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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdiv0api 8701 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
 |-  A  e.  CC   &    |-  A #  0   =>    |-  ( 0  /  A )  =  0
 
Theoremdivclapzi 8702 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( A  /  B )  e.  CC )
 
Theoremdivcanap1zi 8703 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 8704 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 8705 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 8706 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 8707 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 8708 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 8709 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  e.  CC
 
Theoremdivcanap2i 8710 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 8711 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 8712 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 8713 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 8714 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 8715 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 8716 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 8717 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 8718 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 8719 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 8720 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 8721 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 8722 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 8723 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 8724 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 8725 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 8726 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 8727 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 8728 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 8729 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 8730 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 8731 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrerecclapi 8732 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclapzi 8733 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 8734 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 8735 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremrecclapd 8736 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  CC )
 
Theoremrecap0d 8737 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 8738 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 8739 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 8740 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 8741 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 8742 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 0  /  A )  =  0 )
 
Theoremapmul1 8743 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 8744 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 8745 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 8746 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 8747 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 8748 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 8749 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 8750 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 8751 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 8752 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 8753 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 8754 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8660. Generalization of diveqap1d 8753. (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 8755 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8637. (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 8756 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 8757 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 8758 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 8759 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 8760 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 8761 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 8762 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 8763 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 8764 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 8765 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 8766 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 8767 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 8768 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 8769 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 8770 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 8771 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 8772 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 8773 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 8774 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 8775 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 8776 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 8777 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 8778 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 8779 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 8780 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 8781 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 8782 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 8783 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 ) )
 
Theoremdivdirapd 8784 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  /  C )  =  ( ( A  /  C )  +  ( B  /  C ) ) )
 
Theoremdivsubdirapd 8785 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C #  0 )   =>    |-  ( ph  ->  (
 ( A  -  B )  /  C )  =  ( ( A  /  C )  -  ( B  /  C ) ) )
 
Theoremdiv11apd 8786 One-to-one relationship 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  /  C )  =  ( B  /  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremdivmuldivapd 8787 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
 |-  ( 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 ) ) )
 
Theoremdivmuleqapd 8788 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 ) ) )
 
Theoremrerecclapd 8789 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremredivclapd 8790 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR )
 
Theoremdiveqap1bd 8791 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8660. Converse of diveqap1d 8753. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  /  B )  =  1 )
 
Theoremdiv2subap 8792 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 ) ) )
 
Theoremdiv2subapd 8793 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8792. (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 ) ) )
 
Theoremsubrecap 8794 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 ) ) )
 
Theoremsubrecapi 8795 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 ) )
 
Theoremsubrecapd 8796 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 ) ) )
 
Theoremmvllmulapd 8797 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  ( A  x.  B )  =  C )   =>    |-  ( ph  ->  B  =  ( C  /  A ) )
 
Theoremrerecapb 8798* A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
 |-  ( A  e.  RR  ->  ( A #  0  <->  E. x  e.  RR  ( A  x.  x )  =  1 )
 )
 
4.3.9  Ordering on reals (cont.)
 
Theoremltp1 8799 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 8800 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
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