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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremconjmulap 8501 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( 1 
 /  P )  +  ( 1  /  Q ) )  =  1  <->  ( ( P  -  1
 )  x.  ( Q  -  1 ) )  =  1 ) )
 
Theoremrerecclap 8502 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( 1  /  A )  e.  RR )
 
Theoremredivclap 8503 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B #  0 )  ->  ( A  /  B )  e.  RR )
 
Theoremeqneg 8504 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 8505 A complex number equals its negative iff it is zero. Deduction form of eqneg 8504. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  -u A  <->  A  =  0
 ) )
 
Theoremeqnegad 8506 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8504. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  -u A )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiv2negap 8507 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u A  /  -u B )  =  ( A  /  B ) )
 
Theoremdivneg2ap 8508 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( A  /  B )  =  ( A  /  -u B ) )
 
Theoremrecclapzi 8509 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  CC )
 
Theoremrecap0apzi 8510 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( 1  /  A ) #  0 )
 
Theoremrecidapzi 8511 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   =>    |-  ( A #  0  ->  ( A  x.  (
 1  /  A )
 )  =  1 )
 
Theoremdiv1i 8512 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
 |-  A  e.  CC   =>    |-  ( A  / 
 1 )  =  A
 
Theoremeqnegi 8513 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  =  -u A  <->  A  =  0
 )
 
Theoremrecclapi 8514 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  CC
 
Theoremrecidapi 8515 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  x.  (
 1  /  A )
 )  =  1
 
Theoremrecrecapi 8516 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
 
Theoremdividapi 8517 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( A  /  A )  =  1
 
Theoremdiv0api 8518 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclapzi 8519 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1zi 8520 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( A  /  B )  x.  B )  =  A )
 
Theoremdivcanap2zi 8521 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivrecapzi 8522 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivcanap3zi 8523 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcanap4zi 8524 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremrec11api 8525 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A #  0  /\  B #  0 )  ->  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B ) )
 
Theoremdivclapi 8526 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  e.  CC
 
Theoremdivcanap2i 8527 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( B  x.  ( A  /  B ) )  =  A
 
Theoremdivcanap1i 8528 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( A 
 /  B )  x.  B )  =  A
 
Theoremdivrecapi 8529 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  =  ( A  x.  ( 1  /  B ) )
 
Theoremdivcanap3i 8530 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( B  x.  A )  /  B )  =  A
 
Theoremdivcanap4i 8531 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( ( A  x.  B )  /  B )  =  A
 
Theoremdivap0i 8532 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( A  /  B ) #  0
 
Theoremrec11apii 8533 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( ( 1  /  A )  =  (
 1  /  B )  <->  A  =  B )
 
Theoremdivassapzi 8534 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C #  0  ->  ( ( A  x.  B ) 
 /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdivmulapzi 8535 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( B #  0  ->  ( ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremdivdirapzi 8536 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( C #  0  ->  ( ( A  +  B ) 
 /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivdiv23apzi 8537 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( B #  0  /\  C #  0 )  ->  (
 ( A  /  B )  /  C )  =  ( ( A  /  C )  /  B ) )
 
Theoremdivmulapi 8538 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  B #  0   =>    |-  ( ( A  /  B )  =  C  <->  ( B  x.  C )  =  A )
 
Theoremdivdiv32api 8539 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  B #  0   &    |-  C #  0   =>    |-  ( ( A  /  B )  /  C )  =  ( ( A 
 /  C )  /  B )
 
Theoremdivassapi 8540 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) )
 
Theoremdivdirapi 8541 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  +  B )  /  C )  =  ( ( A 
 /  C )  +  ( B  /  C ) )
 
Theoremdiv23api 8542 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  x.  B )  /  C )  =  ( ( A 
 /  C )  x.  B )
 
Theoremdiv11api 8543 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  C #  0   =>    |-  ( ( A  /  C )  =  ( B  /  C )  <->  A  =  B )
 
Theoremdivmuldivapi 8544 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  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 ) )
 
Theoremdivmul13api 8545 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  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 ) )
 
Theoremdivadddivapi 8546 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  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 ) )
 
Theoremdivdivdivapi 8547 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  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 ) )
 
Theoremrerecclapzi 8548 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrerecclapi 8549 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclapzi 8550 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( B #  0  ->  ( A  /  B )  e.  RR )
 
