Theorem List for Intuitionistic Logic Explorer - 7801-7900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | divdivdivap 7801 |
Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by
Jim Kingdon, 25-Feb-2020.)
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Theorem | divcanap5 7802 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divmul13ap 7803 |
Swap the denominators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
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Theorem | divmul24ap 7804 |
Swap the numerators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
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Theorem | divmuleqap 7805 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | recdivap 7806 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divcanap6 7807 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | divdiv32ap 7808 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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# #
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Theorem | divcanap7 7809 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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# #
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Theorem | dmdcanap 7810 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
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Theorem | divdivap1 7811 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divdivap2 7812 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | recdivap2 7813 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | ddcanap 7814 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | divadddivap 7815 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divsubdivap 7816 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | conjmulap 7817 |
Two numbers whose reciprocals sum to 1 are called "conjugates" and
satisfy
this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | rerecclap 7818 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | redivclap 7819 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | eqneg 7820 |
A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | eqnegd 7821 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 7820. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 7822 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 7820. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 7823 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divneg2ap 7824 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recclapzi 7825 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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#
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Theorem | recap0apzi 7826 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
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# # |
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Theorem | recidapzi 7827 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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#
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Theorem | div1i 7828 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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Theorem | eqnegi 7829 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
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Theorem | recclapi 7830 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theorem | recidapi 7831 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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Theorem | recrecapi 7832 |
A number is equal to the reciprocal of its reciprocal. Theorem I.10
of [Apostol] p. 18. (Contributed by
NM, 9-Feb-1995.)
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#
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Theorem | dividapi 7833 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
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Theorem | div0api 7834 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theorem | divclapzi 7835 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divcanap1zi 7836 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap2zi 7837 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divrecapzi 7838 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | divcanap3zi 7839 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap4zi 7840 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | rec11api 7841 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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# #
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Theorem | divclapi 7842 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap2i 7843 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap1i 7844 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divrecapi 7845 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divcanap3i 7846 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap4i 7847 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divap0i 7848 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
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Theorem | rec11apii 7849 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divassapzi 7850 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapzi 7851 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divdirapzi 7852 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divdiv23apzi 7853 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# #
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Theorem | divmulapi 7854 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divdiv32api 7855 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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# #
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Theorem | divassapi 7856 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divdirapi 7857 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div23api 7858 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
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Theorem | div11api 7859 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divmuldivapi 7860 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #
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Theorem | divmul13api 7861 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #
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Theorem | divadddivapi 7862 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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# #
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Theorem | divdivdivapi 7863 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theorem | rerecclapzi 7864 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | rerecclapi 7865 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapzi 7866 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapi 7867 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div1d 7868 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | recclapd 7869 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | recap0d 7870 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
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# |
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Theorem | recidapd 7871 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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Theorem | recidap2d 7872 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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# |
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Theorem | recrecapd 7873 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | dividapd 7874 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | div0apd 7875 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | apmul1 7876 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
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# # #
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Theorem | divclapd 7877 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap1d 7878 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap2d 7879 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divrecapd 7880 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divrecap2d 7881 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divcanap3d 7882 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap4d 7883 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | diveqap0d 7884 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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Theorem | diveqap1d 7885 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | diveqap1ad 7886 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 7793. Generalization of diveqap1d 7885.
(Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | diveqap0ad 7887 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 7770. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | divap1d 7888 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
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Theorem | divap0bd 7889 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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#
# # |
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Theorem | divnegapd 7890 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divneg2apd 7891 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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#
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Theorem | div2negapd 7892 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divap0d 7893 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | recdivapd 7894 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
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#
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Theorem | recdivap2d 7895 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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#
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Theorem | divcanap6d 7896 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
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#
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Theorem | ddcanapd 7897 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
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#
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Theorem | rec11apd 7898 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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#
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Theorem | divmulapd 7899 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | div32apd 7900 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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