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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmsq0i 9601 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( ( A  x.  A )  =  0  <->  A  =  0
 )
 
Theoremmul0ori 9602 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  x.  B )  =  0  <->  ( A  =  0  \/  B  =  0 ) )
 
Theoremmsq0d 9603 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  A )  =  0  <->  A  =  0
 ) )
 
Theoremmul0ord 9604 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  x.  B )  =  0  <->  ( A  =  0  \/  B  =  0 ) ) )
 
Theoremmulne0bd 9605 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  =/=  0  /\  B  =/=  0 )  <-> 
 ( A  x.  B )  =/=  0 ) )
 
Theoremmulne0d 9606 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A  x.  B )  =/=  0 )
 
Theoremmulne0bad 9607 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9606 and consequence of mulne0bd 9605. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  x.  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremmulne0bbd 9608 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9606 and consequence of mulne0bd 9605. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  x.  B )  =/=  0
 )   =>    |-  ( ph  ->  B  =/=  0 )
 
5.3.6  Division
 
Syntaxcdiv 9609 Extend class notation to include division.
 class  /
 
Definitiondf-div 9610* Define division. Theorem divmuli 9700 relates it to multiplication, and divcli 9688 and redivcli 9713 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
 |- 
 /  =  ( x  e.  CC ,  y  e.  ( CC  \  {
 0 } )  |->  (
 iota_ z  e.  CC ( y  x.  z
 )  =  x ) )
 
Theorem1div0 9611 You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that  (/) is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
 |-  ( 1  /  0
 )  =  (/)
 
Theoremdivval 9612* Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( A  /  B )  =  ( iota_ x  e. 
 CC ( B  x.  x )  =  A ) )
 
Theoremdivmul 9613 Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  C )  =  B  <->  ( C  x.  B )  =  A ) )
 
Theoremdivmul2 9614 Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  C )  =  B  <->  A  =  ( C  x.  B ) ) )
 
Theoremdivmul3 9615 Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  C )  =  B  <->  A  =  ( B  x.  C ) ) )
 
Theoremdivcl 9616 Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  CC )
 
Theoremreccl 9617 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  CC )
 
Theoremdivcan2 9618 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivcan1 9619 A cancellation law for division. (Contributed by NM, 5-Jun-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  /  B )  x.  B )  =  A )
 
Theoremdiveq0 9620 A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  /  B )  =  0  <->  A  =  0 ) )
 
Theoremdivne0b 9621 The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( A  =/=  0  <->  ( A  /  B )  =/=  0 ) )
 
Theoremdivne0 9622 The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  B )  =/=  0 )
 
Theoremrecne0 9623 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecid 9624 Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremrecid2 9625 Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( (
 1  /  A )  x.  A )  =  1 )
 
Theoremdivrec 9626 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivrec2 9627 Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( A  /  B )  =  ( (
 1  /  B )  x.  A ) )
 
Theoremdivass 9628 An associative law for division. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv23 9629 A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  x.  B )  /  C )  =  (
 ( A  /  C )  x.  B ) )
 
Theoremdiv32 9630 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  C  e.  CC )  ->  ( ( A  /  B )  x.  C )  =  ( A  x.  ( C  /  B ) ) )
 
Theoremdiv13 9631 A commutative/associative law for division. (Contributed by NM, 22-Apr-2005.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  C  e.  CC )  ->  ( ( A  /  B )  x.  C )  =  ( ( C  /  B )  x.  A ) )
 
Theoremdiv12 9632 A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdivdir 9633 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivcan3 9634 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcan4 9635 A cancellation law for division. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremdiv11 9636 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  C )  =  ( B  /  C ) 
 <->  A  =  B ) )
 
Theoremdivid 9637 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( A  /  A )  =  1 )
 
Theoremdiv0 9638 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 9639 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 )
 
Theoremdiveq1 9640 Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  /  B )  =  1  <->  A  =  B ) )
 
Theoremdivneg 9641 Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  -u ( A  /  B )  =  ( -u A  /  B ) )
 
Theoremdivsubdir 9642 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 ) ) )
 
Theoremrecrec 9643 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 26-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  ( 1  /  A ) )  =  A )
 
Theoremrec11 9644 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B ) )
 
Theoremrec11r 9645 Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( 1  /  A )  =  B  <->  ( 1  /  B )  =  A ) )
 
Theoremdivmuldiv 9646 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( ( 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 ) ) )
 
Theoremdivdivdiv 9647 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( ( 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 ) ) )
 
Theoremdivcan5 9648 Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  A )  /  ( C  x.  B ) )  =  ( A  /  B ) )
 
Theoremdivmul13 9649 Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( 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 ) ) )
 
Theoremdivmul24 9650 Swap the numerators in the product of two ratios. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( 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 ) ) )
 
Theoremdivmuleq 9651 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( ( 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 ) ) )
 
