Theorem List for Intuitionistic Logic Explorer - 8501-8600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | divmuleqap 8501 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![(
(](lp.gif) # ![0 0](0.gif) ![( (](lp.gif)
# ![0 0](0.gif) ![) )](rp.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![D D](_cd.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | recdivap 8502 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap6 8503 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | divdiv32ap 8504 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif)
![( (](lp.gif) ![(
(](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap7 8505 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dmdcanap 8506 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif)
![CC CC](bbc.gif)
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdivap1 8507 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdivap2 8508 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif)
![( (](lp.gif) ![(
(](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | recdivap2 8509 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ddcanap 8510 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | divadddivap 8511 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![(
(](lp.gif) # ![0 0](0.gif) ![( (](lp.gif)
# ![0 0](0.gif) ![) )](rp.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divsubdivap 8512 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![(
(](lp.gif) # ![0 0](0.gif) ![( (](lp.gif)
# ![0 0](0.gif) ![) )](rp.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | conjmulap 8513 |
Two numbers whose reciprocals sum to 1 are called "conjugates" and
satisfy
this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif)
![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![P P](_cp.gif) ![( (](lp.gif) ![Q Q](_cq.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![1 1](1.gif) ![( (](lp.gif) ![1 1](1.gif) ![) )](rp.gif) ![1 1](1.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | rerecclap 8514 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![A A](_ca.gif)
![RR RR](bbr.gif) ![) )](rp.gif) |
|
Theorem | redivclap 8515 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif)
![B B](_cb.gif) ![RR RR](bbr.gif) ![) )](rp.gif) |
|
Theorem | eqneg 8516 |
A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
|
![( (](lp.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![0 0](0.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | eqnegd 8517 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8516. (Contributed by David Moews, 28-Feb-2017.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![( (](lp.gif)
![-u -u](shortminus.gif)
![0 0](0.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | eqnegad 8518 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8516. (Contributed by David Moews,
28-Feb-2017.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![A A](_ca.gif) ![( (](lp.gif) ![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | div2negap 8519 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![-u -u](shortminus.gif) ![B B](_cb.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divneg2ap 8520 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![-u -u](shortminus.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![-u -u](shortminus.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | recclapzi 8521 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![A A](_ca.gif)
![CC CC](bbc.gif) ![) )](rp.gif) |
|
Theorem | recap0apzi 8522 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![A A](_ca.gif) # ![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | recidapzi 8523 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | div1i 8524 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
|
![( (](lp.gif) ![1
1](1.gif) ![A A](_ca.gif) |
|
Theorem | eqnegi 8525 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
|
![( (](lp.gif) ![-u -u](shortminus.gif)
![0 0](0.gif) ![)
)](rp.gif) |
|
Theorem | recclapi 8526 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
|
# ![( (](lp.gif) ![A A](_ca.gif)
![CC CC](bbc.gif) |
|
Theorem | recidapi 8527 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
|
# ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![1 1](1.gif) |
|
Theorem | recrecapi 8528 |
A number is equal to the reciprocal of its reciprocal. Theorem I.10
of [Apostol] p. 18. (Contributed by
NM, 9-Feb-1995.)
|
# ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![A A](_ca.gif) |
|
Theorem | dividapi 8529 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
|
# ![( (](lp.gif) ![A A](_ca.gif)
![1 1](1.gif) |
|
Theorem | div0api 8530 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
|
# ![( (](lp.gif) ![A A](_ca.gif)
![0 0](0.gif) |
|
Theorem | divclapzi 8531 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif)
![B B](_cb.gif) ![CC CC](bbc.gif) ![) )](rp.gif) |
|
Theorem | divcanap1zi 8532 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![B B](_cb.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divcanap2zi 8533 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divrecapzi 8534 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif)
![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap3zi 8535 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divcanap4zi 8536 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | rec11api 8537 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
|
![( (](lp.gif) ![( (](lp.gif) # # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif)
![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divclapi 8538 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# ![( (](lp.gif) ![B B](_cb.gif)
![CC CC](bbc.gif) |
|
Theorem | divcanap2i 8539 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![A A](_ca.gif) |
|
Theorem | divcanap1i 8540 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif) ![A A](_ca.gif) |
|
Theorem | divrecapi 8541 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
# ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap3i 8542 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![A A](_ca.gif) |
|
Theorem | divcanap4i 8543 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif)
![A A](_ca.gif) |
|
Theorem | divap0i 8544 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
|
# # ![( (](lp.gif) ![B B](_cb.gif) # ![0 0](0.gif) |
|
Theorem | rec11apii 8545 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
# # ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif)
![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | divassapzi 8546 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
![( (](lp.gif) #
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divmulapzi 8547 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
![( (](lp.gif) #
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![C C](_cc.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdirapzi 8548 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
![( (](lp.gif) #
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdiv23apzi 8549 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
|
![( (](lp.gif) ![(
(](lp.gif) # # ![0 0](0.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divmulapi 8550 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divdiv32api 8551 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
# # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | divassapi 8552 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdirapi 8553 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | div23api 8554 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | div11api 8555 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![( (](lp.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | divmuldivapi 8556 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divmul13api 8557 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divadddivapi 8558 |
Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdivdivapi 8559 |
Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
|
# # # ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | rerecclapzi 8560 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
![( (](lp.gif) # ![( (](lp.gif) ![A A](_ca.gif)
![RR RR](bbr.gif) ![) )](rp.gif) |
|
Theorem | rerecclapi 8561 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# ![( (](lp.gif) ![A A](_ca.gif)
![RR RR](bbr.gif) |
|
Theorem | redivclapzi 8562 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
![( (](lp.gif) # ![( (](lp.gif)
![B B](_cb.gif) ![RR RR](bbr.gif) ![) )](rp.gif) |
|
Theorem | redivclapi 8563 |
Closure law for division of reals. (Contributed by Jim Kingdon,
9-Mar-2020.)