Theoremredivclapi 8551 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  B #  0   =>    |-  ( A  /  B )  e.  RR
 
Theoremdiv1d 8552 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremrecclapd 8553 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  CC )
 
Theoremrecap0d 8554 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A ) #  0 )
 
Theoremrecidapd 8555 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( A  x.  ( 1  /  A ) )  =  1 )
 
Theoremrecidap2d 8556 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 ( 1  /  A )  x.  A )  =  1 )
 
Theoremrecrecapd 8557 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  ( 1  /  A ) )  =  A )
 
Theoremdividapd 8558 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( A  /  A )  =  1 )
 
Theoremdiv0apd 8559 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 0  /  A )  =  0 )
 
Theoremapmul1 8560 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
 
Theoremapmul2 8561 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) 
 ->  ( A #  B  <->  ( C  x.  A ) #  ( C  x.  B ) ) )
 
Theoremdivclapd 8562 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1d 8563 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B )  x.  B )  =  A )
 
Theoremdivcanap2d 8564 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( B  x.  ( A  /  B ) )  =  A )
 
Theoremdivrecapd 8565 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
 
Theoremdivrecap2d 8566 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  =  (
 ( 1  /  B )  x.  A ) )
 
Theoremdivcanap3d 8567 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( B  x.  A )  /  B )  =  A )
 
Theoremdivcanap4d 8568 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A  x.  B )  /  B )  =  A )
 
Theoremdiveqap0d 8569 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  ( A  /  B )  =  0
 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremdiveqap1d 8570 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  ( A  /  B )  =  1
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdiveqap1ad 8571 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8477. Generalization of diveqap1d 8570. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B )  =  1  <->  A  =  B ) )
 
Theoremdiveqap0ad 8572 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8454. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B )  =  0  <->  A  =  0
 ) )
 
Theoremdivap1d 8573 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  A #  B )   =>    |-  ( ph  ->  ( A  /  B ) #  1 )
 
Theoremdivap0bd 8574 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A #  0  <->  ( A  /  B ) #  0 ) )
 
Theoremdivnegapd 8575 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  -u ( A  /  B )  =  ( -u A  /  B ) )
 
Theoremdivneg2apd 8576 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  -u ( A  /  B )  =  ( A  /  -u B ) )
 
Theoremdiv2negapd 8577 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( -u A  /  -u B )  =  ( A  /  B ) )
 
Theoremdivap0d 8578 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B ) #  0 )
 
Theoremrecdivapd 8579 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( 1  /  ( A  /  B ) )  =  ( B 
 /  A ) )
 
Theoremrecdivap2d 8580 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( 1 
 /  A )  /  B )  =  (
 1  /  ( A  x.  B ) ) )
 
Theoremdivcanap6d 8581 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( ( A 
 /  B )  x.  ( B  /  A ) )  =  1
 )
 
Theoremddcanapd 8582 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  ( A  /  B ) )  =  B )
 
Theoremrec11apd 8583 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  ( 1  /  A )  =  (
 1  /  B )
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmulapd 8584 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  =  C  <->  ( B  x.  C )  =  A ) )
 
Theoremapdivmuld 8585 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B ) #  C  <->  ( B  x.  C ) #  A )
 )
 
Theoremdiv32apd 8586 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( A  x.  ( C  /  B ) ) )
 
Theoremdiv13apd 8587 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  (
 ( A  /  B )  x.  C )  =  ( ( C  /  B )  x.  A ) )
 
Theoremdivdiv32apd 8588 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdivcanap5d 8589 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdivcanap5rd 8590 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdivcanap7d 8591 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdmdcanapd 8592 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdmdcanap2d 8593 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdivdivap1d 8594 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) ) )
 
Theoremdivdivap2d 8595 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( 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 ) )
 
Theoremdivmulap2d 8596 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( C  x.  B ) ) )
 
Theoremdivmulap3d 8597 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  /  C )  =  B  <->  A  =  ( B  x.  C ) ) )
 
Theoremdivassapd 8598 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( A  x.  ( B  /  C ) ) )
 
Theoremdiv12apd 8599 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  ( A  x.  ( B  /  C ) )  =  ( B  x.  ( A  /  C ) ) )
 
Theoremdiv23apd 8600 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  x.  B )  /  C )  =  ( ( A  /  C )  x.  B ) )
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