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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcan1i 9501 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 9502 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 9503 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 9504 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 9505 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 9506 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 9507 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 9508 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 9509 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 9510 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 9511 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 9512 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 9513 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 9514 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 9515 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 9516 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 9517 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 9518 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 9519 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 9520 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 9521 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrereccli 9522 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclzi 9523 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 9524 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 9525 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremreccld 9526 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0d 9527 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 9528 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 9529 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 9530 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 9531 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 9532 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 0  /  A )  =  0 )
 
Theoremdivcld 9533 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 9534 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 9535 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 9536 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 9537 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 9538 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 9539 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 9540 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 9541 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 9542 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 9451. Generalization of diveq1d 9541. (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 9543 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 9431. (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 9544 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 9541. (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 9545 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 9546 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 9547 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 9548 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 9549 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 9550 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 9551 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 9552 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 9553 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 9554 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 9555 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 9556 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 9557 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 9558 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 9559 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 9560 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 9561 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 9562 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 9563 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 9564 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 9565 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 9566 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 9567 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 9568 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 9569 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 9570 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 9571 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 9572 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 ) ) )
 
Theoremdiv11d 9573 One-to-one relationship 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  /  C )  =  ( B  /  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuldivd 9574 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( C  /  D ) )  =  ( ( A  x.  C )  /  ( B  x.  D ) ) )
 
Theoremdivmul13d 9575 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( C  /  D ) )  =  ( ( C  /  B )  x.  ( A  /  D ) ) )
 
Theoremdivmul24d 9576 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  ( C  /  D ) )  =  ( ( A  /  D )  x.  ( C  /  B ) ) )
 
Theoremdivadddivd 9577 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  +  ( C  /  D ) )  =  ( ( ( A  x.  D )  +  ( C  x.  B ) )  /  ( B  x.  D ) ) )
 
Theoremdivsubdivd 9578 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  -  ( C  /  D ) )  =  ( ( ( A  x.  D )  -  ( C  x.  B ) )  /  ( B  x.  D ) ) )
 
Theoremdivmuleqd 9579 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  ( C  /  D )  <->  ( A  x.  D )  =  ( C  x.  B ) ) )
 
Theoremdivdivdivd 9580 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  D  =/=  0
 )   &    |-  ( ph  ->  C  =/=  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  /  ( C  /  D ) )  =  ( ( A  x.  D )  /  ( B  x.  C ) ) )
 
Theoremdiveq1bd 9581 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 9451. Converse of diveq1d 9541. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  /  B )  =  1 )
 
Theoremdiv2sub 9582 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C  =/=  D ) )  ->  (
 ( A  -  B )  /  ( C  -  D ) )  =  ( ( B  -  A )  /  ( D  -  C ) ) )
 
Theoremdiv2subd 9583 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 9582. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  C  =/=  D )   =>    |-  ( ph  ->  ( ( A  -  B )  /  ( C  -  D ) )  =  ( ( B  -  A )  /  ( D  -  C ) ) )
 
Theoremrereccld 9584 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  RR )
 
Theoremredivcld 9585 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 RR )
 
Theoremsubrec 9586 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( 1  /  A )  -  (
 1  /  B )
 )  =  ( ( B  -  A ) 
 /  ( A  x.  B ) ) )
 
Theoremsubreci 9587 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A  =/=  0   &    |-  B  =/=  0   =>    |-  ( ( 1  /  A )  -  (
 1  /  B )
 )  =  ( ( B  -  A ) 
 /  ( A  x.  B ) )
 
Theoremsubrecd 9588 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A )  -  ( 1  /  B ) )  =  ( ( B  -  A )  /  ( A  x.  B ) ) )
 
5.3.7  Ordering on reals (cont.)
 
Theoremelimgt0 9589 Hypothesis for weak deduction theorem to eliminate  0  <  A. (Contributed by NM, 15-May-1999.)
 |-  0  <  if (
 0  <  A ,  A ,  1 )
 
Theoremelimge0 9590 Hypothesis for weak deduction theorem to eliminate  0  <_  A. (Contributed by NM, 30-Jul-1999.)
 |-  0  <_  if (
 0  <_  A ,  A ,  0 )
 
Theoremltp1 9591 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 9592 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1 9593 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <  A )
 
Theoremlem1 9594 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <_  A )
 
Theoremletrp1 9595 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  ( B  +  1 ) )
 
Theoremp1le 9596 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  1 )  <_  B )  ->  A  <_  B )
 
Theoremrecgt0 9597 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  0  <  (
 1  /  A )
 )
 
Theoremprodgt0 9598 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  B )
 
Theoremprodgt02 9599 Infer that a multiplier is positive from a nonnegative muliplicand and positive product. (Contributed by NM, 24-Apr-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  B  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  A )
 
Theoremprodge0 9600 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <_  ( A  x.  B ) ) ) 
 ->  0  <_  B )
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