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremdiv11ap 8604 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( A  /  C )  =  ( B  /  C )  <->  A  =  B ) )
 
Theoremdividap 8605 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( A  /  A )  =  1 )
 
Theoremdiv0ap 8606 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 8607 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  /  1
 )  =  A )
 
Theorem1div1e1 8608 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  ( 1  /  1
 )  =  1
 
Theoremdiveqap1 8609 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  /  B )  =  1  <->  A  =  B ) )
 
Theoremdivnegap 8610 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( A  /  B )  =  ( -u A  /  B ) )
 
Theoremmuldivdirap 8611 Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( ( C  x.  A )  +  B )  /  C )  =  ( A  +  ( B  /  C ) ) )
 
Theoremdivsubdirap 8612 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( A  -  B )  /  C )  =  ( ( A 
 /  C )  -  ( B  /  C ) ) )
 
Theoremrecrecap 8613 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 1  /  (
 1  /  A )
 )  =  A )
 
Theoremrec11ap 8614 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B ) )
 
Theoremrec11rap 8615 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  /  A )  =  B  <->  ( 1  /  B )  =  A ) )
 
Theoremdivmuldivap 8616 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D ) )  =  (
 ( A  x.  B )  /  ( C  x.  D ) ) )
 
Theoremdivdivdivap 8617 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( ( C  e.  CC  /\  C #  0 ) 
 /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  (
 ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C ) ) )
 
Theoremdivcanap5 8618 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( C  x.  A )  /  ( C  x.  B ) )  =  ( A  /  B ) )
 
Theoremdivmul13ap 8619 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D ) )  =  (
 ( B  /  C )  x.  ( A  /  D ) ) )
 
Theoremdivmul24ap 8620 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D ) )  =  (
 ( A  /  D )  x.  ( B  /  C ) ) )
 
Theoremdivmuleqap 8621 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  =  ( B  /  D ) 
 <->  ( A  x.  D )  =  ( B  x.  C ) ) )
 
Theoremrecdivap 8622 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
 
Theoremdivcanap6 8623 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( A  /  B )  x.  ( B  /  A ) )  =  1 )
 
Theoremdivdiv32ap 8624 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  (
 ( A  /  C )  /  B ) )
 
Theoremdivcanap7 8625 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  C )  /  ( B  /  C ) )  =  ( A 
 /  B ) )
 
Theoremdmdcanap 8626 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 )  /\  C  e.  CC )  ->  ( ( A  /  B )  x.  ( C  /  A ) )  =  ( C  /  B ) )
 
Theoremdivdivap1 8627 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( A  /  ( B  x.  C ) ) )
 
Theoremdivdivap2 8628 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  ( B  /  C ) )  =  (
 ( A  x.  C )  /  B ) )
 
Theoremrecdivap2 8629 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  /  A )  /  B )  =  ( 1  /  ( A  x.  B ) ) )
 
Theoremddcanap 8630 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  /  ( A  /  B ) )  =  B )
 
Theoremdivadddivap 8631 Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  +  ( B  /  D ) )  =  ( ( ( A  x.  D )  +  ( B  x.  C ) )  /  ( C  x.  D ) ) )
 
Theoremdivsubdivap 8632 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
 ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  -  ( B  /  D ) )  =  ( ( ( A  x.  D )  -  ( B  x.  C ) )  /  ( C  x.  D ) ) )
 
Theoremconjmulap 8633 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( 1 
 /  P )  +  ( 1  /  Q ) )  =  1  <->  ( ( P  -  1
 )  x.  ( Q  -  1 ) )  =  1 ) )
 
Theoremrerecclap 8634 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( 1  /  A )  e.  RR )
 
Theoremredivclap 8635 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B #  0 )  ->  ( A  /  B )  e.  RR )
 
Theoremeqneg 8636 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  =  -u A 
 <->  A  =  0 ) )
 
Theoremeqnegd 8637 A complex number equals its negative iff it is zero. Deduction form of eqneg 8636. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 8638 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8636. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2negap 8639 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u A  /  -u B )  =  ( A  /  B ) )
 
Theoremdivneg2ap 8640 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( A  /  B )  =  ( A  /  -u B ) )
 
Theoremrecclapzi 8641 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecap0apzi 8642 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( 1  /  A ) #  0 )
 
Theoremrecidapzi 8643 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( A  x.  (
 1  /  A )
 )  =  1 )
 
Theoremdiv1i 8644 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 8645 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremrecclapi 8646 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidapi 8647 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecrecapi 8648 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 1  /  (
 1  /  A )
 )  =  A
 
