Theorem List for Intuitionistic Logic Explorer - 8401-8500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | leltapd 8401 |
implies 'less
than' is 'apart'. (Contributed by Jim Kingdon,
13-Aug-2021.)
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#
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Theorem | ap0gt0 8402 |
A nonnegative number is apart from zero if and only if it is positive.
(Contributed by Jim Kingdon, 11-Aug-2021.)
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# |
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Theorem | ap0gt0d 8403 |
A nonzero nonnegative number is positive. (Contributed by Jim
Kingdon, 11-Aug-2021.)
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# |
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Theorem | apsub1 8404 |
Subtraction respects apartness. Analogue of subcan2 7987 for apartness.
(Contributed by Jim Kingdon, 6-Jan-2022.)
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# #
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Theorem | subap0 8405 |
Two numbers being apart is equivalent to their difference being apart from
zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
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# # |
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Theorem | subap0d 8406 |
Two numbers apart from each other have difference apart from zero.
(Contributed by Jim Kingdon, 12-Aug-2021.)
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#
#
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Theorem | cnstab 8407 |
Equality of complex numbers is stable. Stability here means
as defined at df-stab 816. This theorem for real
numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim
Kingdon, 1-Aug-2023.)
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STAB |
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Theorem | aprcl 8408 |
Reverse closure for apartness. (Contributed by Jim Kingdon,
19-Dec-2023.)
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#
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Theorem | apsscn 8409* |
The points apart from a given point are complex numbers. (Contributed
by Jim Kingdon, 19-Dec-2023.)
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#
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Theorem | lt0ap0 8410 |
A number which is less than zero is apart from zero. (Contributed by Jim
Kingdon, 25-Feb-2024.)
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# |
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Theorem | lt0ap0d 8411 |
A real number less than zero is apart from zero. Deduction form.
(Contributed by Jim Kingdon, 24-Feb-2024.)
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# |
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4.3.7 Reciprocals
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Theorem | recextlem1 8412 |
Lemma for recexap 8414. (Contributed by Eric Schmidt, 23-May-2007.)
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Theorem | recexaplem2 8413 |
Lemma for recexap 8414. (Contributed by Jim Kingdon, 20-Feb-2020.)
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#
# |
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Theorem | recexap 8414* |
Existence of reciprocal of nonzero complex number. (Contributed by Jim
Kingdon, 20-Feb-2020.)
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#
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Theorem | mulap0 8415 |
The product of two numbers apart from zero is apart from zero. Lemma
2.15 of [Geuvers], p. 6. (Contributed
by Jim Kingdon, 22-Feb-2020.)
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#
#
# |
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Theorem | mulap0b 8416 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
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# # # |
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Theorem | mulap0i 8417 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
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# # # |
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Theorem | mulap0bd 8418 |
The product of two numbers apart from zero is apart from zero. Exercise
11.11 of [HoTT], p. (varies).
(Contributed by Jim Kingdon,
24-Feb-2020.)
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# # #
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Theorem | mulap0d 8419 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
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#
#
#
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Theorem | mulap0bad 8420 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 8419 and consequence of mulap0bd 8418.
(Contributed by Jim Kingdon, 24-Feb-2020.)
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#
#
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Theorem | mulap0bbd 8421 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 8419 and consequence of mulap0bd 8418.
(Contributed by Jim Kingdon, 24-Feb-2020.)
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#
#
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Theorem | mulcanapd 8422 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
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#
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Theorem | mulcanap2d 8423 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
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#
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Theorem | mulcanapad 8424 |
Cancellation of a nonzero factor on the left in an equation. One-way
deduction form of mulcanapd 8422. (Contributed by Jim Kingdon,
21-Feb-2020.)
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# |
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Theorem | mulcanap2ad 8425 |
Cancellation of a nonzero factor on the right in an equation. One-way
deduction form of mulcanap2d 8423. (Contributed by Jim Kingdon,
21-Feb-2020.)
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# |
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Theorem | mulcanap 8426 |
Cancellation law for multiplication (full theorem form). (Contributed by
Jim Kingdon, 21-Feb-2020.)
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#
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Theorem | mulcanap2 8427 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
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#
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Theorem | mulcanapi 8428 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
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#
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Theorem | muleqadd 8429 |
Property of numbers whose product equals their sum. Equation 5 of
[Kreyszig] p. 12. (Contributed by NM,
13-Nov-2006.)
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Theorem | receuap 8430* |
Existential uniqueness of reciprocals. (Contributed by Jim Kingdon,
21-Feb-2020.)
