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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcan1 9401 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 9402 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 9403 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 9404 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 9405 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 9406 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 9407 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 9408 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 9409 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 9410 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 9411 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 9412 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 9413 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 9414 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 9415 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 9416 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 9417 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 9418 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 9419 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( A  /  A )  =  1 )
 
Theoremdiv0 9420 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 9421 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 9422 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 9423 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 9424 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 9425 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 9426 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 9427 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 9428 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 9429 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 9430 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 9431 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 9432 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 9433 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 9434 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 9435 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 9436 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 9437 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 9438 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 9439 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 9440 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 9441 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 9442 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 9443 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 9444 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 9445 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 9446 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 9447 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 9448 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 9449 A complex number equals its negative iff it is zero. Deduction form of eqneg 9448. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 9450 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9448. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2neg 9451 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 9452 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 9453 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0zi 9454 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecidzi 9455 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremdiv1i 9456 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 9457 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremreccli 9458 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidi 9459 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecreci 9460 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 9461 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0i 9462 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclzi 9463 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 9464 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 9465 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 9466 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 9467 A cancellation law for division. (Eliminates a hypothesis of divcan3i 9474 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 9468 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 9469 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 9470 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 9471 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 9472 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 9473 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 9474 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 9475 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 9476 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 9477 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 9478 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 9479 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 9480 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 9481 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 9482 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 )
 
Theoremdivdiv32i 9483 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 )
 
Theoremdivassi 9484 An associative law for division. (Contributed by NM, 15-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C  =/=  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) )
 
Theoremdivdiri 9485 Distribution of division over addition. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C  =/=  0   =>    |-  ( ( A  +  B )  /  C )  =  ( ( A 
 /  C )  +  ( B  /  C ) )
 
Theoremdiv23i 9486 A commutative/associative law for division. (Contributed by NM, 3-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C  =/=  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( ( A 
 /  C )  x.  B )
 
Theoremdiv11i 9487 One-to-one relationship for division. (Contributed by NM, 20-Aug-2001.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C  =/=  0   =>    |-  ( ( A  /  C )  =  ( B  /  C )  <->  A  =  B )
 
Theoremdivmuldivi 9488 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 16-Feb-1995.)
 |-  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 ) )
 
Theoremdivmul13i 9489 Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999.)
 |-  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 ) )
 
Theoremdivadddivi 9490 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 21-Feb-1995.)
 |-  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 ) )
 
Theoremdivdivdivi 9491 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.)
 |-  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 ) )
 
Theoremrerecclzi 9492 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrereccli 9493 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclzi 9494 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( B  =/=  0  ->  ( A  /  B )  e.  RR )
 
Theoremredivcli 9495 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  B  =/=  0   =>    |-  ( A  /  B )  e. 
 RR
 
Theoremdiv1d 9496 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremreccld 9497 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0d 9498 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  =/=  0 )
 
Theoremrecidd 9499 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( A  x.  ( 1 
 /  A ) )  =  1 )
 
Theoremrecid2d 9500 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( ( 1  /  A )  x.  A )  =  1 )
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