Theorem List for Intuitionistic Logic Explorer - 8601-8700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | divcanap5 8601 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divmul13ap 8602 |
Swap the denominators in the product of two ratios. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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Theorem | divmul24ap 8603 |
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 8604 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | recdivap 8605 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divcanap6 8606 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | divdiv32ap 8607 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | divcanap7 8608 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | dmdcanap 8609 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
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Theorem | divdivap1 8610 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divdivap2 8611 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | recdivap2 8612 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | ddcanap 8613 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | divadddivap 8614 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | divsubdivap 8615 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | conjmulap 8616 |
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 8617 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | redivclap 8618 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | eqneg 8619 |
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 8620 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8619. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 8621 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8619. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 8622 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divneg2ap 8623 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recclapzi 8624 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recap0apzi 8625 |
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 8626 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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#
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Theorem | div1i 8627 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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Theorem | eqnegi 8628 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
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Theorem | recclapi 8629 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theorem | recidapi 8630 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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Theorem | recrecapi 8631 |
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 8632 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
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Theorem | div0api 8633 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theorem | divclapzi 8634 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divcanap1zi 8635 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap2zi 8636 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divrecapzi 8637 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | divcanap3zi 8638 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap4zi 8639 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | rec11api 8640 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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Theorem | divclapi 8641 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap2i 8642 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap1i 8643 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# |
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Theorem | divrecapi 8644 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divcanap3i 8645 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap4i 8646 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divap0i 8647 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
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Theorem | rec11apii 8648 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divassapzi 8649 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapzi 8650 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divdirapzi 8651 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divdiv23apzi 8652 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divmulapi 8653 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divdiv32api 8654 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divassapi 8655 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divdirapi 8656 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div23api 8657 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
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Theorem | div11api 8658 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divmuldivapi 8659 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divmul13api 8660 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | divadddivapi 8661 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theorem | divdivdivapi 8662 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
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Theorem | rerecclapzi 8663 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | rerecclapi 8664 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapzi 8665 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | redivclapi 8666 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
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Theorem | div1d 8667 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | recclapd 8668 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | recap0d 8669 |
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 8670 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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Theorem | recidap2d 8671 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
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Theorem | recrecapd 8672 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | dividapd 8673 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | div0apd 8674 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | apmul1 8675 |
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 | apmul2 8676 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
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Theorem | divclapd 8677 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap1d 8678 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap2d 8679 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divrecapd 8680 |
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 8681 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | divcanap3d 8682 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | divcanap4d 8683 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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Theorem | diveqap0d 8684 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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Theorem | diveqap1d 8685 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | diveqap1ad 8686 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8592. Generalization of diveqap1d 8685.
(Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | diveqap0ad 8687 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8569. (Contributed by Jim Kingdon, 19-Mar-2020.)
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Theorem | divap1d 8688 |
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 8689 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
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Theorem | divnegapd 8690 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divneg2apd 8691 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | div2negapd 8692 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
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Theorem | divap0d 8693 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
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Theorem | recdivapd 8694 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | recdivap2d 8695 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | divcanap6d 8696 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | ddcanapd 8697 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | rec11apd 8698 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
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Theorem | divmulapd 8699 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
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Theorem | apdivmuld 8700 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 26-Dec-2022.)
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