|
# ![( (](lp.gif) ![B B](_cb.gif)
![RR RR](bbr.gif) |
|
Theorem | div1d 8564 |
A number divided by 1 is itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![( (](lp.gif) ![1 1](1.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | recclapd 8565 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![CC CC](bbc.gif) ![) )](rp.gif) |
|
Theorem | recap0d 8566 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) # ![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | recidapd 8567 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | recidap2d 8568 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | recrecapd 8569 |
A number is equal to the reciprocal of its reciprocal. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif)
![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | dividapd 8570 |
A number divided by itself is one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | div0apd 8571 |
Division into zero is zero. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif)
![A A](_ca.gif) ![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | apmul1 8572 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) # ![( (](lp.gif) ![C C](_cc.gif) #
![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | apmul2 8573 |
Multiplication of both sides of complex apartness by a complex number
apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) # ![0 0](0.gif) ![) )](rp.gif) ![( (](lp.gif) # ![( (](lp.gif) ![A A](_ca.gif) #
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divclapd 8574 |
Closure law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![CC CC](bbc.gif) ![) )](rp.gif) |
|
Theorem | divcanap1d 8575 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divcanap2d 8576 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divrecapd 8577 |
Relationship between division and reciprocal. Theorem I.9 of
[Apostol] p. 18. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divrecap2d 8578 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap3d 8579 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | divcanap4d 8580 |
A cancellation law for division. (Contributed by Jim Kingdon,
29-Feb-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | diveqap0d 8581 |
If a ratio is zero, the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![0 0](0.gif) ![( (](lp.gif) ![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | diveqap1d 8582 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![1 1](1.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | diveqap1ad 8583 |
The quotient of two complex numbers is one iff they are equal.
Deduction form of diveqap1 8489. Generalization of diveqap1d 8582.
(Contributed by Jim Kingdon, 19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | diveqap0ad 8584 |
A fraction of complex numbers is zero iff its numerator is. Deduction
form of diveqap0 8466. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![0 0](0.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | divap1d 8585 |
If two complex numbers are apart, their quotient is apart from one.
(Contributed by Jim Kingdon, 20-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) #
![1 1](1.gif) ![) )](rp.gif) |
|
Theorem | divap0bd 8586 |
A ratio is zero iff the numerator is zero. (Contributed by Jim
Kingdon, 19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) # ![( (](lp.gif) ![B B](_cb.gif) # ![0 0](0.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divnegapd 8587 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![-u -u](shortminus.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divneg2apd 8588 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![-u -u](shortminus.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | div2negapd 8589 |
Quotient of two negatives. (Contributed by Jim Kingdon,
19-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![-u -u](shortminus.gif) ![-u -u](shortminus.gif) ![B B](_cb.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divap0d 8590 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) #
![0 0](0.gif) ![) )](rp.gif) |
|
Theorem | recdivapd 8591 |
The reciprocal of a ratio. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | recdivap2d 8592 |
Division into a reciprocal. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![( (](lp.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divcanap6d 8593 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![A A](_ca.gif) ![) )](rp.gif)
![1 1](1.gif) ![)
)](rp.gif) |
|
Theorem | ddcanapd 8594 |
Cancellation in a double division. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | rec11apd 8595 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
3-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) #
![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif)
![( (](lp.gif)
![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | divmulapd 8596 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![C C](_cc.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | apdivmuld 8597 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 26-Dec-2022.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) # ![( (](lp.gif) ![C C](_cc.gif) #
![A A](_ca.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | div32apd 8598 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | div13apd 8599 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 8-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | divdiv32apd 8600 |
Swap denominators in a division. (Contributed by Jim Kingdon,
8-Mar-2020.)
|
![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) ![CC CC](bbc.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) # ![0 0](0.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |