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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulne0bad 9301 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9300 and consequence of mulne0bd 9299. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  x.  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremmulne0bbd 9302 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9300 and consequence of mulne0bd 9299. (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 9303 Extend class notation to include division.
 class  /
 
Definitiondf-div 9304* Define division. Theorem divmuli 9394 relates it to multiplication, and divcli 9382 and redivcli 9407 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.)
 |- 
 /  =  ( x  e.  CC ,  y  e.  ( CC  \  {
 0 } )  |->  (
 iota_ z  e.  CC ( y  x.  z
 )  =  x ) )
 
Theorem1div0 9305 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.)
 |-  ( 1  /  0
 )  =  (/)
 
Theoremdivval 9306* 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 9307 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 9308 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 9309 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 9310 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 9311 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  CC )
 
Theoremdivcan2 9312 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 9313 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 9314 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 9315 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 9316 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 9317 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 9318 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 9319 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 9320 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 9321 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 9322 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 9323 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 9324 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 9325 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 9326 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 9327 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 9328 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 9329 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 9330 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 9331 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( A  /  A )  =  1 )
 
Theoremdiv0 9332 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 0  /  A )  =  0 )
 
Theoremdiv1 9333 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 9334 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 9335 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 9336 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 9337 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 9338 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 9339 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 9340 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 9341 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 9342 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 9343 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 9344 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 9345 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 9346 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 9347 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 9348 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 9349 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 9350 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 9351 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 9352 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 9353 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 9354 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 9355 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 9356 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 9357 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 9358 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 9359 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 9360 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 9361 A complex number equals its negative iff it is zero. Deduction form of eqneg 9360. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 9362 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9360. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2neg 9363 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 9364 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 9365 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0zi 9366 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecidzi 9367 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremdiv1i 9368 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 9369 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremreccli 9370 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidi 9371 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecreci 9372 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 9373 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0i 9374 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclzi 9375 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 9376 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 9377 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 9378 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 9379 A cancellation law for division. (Eliminates a hypothesis of divcan3i 9386 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 9380 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 9381 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 9382 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 9383 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 9384 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 9385 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 9386 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 9387 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 9388 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 9389 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 9390 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 9391 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 9392 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 9393 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 9394 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 9395 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 9396 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 9397 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 9398 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 9399 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 9400 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 ) )
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