Theorem List for Intuitionistic Logic Explorer - 7701-7800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | recdivap2 7701 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | ddcanap 7702 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | divadddivap 7703 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divsubdivap 7704 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | conjmulap 7705 |
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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Theorem | rerecclap 7706 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | redivclap 7707 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | eqneg 7708 |
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 7709 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 7708. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 7710 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 7708. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 7711 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divneg2ap 7712 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recclapzi 7713 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recap0apzi 7714 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | recidapzi 7715 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | div1i 7716 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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Theorem | eqnegi 7717 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
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Theorem | recclapi 7718 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theorem | recidapi 7719 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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Theorem | recrecapi 7720 |
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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Theorem | dividapi 7721 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
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Theorem | div0api 7722 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theorem | divclapzi 7723 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divcanap1zi 7724 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap2zi 7725 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divrecapzi 7726 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | divcanap3zi 7727 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap4zi 7728 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | rec11api 7729 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theorem | divclapi 7730 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap2i 7731 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap1i 7732 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divrecapi 7733 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divcanap3i 7734 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap4i 7735 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divap0i 7736 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
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Theorem | rec11apii 7737 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divassapzi 7738 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapzi 7739 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divdirapzi 7740 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divdiv23apzi 7741 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapi 7742 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divdiv32api 7743 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divassapi 7744 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divdirapi 7745 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div23api 7746 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
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Theorem | div11api 7747 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divmuldivapi 7748 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divmul13api 7749 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divadddivapi 7750 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theorem | divdivdivapi 7751 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theorem | rerecclapzi 7752 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | rerecclapi 7753 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapzi 7754 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapi 7755 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div1d 7756 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | recclapd 7757 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | recap0d 7758 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
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Theorem | recidapd 7759 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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Theorem | recidap2d 7760 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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Theorem | recrecapd 7761 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | dividapd 7762 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | div0apd 7763 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | apmul1 7764 |
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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Theorem | divclapd 7765 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap1d 7766 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap2d 7767 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divrecapd 7768 |
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 7769 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divcanap3d 7770 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap4d 7771 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | diveqap0d 7772 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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Theorem | diveqap1d 7773 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | diveqap1ad 7774 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 7682. Generalization of diveqap1d 7773.
(Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | diveqap0ad 7775 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 7661. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | divap1d 7776 |
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 7777 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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Theorem | divnegapd 7778 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divneg2apd 7779 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | div2negapd 7780 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divap0d 7781 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | recdivapd 7782 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | recdivap2d 7783 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | divcanap6d 7784 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | ddcanapd 7785 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | rec11apd 7786 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | divmulapd 7787 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | div32apd 7788 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | div13apd 7789 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | divdiv32apd 7790 |
Swap denominators in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
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Theorem | divcanap5d 7791 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | divcanap5rd 7792 |
Cancellation of common factor in a ratio. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | divcanap7d 7793 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
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Theorem | dmdcanapd 7794 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | dmdcanap2d 7795 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | divdivap1d 7796 |
Division into a fraction. (Contributed by Jim Kingdon,
8-Mar-2020.)
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Theorem | divdivap2d 7797 |
Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
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Theorem | divmulap2d 7798 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
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Theorem | divmulap3d 7799 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 2-Mar-2020.)
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Theorem | divassapd 7800 |
An associative law for division. (Contributed by Jim Kingdon,
2-Mar-2020.)
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