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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivmul13i 9401 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 9402 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 9403 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 9404 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   =>    |-  ( A  =/=  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrereccli 9405 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  RR   &    |-  A  =/=  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclzi 9406 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 9407 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 9408 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremreccld 9409 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  CC )
 
Theoremrecne0d 9410 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 9411 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 9412 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 9413 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 9414 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 9415 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 0  /  A )  =  0 )
 
Theoremdivcld 9416 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 9417 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 9418 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 9419 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 9420 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 9421 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 9422 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 9423 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 9424 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 9425 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 9334. Generalization of diveq1d 9424. (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 9426 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 9314. (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 9427 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 9424. (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 9428 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 9429 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 9430 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 9431 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 9432 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 9433 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 9434 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 9435 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 9436 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 9437 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 9438 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 9439 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 9440 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 9441 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 9442 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 9443 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 9444 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 9445 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 9446 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 9447 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 9448 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 9449 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 9450 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 9451 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 9452 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 9453 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 9454 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 9455 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 9456 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 9457 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 9458 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 9459 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 9460 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 9461 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 9462 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 9463 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 9464 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 9334. Converse of diveq1d 9424. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0 )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  /  B )  =  1 )
 
Theoremdiv2sub 9465 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 9466 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 9465. (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 9467 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( 1  /  A )  e.  RR )
 
Theoremredivcld 9468 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 9469 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 9470 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 9471 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 9472 Hypothesis for weak deduction theorem to eliminate  0  <  A. (Contributed by NM, 15-May-1999.)
 |-  0  <  if (
 0  <  A ,  A ,  1 )
 
Theoremelimge0 9473 Hypothesis for weak deduction theorem to eliminate  0  <_  A. (Contributed by NM, 30-Jul-1999.)
 |-  0  <_  if (
 0  <_  A ,  A ,  0 )
 
Theoremltp1 9474 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 9475 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1 9476 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <  A )
 
Theoremlem1 9477 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 9478 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 9479 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 9480 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 9481 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 9482 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 9483 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 )
 
Theoremprodge02 9484 Infer that a multiplier is nonnegative from a positive muliplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  B  /\  0  <_  ( A  x.  B ) ) ) 
 ->  0  <_  A )
 
Theoremltmul1a 9485 Lemma for ltmul1 9486. Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  <  B )  ->  ( A  x.  C )  <  ( B  x.  C ) )
 
Theoremltmul1 9486 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremltmul2 9487 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1 9488 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2 9489 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremlemul1a 9490 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2a 9491 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  ->  ( C  x.  A )  <_  ( C  x.  B ) )
 
Theoremltmul12a 9492 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
 |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B ) )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  < 
 ( B  x.  D ) )
 
Theoremlemul12b 9493 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <_  D )
 ) )  ->  (
 ( A  <_  B  /\  C  <_  D )  ->  ( A  x.  C )  <_  ( B  x.  D ) ) )
 
Theoremlemul12a 9494 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
 |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR )  /\  ( ( C  e.  RR  /\  0  <_  C )  /\  D  e.  RR ) )  ->  ( ( A  <_  B  /\  C  <_  D )  ->  ( A  x.  C )  <_  ( B  x.  D ) ) )
 
Theoremmulgt1 9495 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 1  <  A  /\  1  <  B ) ) 
 ->  1  <  ( A  x.  B ) )
 
Theoremltmulgt11 9496 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A ) 
 ->  ( 1  <  B  <->  A  <  ( A  x.  B ) ) )
 
Theoremltmulgt12 9497 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  A ) 
 ->  ( 1  <  B  <->  A  <  ( B  x.  A ) ) )
 
Theoremlemulge11 9498 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  1  <_  B ) ) 
 ->  A  <_  ( A  x.  B ) )
 
Theoremlemulge12 9499 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  1  <_  B ) ) 
 ->  A  <_  ( B  x.  A ) )
 
Theoremltdiv1 9500 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
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