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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivclap 7701 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  e.  CC )
 
Theoremrecclap 7702 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 1  /  A )  e.  CC )
 
Theoremdivcanap2 7703 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivcanap1 7704 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  /  B )  x.  B )  =  A )
 
Theoremdiveqap0 7705 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  /  B )  =  0  <->  A  =  0 ) )
 
Theoremdivap0b 7706 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A #  0  <->  ( A  /  B ) #  0 )
 )
 
Theoremdivap0 7707 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  /  B ) #  0 )
 
Theoremrecap0 7708 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 1  /  A ) #  0 )
 
Theoremrecidap 7709 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( A  x.  (
 1  /  A )
 )  =  1 )
 
Theoremrecidap2 7710 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( ( 1  /  A )  x.  A )  =  1 )
 
Theoremdivrecap 7711 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivrecap2 7712 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( ( 1 
 /  B )  x.  A ) )
 
Theoremdivassap 7713 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv23ap 7714 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( ( A  x.  B )  /  C )  =  ( ( A 
 /  C )  x.  B ) )
 
Theoremdiv32ap 7715 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  C  e.  CC )  ->  ( ( A 
 /  B )  x.  C )  =  ( A  x.  ( C 
 /  B ) ) )
 
Theoremdiv13ap 7716 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  C  e.  CC )  ->  ( ( A 
 /  B )  x.  C )  =  ( ( C  /  B )  x.  A ) )
 
Theoremdiv12ap 7717 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdivdirap 7718 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 7719 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 7720 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 7721 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 7722 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( A  /  A )  =  1 )
 
Theoremdiv0ap 7723 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 7724 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 7725 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  ( 1  /  1
 )  =  1
 
Theoremdiveqap1 7726 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 7727 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 7728 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 7729 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 7730 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 7731 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 7732 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 7733 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 7734 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 7735 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 7736 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 7737 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 7738 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 7739 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 7740 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 7741 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 7742 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 7743 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 7744 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 7745 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 7746 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 7747 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 7748 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 7749 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 7750 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 7751 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( 1  /  A )  e.  RR )
 
Theoremredivclap 7752 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 7753 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 7754 A complex number equals its negative iff it is zero. Deduction form of eqneg 7753. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 7755 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7753. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2negap 7756 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 7757 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 7758 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecap0apzi 7759 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 7760 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 7761 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 7762 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremrecclapi 7763 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidapi 7764 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecrecapi 7765 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 7766 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0api 7767 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclapzi 7768 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1zi 7769 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 7770 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 7771 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 7772 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 7773 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 7774 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 7775 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  e.  CC
 
Theoremdivcanap2i 7776 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 7777 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 7778 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 7779 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 7780 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 7781 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 7782 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 7783 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 7784 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 7785 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 7786 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 7787 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 7788 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 7789 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 7790 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 7791 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 7792 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 7793 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 7794 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 7795 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 7796 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 7797 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrerecclapi 7798 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclapzi 7799 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 7800 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
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