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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcanap3zi 7801 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 7802 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 7803 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 7804 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  B #  0   =>    |-  ( A  /  B )  e.  CC
 
Theoremdivcanap2i 7805 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 7806 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 7807 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 7808 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 7809 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 7810 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 7811 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 7812 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 7813 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 7814 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 7815 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 7816 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 7817 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 7818 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 7819 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 7820 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 7821 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 7822 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 7823 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 7824 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 7825 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 7826 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   =>    |-  ( A #  0  ->  ( 1  /  A )  e.  RR )
 
Theoremrerecclapi 7827 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  A  e.  RR   &    |-  A #  0   =>    |-  ( 1  /  A )  e.  RR
 
Theoremredivclapzi 7828 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 7829 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 7830 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  1 )  =  A )
 
Theoremrecclapd 7831 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  CC )
 
Theoremrecap0d 7832 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 7833 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 7834 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 7835 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 7836 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 7837 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 0  /  A )  =  0 )
 
Theoremapmul1 7838 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 ) ) )
 
Theoremdivclapd 7839 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 7840 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 7841 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 7842 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 7843 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 7844 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 7845 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 7846 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 7847 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 7848 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7755. Generalization of diveqap1d 7847. (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 7849 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7734. (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 7850 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 7851 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 7852 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 7853 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 7854 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 7855 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 7856 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 7857 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 7858 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 7859 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 7860 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 7861 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 ) )
 
Theoremdiv32apd 7862 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 7863 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 7864 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 7865 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 7866 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 7867 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 7868 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 7869 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 7870 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 7871 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 7872 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 7873 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 7874 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 7875 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 7876 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 7877 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 7878 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 7879 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 7880 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 ) ) )
 
Theoremrerecclapd 7881 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremredivclapd 7882 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 )
 
Theoremmvllmulapd 7883 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 ) )
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7884 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
 |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
 
Theoremlep1 7885 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
 |-  ( A  e.  RR  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1 7886 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
 |-  ( A  e.  RR  ->  ( A  -  1
 )  <  A )
 
Theoremlem1 7887 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 7888 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 7889 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 7890 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 )
 )
 
Theoremprodgt0gt0 7891 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7892 for the same theorem with  0  < 
A replaced by the weaker condition 
0  <_  A. (Contributed by Jim Kingdon, 29-Feb-2020.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  B )
 
Theoremprodgt0 7892 Infer that a multiplicand is positive from a nonnegative multiplier 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 7893 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  B  /\  0  <  ( A  x.  B ) ) ) 
 ->  0  <  A )
 
Theoremprodge0 7894 Infer that a multiplicand is nonnegative from a positive multiplier 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 7895 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  B  /\  0  <_  ( A  x.  B ) ) ) 
 ->  0  <_  A )
 
Theoremltmul2 7896 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 ) ) )
 
Theoremlemul2 7897 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 7898 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (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 7899 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 7900 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 ) )
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