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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivsubdiv 10301 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 10302 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 10303 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 10304 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 10305 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 10306 A complex number equals its negative iff it is zero. Deduction form of eqneg 10305. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 10307 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 10305. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2neg 10308 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 10309 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 10310 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0zi 10311 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  =/=  0
 )
 
Theoremrecidzi 10312 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0  ->  ( A  x.  ( 1  /  A ) )  =  1
 )
 
Theoremdiv1i 10313 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 10314 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremreccli 10315 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidi 10316 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecreci 10317 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 10318 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0i 10319 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclzi 10320 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 10321 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 10322 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 10323 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 10324 A cancellation law for division. (Eliminates a hypothesis of divcan3i 10331 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 10325 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 10326 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 10327 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 10328 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 10329 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 10330 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 10331 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 10332 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 10333 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 10334 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 10335 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 10336 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 10337 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 10338 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 10339 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 10340 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 10341 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 10342 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 10343 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 10344 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 10345 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 10346 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 10347 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 10348 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 10349 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrereccli 10350 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclzi 10351 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 10352 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 10353 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremreccld 10354 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0d 10355 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 10356 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 10357 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 )
 
Theoremrecrecd 10358 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  ( 1 
 /  A ) )  =  A )
 
Theoremdividd 10359 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( A  /  A )  =  1 )
 
Theoremdiv0d 10360 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 0  /  A )  =  0 )
 
Theoremdivcld 10361 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 CC )
 
Theoremdivcan1d 10362 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  B )  =  A )
 
Theoremdivcan2d 10363 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivrecd 10364 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  /  B )  =  ( A  x.  (
 1  /  B )
 ) )
 
Theoremdivrec2d 10365 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  /  B )  =  ( ( 1  /  B )  x.  A ) )
 
Theoremdivcan3d 10366 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( B  x.  A )  /  B )  =  A )
 
Theoremdivcan4d 10367 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  B )  =  A )
 
Theoremdiveq0d 10368 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   &    |-  ( ph  ->  ( A  /  B )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiveq1d 10369 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   &    |-  ( ph  ->  ( A  /  B )  =  1 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdiveq1ad 10370 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 10279. Generalization of diveq1d 10369. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  1  <->  A  =  B ) )
 
Theoremdiveq0ad 10371 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 10258. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  0  <->  A  =  0
 ) )
 
Theoremdivne1d 10372 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 10369. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  /  B )  =/=  1 )
 
Theoremdivne0bd 10373 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  =/=  0  <->  ( A  /  B )  =/=  0
 ) )
 
Theoremdivnegd 10374 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  -u ( A  /  B )  =  ( -u A  /  B ) )
 
Theoremdivneg2d 10375 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  -u ( A  /  B )  =  ( A  /  -u B ) )
 
Theoremdiv2negd 10376 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( -u A  /  -u B )  =  ( A  /  B ) )
 
Theoremdivne0d 10377 The ratio of 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  /  B )  =/=  0 )
 
Theoremrecdivd 10378 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 1  /  ( A  /  B ) )  =  ( B  /  A ) )
 
Theoremrecdiv2d 10379 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A )  /  B )  =  ( 1  /  ( A  x.  B ) ) )
 
Theoremdivcan6d 10380 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( B  /  A ) )  =  1 )
 
Theoremddcand 10381 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A  /  ( A  /  B ) )  =  B )
 
Theoremrec11d 10382 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  ( 1  /  A )  =  ( 1  /  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuld 10383 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremdiv32d 10384 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( A  x.  ( C  /  B ) ) )
 
Theoremdiv13d 10385 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( ( C  /  B )  x.  A ) )
 
Theoremdivdiv32d 10386 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  /  B )  /  C )  =  ( ( A  /  C )  /  B ) )
 
Theoremdivcan5d 10387 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( C  x.  A )  /  ( C  x.  B ) )  =  ( A  /  B ) )
 
Theoremdivcan5rd 10388 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  x.  C )  /  ( B  x.  C ) )  =  ( A  /  B ) )
 
Theoremdivcan7d 10389 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  /  C )  /  ( B  /  C ) )  =  ( A  /  B ) )
 
Theoremdmdcand 10390 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( B  /  C )  x.  ( A  /  B ) )  =  ( A  /  C ) )
 
Theoremdmdcan2d 10391 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  /  B )  x.  ( B  /  C ) )  =  ( A  /  C ) )
 
Theoremdivdiv1d 10392 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( ( A  /  B )  /  C )  =  ( A  /  ( B  x.  C ) ) )
 
Theoremdivdiv2d 10393 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( A  /  ( B 
 /  C ) )  =  ( ( A  x.  C )  /  B ) )
 
Theoremdivmul2d 10394 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( C  x.  B ) ) )
 
Theoremdivmul3d 10395 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( B  x.  C ) ) )
 
Theoremdivassd 10396 An associative law for division. (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.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv12d 10397 A commutative/associative law for division. (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.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdiv23d 10398 A commutative/associative law for division. (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.  B )  /  C )  =  ( ( A  /  C )  x.  B ) )
 
Theoremdivdird 10399 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivsubdird 10400 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  -  B )  /  C )  =  ( ( A  /  C )  -  ( B  /  C ) ) )
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