Theorem List for Intuitionistic Logic Explorer - 8401-8500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | divvalap 8401* |
Value of division: the (unique) element such that
. This is meaningful only when is apart from
zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
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Theorem | divmulap 8402 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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Theorem | divmulap2 8403 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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Theorem | divmulap3 8404 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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Theorem | divclap 8405 |
Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
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Theorem | recclap 8406 |
Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
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#
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Theorem | divcanap2 8407 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
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Theorem | divcanap1 8408 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
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#
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Theorem | diveqap0 8409 |
A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon,
22-Feb-2020.)
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#
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Theorem | divap0b 8410 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
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# #
#
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Theorem | divap0 8411 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
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#
# |
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Theorem | recap0 8412 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
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# # |
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Theorem | recidap 8413 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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#
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Theorem | recidap2 8414 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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Theorem | divrecap 8415 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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#
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Theorem | divrecap2 8416 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 25-Feb-2020.)
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#
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Theorem | divassap 8417 |
An associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | div23ap 8418 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | div32ap 8419 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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# |
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Theorem | div13ap 8420 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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# |
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Theorem | div12ap 8421 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divmulassap 8422 |
An associative law for division and multiplication. (Contributed by Jim
Kingdon, 24-Jan-2022.)
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Theorem | divmulasscomap 8423 |
An associative/commutative law for division and multiplication.
(Contributed by Jim Kingdon, 24-Jan-2022.)
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Theorem | divdirap 8424 |
Distribution of division over addition. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divcanap3 8425 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divcanap4 8426 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | div11ap 8427 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | dividap 8428 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | div0ap 8429 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
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Theorem | div1 8430 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
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Theorem | 1div1e1 8431 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
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Theorem | diveqap1 8432 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divnegap 8433 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | muldivdirap 8434 |
Distribution of division over addition with a multiplication.
(Contributed by Jim Kingdon, 11-Nov-2021.)
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Theorem | divsubdirap 8435 |
Distribution of division over subtraction. (Contributed by NM,
4-Mar-2005.)
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Theorem | recrecap 8436 |
A number is equal to the reciprocal of its reciprocal. (Contributed by
Jim Kingdon, 25-Feb-2020.)
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Theorem | rec11ap 8437 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
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#
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Theorem | rec11rap 8438 |
Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
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#
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Theorem | divmuldivap 8439 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | divdivdivap 8440 |
Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by
Jim Kingdon, 25-Feb-2020.)
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# # #
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Theorem | divcanap5 8441 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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Theorem | divmul13ap 8442 |
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 8443 |
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 8444 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | recdivap 8445 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | divcanap6 8446 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | divdiv32ap 8447 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | divcanap7 8448 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | dmdcanap 8449 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
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Theorem | divdivap1 8450 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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# #
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Theorem | divdivap2 8451 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | recdivap2 8452 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | ddcanap 8453 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | divadddivap 8454 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
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Theorem | divsubdivap 8455 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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Theorem | conjmulap 8456 |
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 8457 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | redivclap 8458 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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Theorem | eqneg 8459 |
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 8460 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8459. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 8461 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8459. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 8462 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divneg2ap 8463 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | recclapzi 8464 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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#
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Theorem | recap0apzi 8465 |
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 8466 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | div1i 8467 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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Theorem | eqnegi 8468 |
A number equal to its negative is zero. (Contributed by NM,
29-May-1999.)
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Theorem | recclapi 8469 |
Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
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Theorem | recidapi 8470 |
Multiplication of a number and its reciprocal. (Contributed by NM,
9-Feb-1995.)
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Theorem | recrecapi 8471 |
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 8472 |
A number divided by itself is one. (Contributed by NM,
9-Feb-1995.)
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Theorem | div0api 8473 |
Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
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Theorem | divclapzi 8474 |
Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
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Theorem | divcanap1zi 8475 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap2zi 8476 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divrecapzi 8477 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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Theorem | divcanap3zi 8478 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | divcanap4zi 8479 |
A cancellation law for division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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Theorem | rec11api 8480 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
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# #
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Theorem | divclapi 8481 |
Closure law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#
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Theorem | divcanap2i 8482 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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Theorem | divcanap1i 8483 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# |
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Theorem | divrecapi 8484 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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Theorem | divcanap3i 8485 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#
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Theorem | divcanap4i 8486 |
A cancellation law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#
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Theorem | divap0i 8487 |
The ratio of numbers apart from zero is apart from zero. (Contributed
by Jim Kingdon, 28-Feb-2020.)
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# # # |
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Theorem | rec11apii 8488 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# #
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Theorem | divassapzi 8489 |
An associative law for division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#
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Theorem | divmulapzi 8490 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 28-Feb-2020.)
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#
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Theorem | divdirapzi 8491 |
Distribution of division over addition. (Contributed by Jim Kingdon,
28-Feb-2020.)
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#
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Theorem | divdiv23apzi 8492 |
Swap denominators in a division. (Contributed by Jim Kingdon,
28-Feb-2020.)
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# #
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Theorem | divmulapi 8493 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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#
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Theorem | divdiv32api 8494 |
Swap denominators in a division. (Contributed by Jim Kingdon,
29-Feb-2020.)
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# #
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Theorem | divassapi 8495 |
An associative law for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#
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Theorem | divdirapi 8496 |
Distribution of division over addition. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#
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Theorem | div23api 8497 |
A commutative/associative law for division. (Contributed by Jim
Kingdon, 9-Mar-2020.)
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#
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Theorem | div11api 8498 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
9-Mar-2020.)
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#
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Theorem | divmuldivapi 8499 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #
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Theorem | divmul13api 8500 |
Swap denominators of two ratios. (Contributed by Jim Kingdon,
9-Mar-2020.)
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# #
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