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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulcand 9401 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( C  x.  A )  =  ( C  x.  B )  <->  A  =  B ) )
 
Theoremmulcan2d 9402 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  x.  C )  =  ( B  x.  C )  <->  A  =  B ) )
 
Theoremmulcanad 9403 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9401. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( C  x.  A )  =  ( C  x.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremmulcan2ad 9404 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9402. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( A  x.  C )  =  ( B  x.  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremmulcan 9405 Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  A )  =  ( C  x.  B ) 
 <->  A  =  B ) )
 
Theoremmulcan2 9406 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  x.  C )  =  ( B  x.  C ) 
 <->  A  =  B ) )
 
Theoremmulcani 9407 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C  =/=  0   =>    |-  ( ( C  x.  A )  =  ( C  x.  B )  <->  A  =  B )
 
Theoremmul0or 9408 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <->  ( A  =  0  \/  B  =  0 ) ) )
 
Theoremmulne0b 9409 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  =/=  0  /\  B  =/=  0 )  <->  ( A  x.  B )  =/=  0
 ) )
 
Theoremmulne0 9410 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( A  x.  B )  =/=  0 )
 
Theoremmulne0i 9411 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A  =/=  0   &    |-  B  =/=  0   =>    |-  ( A  x.  B )  =/=  0
 
Theoremmuleqadd 9412 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  ( A  +  B ) 
 <->  ( ( A  -  1 )  x.  ( B  -  1 ) )  =  1 ) )
 
Theoremreceu 9413* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  E! x  e.  CC  ( B  x.  x )  =  A )
 
Theoremmulnzcnopr 9414 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)
 |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) ) : ( ( CC  \  { 0 } )  X.  ( CC  \  {
 0 } ) ) --> ( CC  \  {
 0 } )
 
Theoremmsq0i 9415 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 9416 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 9417 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 9418 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 9419 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 9420 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 9421 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9420 and consequence of mulne0bd 9419. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  x.  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremmulne0bbd 9422 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9420 and consequence of mulne0bd 9419. (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 9423 Extend class notation to include division.
 class  /
 
Definitiondf-div 9424* Define division. Theorem divmuli 9514 relates it to multiplication, and divcli 9502 and redivcli 9527 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 9425 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 9426* 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 9427 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 9428 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 9429 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 9430 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 9431 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  CC )
 
Theoremdivcan2 9432 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 9433 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 9434 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 9435 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 9436 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 9437 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 9438 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 9439 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 9440 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 9441 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 9442 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 9443 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 9444 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 9445 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 9446 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 9447 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 9448 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 9449 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 9450 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 9451 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( A  /  A )  =  1 )
 
Theoremdiv0 9452 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 9453 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 9454 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 9455 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 9456 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 9457 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 9458 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 9459 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 9460 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 9461 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 9462 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 9463 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 9464 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 9465 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 9466 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 9467 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 9468 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 9469 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 9470 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 9471 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 9472 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 9473 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 9474 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 9475 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 9476 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 9477 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 9478 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 9479 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 9480 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 9481 A complex number equals its negative iff it is zero. Deduction form of eqneg 9480. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 9482 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9480. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2neg 9483 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 9484 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 9485 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0zi 9486 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecidzi 9487 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremdiv1i 9488 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 9489 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremreccli 9490 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidi 9491 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecreci 9492 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 9493 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0i 9494 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclzi 9495 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 9496 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 9497 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 9498 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 9499 A cancellation law for division. (Eliminates a hypothesis of divcan3i 9506 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 9500 A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( ( A  x.  B )  /  B )  =  A )
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