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#
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Theorem | mul0eqap 8431 |
If two numbers are apart from each other and their product is zero, one
of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
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#
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4.3.8 Division
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Syntax | cdiv 8432 |
Extend class notation to include division.
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Definition | df-div 8433* |
Define division. Theorem divmulap 8435 relates it to multiplication, and
divclap 8438 and redivclap 8491 prove its closure laws. (Contributed by NM,
2-Feb-1995.) Use divvalap 8434 instead. (Revised by Mario Carneiro,
1-Apr-2014.) (New usage is discouraged.)
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Theorem | divvalap 8434* |
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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#
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Theorem | divmulap 8435 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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#
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Theorem | divmulap2 8436 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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#
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Theorem | divmulap3 8437 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
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#
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Theorem | divclap 8438 |
Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
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#
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Theorem | recclap 8439 |
Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
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#
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Theorem | divcanap2 8440 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
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#
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Theorem | divcanap1 8441 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
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#
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Theorem | diveqap0 8442 |
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 8443 |
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 8444 |
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 8445 |
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 8446 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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#
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Theorem | recidap2 8447 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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#
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Theorem | divrecap 8448 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
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#
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Theorem | divrecap2 8449 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 25-Feb-2020.)
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#
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Theorem | divassap 8450 |
An associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | div23ap 8451 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | div32ap 8452 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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# |
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Theorem | div13ap 8453 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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# |
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Theorem | div12ap 8454 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | divmulassap 8455 |
An associative law for division and multiplication. (Contributed by Jim
Kingdon, 24-Jan-2022.)
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#
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Theorem | divmulasscomap 8456 |
An associative/commutative law for division and multiplication.
(Contributed by Jim Kingdon, 24-Jan-2022.)
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#
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Theorem | divdirap 8457 |
Distribution of division over addition. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | divcanap3 8458 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | divcanap4 8459 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | div11ap 8460 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | dividap 8461 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | div0ap 8462 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
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#
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Theorem | div1 8463 |
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 8464 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
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Theorem | diveqap1 8465 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | divnegap 8466 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
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Theorem | muldivdirap 8467 |
Distribution of division over addition with a multiplication.
(Contributed by Jim Kingdon, 11-Nov-2021.)
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#
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Theorem | divsubdirap 8468 |
Distribution of division over subtraction. (Contributed by NM,
4-Mar-2005.)
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#
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Theorem | recrecap 8469 |
A number is equal to the reciprocal of its reciprocal. (Contributed by
Jim Kingdon, 25-Feb-2020.)
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#
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Theorem | rec11ap 8470 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
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#
#
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Theorem | rec11rap 8471 |
Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
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#
#
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Theorem | divmuldivap 8472 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
25-Feb-2020.)
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#
#
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Theorem | divdivdivap 8473 |
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 8474 |
Cancellation of common factor in a ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
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# #
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Theorem | divmul13ap 8475 |
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 8476 |
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 8477 |
Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
#
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Theorem | recdivap 8478 |
The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
# |
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Theorem | divcanap6 8479 |
Cancellation of inverted fractions. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
#
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Theorem | divdiv32ap 8480 |
Swap denominators in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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# #
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Theorem | divcanap7 8481 |
Cancel equal divisors in a division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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# #
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Theorem | dmdcanap 8482 |
Cancellation law for division and multiplication. (Contributed by Jim
Kingdon, 26-Feb-2020.)
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#
#
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Theorem | divdivap1 8483 |
Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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# #
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Theorem | divdivap2 8484 |
Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
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# #
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Theorem | recdivap2 8485 |
Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
#
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Theorem | ddcanap 8486 |
Cancellation in a double division. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
#
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Theorem | divadddivap 8487 |
Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
#
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Theorem | divsubdivap 8488 |
Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
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#
#
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Theorem | conjmulap 8489 |
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 8490 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | redivclap 8491 |
Closure law for division of reals. (Contributed by Jim Kingdon,
26-Feb-2020.)
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#
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Theorem | eqneg 8492 |
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 8493 |
A complex number equals its negative iff it is zero. Deduction form of
eqneg 8492. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | eqnegad 8494 |
If a complex number equals its own negative, it is zero. One-way
deduction form of eqneg 8492. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | div2negap 8495 |
Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
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#
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Theorem | divneg2ap 8496 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
27-Feb-2020.)
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#
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Theorem | recclapzi 8497 |
Closure law for reciprocal. (Contributed by Jim Kingdon,
27-Feb-2020.)
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#
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Theorem | recap0apzi 8498 |
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 8499 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 27-Feb-2020.)
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#
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Theorem | div1i 8500 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
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