Theoremdividapi 8649 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0api 8650 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclapzi 8651 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1zi 8652 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( A  /  B )  x.  B )  =  A )
 
Theoremdivcanap2zi 8653 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivrecapzi 8654 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivcanap3zi 8655 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcanap4zi 8656 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremrec11api 8657 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A #  0  /\  B #  0 )  ->  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B ) )
 
Theoremdivclapi 8658 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  e.  CC
 
Theoremdivcanap2i 8659 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( B  x.  ( A  /  B ) )  =  A
 
Theoremdivcanap1i 8660 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( A 
 /  B )  x.  B )  =  A
 
Theoremdivrecapi 8661 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  =  ( A  x.  ( 1  /  B ) )
 
Theoremdivcanap3i 8662 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( B  x.  A )  /  B )  =  A
 
Theoremdivcanap4i 8663 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( A  x.  B )  /  B )  =  A
 
Theoremdivap0i 8664 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( A  /  B ) #  0
 
Theoremrec11apii 8665 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B )
 
Theoremdivassapzi 8666 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C #  0  ->  ( ( A  x.  B ) 
 /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdivmulapzi 8667 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( B #  0  ->  ( ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremdivdirapzi 8668 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C #  0  ->  ( ( A  +  B ) 
 /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivdiv23apzi 8669 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( B #  0  /\  C #  0 )  ->  (
 ( A  /  B )  /  C )  =  ( ( A  /  C )  /  B ) )
 
Theoremdivmulapi 8670 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  B #  0   =>    |-  ( ( A  /  B )  =  C  <->  ( B  x.  C )  =  A )
 
Theoremdivdiv32api 8671 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  B #  0   &    |-  C #  0   =>    |-  ( ( A  /  B )  /  C )  =  ( ( A 
 /  C )  /  B )
 
Theoremdivassapi 8672 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) )
 
Theoremdivdirapi 8673 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  +  B )  /  C )  =  ( ( A 
 /  C )  +  ( B  /  C ) )
 
Theoremdiv23api 8674 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( ( A 
 /  C )  x.  B )
 
Theoremdiv11api 8675 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  /  C )  =  ( B  /  C )  <->  A  =  B )
 
Theoremdivmuldivapi 8676 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  B #  0   &    |-  D #  0   =>    |-  ( ( A  /  B )  x.  ( C  /  D ) )  =  ( ( A  x.  C )  /  ( B  x.  D ) )
 
Theoremdivmul13api 8677 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  B #  0   &    |-  D #  0   =>    |-  ( ( A  /  B )  x.  ( C  /  D ) )  =  ( ( C 
 /  B )  x.  ( A  /  D ) )
 
Theoremdivadddivapi 8678 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  B #  0   &    |-  D #  0   =>    |-  ( ( A  /  B )  +  ( C  /  D ) )  =  ( ( ( A  x.  D )  +  ( C  x.  B ) )  /  ( B  x.  D ) )
 
Theoremdivdivdivapi 8679 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   &    |-  B #  0   &    |-  D #  0   &    |-  C #  0   =>    |-  ( ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C ) )
 
Theoremrerecclapzi 8680 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrerecclapi 8681 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclapzi 8682 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( B #  0  ->  ( A  /  B )  e.  RR )
 
Theoremredivclapi 8683 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  B #  0   =>    |-  ( A  /  B )  e.  RR
 
Theoremdiv1d 8684 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremrecclapd 8685 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  CC )
 
Theoremrecap0d 8686 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A ) #  0 )
 
Theoremrecidapd 8687 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( A  x.  ( 1  /  A ) )  =  1 )
 
Theoremrecidap2d 8688 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A )  x.  A )  =  1 )
 
Theoremrecrecapd 8689 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  ( 1  /  A ) )  =  A )
 
Theoremdividapd 8690 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( A  /  A )  =  1 )
 
Theoremdiv0apd 8691 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 0  /  A )  =  0 )
 
Theoremapmul1 8692 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
 
Theoremapmul2 8693 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( A #  B  <->  ( C  x.  A ) #  ( C  x.  B ) ) )
 
Theoremdivclapd 8694 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1d 8695 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B )  x.  B )  =  A )
 
Theoremdivcanap2d 8696 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivrecapd 8697 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivrecap2d 8698 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  =  (
 ( 1  /  B )  x.  A ) )
 
Theoremdivcanap3d 8699 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcanap4d 8700 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A  x.  B )  /  B )  =  A )
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