Theoremrecdiv 9652 The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( 1  /  ( A  /  B ) )  =  ( B  /  A ) )
 
Theoremdivcan6 9653 Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  /  B )  x.  ( B  /  A ) )  =  1 )
 
Theoremdivdiv32 9654 Swap denominators in a division. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( ( A 
 /  C )  /  B ) )
 
Theoremdivcan7 9655 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  C )  /  ( B  /  C ) )  =  ( A  /  B ) )
 
Theoremdmdcan 9656 Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  (
 ( A  /  B )  x.  ( C  /  A ) )  =  ( C  /  B ) )
 
Theoremdivdiv1 9657 Division into a fraction. (Contributed by NM, 31-Dec-2007.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  /  B )  /  C )  =  ( A  /  ( B  x.  C ) ) )
 
Theoremdivdiv2 9658 Division by a fraction. (Contributed by NM, 27-Dec-2008.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  ( B  /  C ) )  =  ( ( A  x.  C )  /  B ) )
 
Theoremrecdiv2 9659 Division into a reciprocal. (Contributed by NM, 19-Oct-2007.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( 1  /  A )  /  B )  =  ( 1  /  ( A  x.  B ) ) )
 
Theoremddcan 9660 Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  /  ( A  /  B ) )  =  B )
 
Theoremdivadddiv 9661 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( 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 ) ) )
 
Theoremdivsubdiv 9662 Subtraction of two ratios. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  ( ( ( 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 ) ) )
 
Theoremconjmul 9663 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.)
 |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  ( ( ( 1 
 /  P )  +  ( 1  /  Q ) )  =  1  <->  ( ( P  -  1
 )  x.  ( Q  -  1 ) )  =  1 ) )
 
Theoremrereccl 9664 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  RR )
 
Theoremredivcl 9665 Closure law for division of reals. (Contributed by NM, 27-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  RR )
 
Theoremeqneg 9666 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 9667 A complex number equals its negative iff it is zero. Deduction form of eqneg 9666. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 9668 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9666. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2neg 9669 Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( -u A  /  -u B )  =  ( A  /  B ) )
 
Theoremdivneg2 9670 Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  -u ( A  /  B )  =  ( A  /  -u B ) )
 
Theoremrecclzi 9671 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0zi 9672 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecidzi 9673 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremdiv1i 9674 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 9675 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremreccli 9676 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidi 9677 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecreci 9678 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
 
Theoremdividi 9679 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0i 9680 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclzi 9681 Closure law for division. (Contributed by NM, 7-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( A  /  B )  e.  CC )
 
Theoremdivcan1zi 9682 A cancellation law for division. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( ( A  /  B )  x.  B )  =  A )
 
Theoremdivcan2zi 9683 A cancellation law for division. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivreczi 9684 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 11-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivcan3zi 9685 A cancellation law for division. (Eliminates a hypothesis of divcan3i 9692 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcan4zi 9686 A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremrec11i 9687 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  =/=  0  /\  B  =/=  0
 )  ->  ( (
 1  /  A )  =  ( 1  /  B ) 
 <->  A  =  B ) )
 
Theoremdivcli 9688 Closure law for division. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  ( A  /  B )  e. 
 CC
 
Theoremdivcan2i 9689 A cancellation law for division. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  ( B  x.  ( A  /  B ) )  =  A
 
Theoremdivcan1i 9690 A cancellation law for division. (Contributed by NM, 18-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  (
 ( A  /  B )  x.  B )  =  A
 
Theoremdivreci 9691 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  ( A  /  B )  =  ( A  x.  (
 1  /  B )
 )
 
Theoremdivcan3i 9692 A cancellation law for division. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  (
 ( B  x.  A )  /  B )  =  A
 
Theoremdivcan4i 9693 A cancellation law for division. (Contributed by NM, 18-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B  =/=  0   =>    |-  (
 ( A  x.  B )  /  B )  =  A
 
Theoremdivne0i 9694 The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A  =/=  0   &    |-  B  =/=  0   =>    |-  ( A  /  B )  =/=  0
 
Theoremrec11ii 9695 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A  =/=  0   &    |-  B  =/=  0   =>    |-  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B )
 
Theoremdivasszi 9696 An associative law for division. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C  =/=  0  ->  (
 ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdivmulzi 9697 Relationship between division and multiplication. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( B  =/=  0  ->  (
 ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremdivdirzi 9698 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C  =/=  0  ->  (
 ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivdiv23zi 9699 Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( B  =/=  0  /\  C  =/=  0 ) 
 ->  ( ( A  /  B )  /  C )  =  ( ( A 
 /  C )  /  B ) )
 
Theoremdivmuli 9700 Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  B  =/=  0   =>    |-  ( ( A  /  B )  =  C  <->  ( B  x.  C )  =  A )
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