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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqr2irr 12401 The square root of 2 is irrational. See zsqrelqelz 12703 for a generalization to all non-square integers. The proof's core is proven in sqr2irrlem 12400, which shows that if , then and are even, so and are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremsqr2re 12402 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)

6.1.2  Some Number sets are chains of proper subsets

Theoremnthruc 12403 The sequence , , , , and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to but not , one-half belongs to but not , the square root of 2 belongs to but not , and finally that the imaginary number belongs to but not . See nthruz 12404 for a further refinement. (Contributed by NM, 12-Jan-2002.)

Theoremnthruz 12404 The sequence , , and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to but not and minus one belongs to but not . This theorem refines the chain of proper subsets nthruc 12403. (Contributed by NM, 9-May-2004.)

6.1.3  The divides relation

Syntaxcdivides 12405 Extend the definition of a class to include the divides relation. See df-divides 12406.

Definitiondf-divides 12406* Define the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivides 12407* Define the divides relation. means divides into with no remainder. For example, (ex-dvds 20648). As proven in divides3 12409, . See divides 12407 and divides2 12408 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivides2 12408 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdivides3 12409 One nonzero integer divides another integer if and only if the remainder upon division is zero. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)

Theoremdvdszrcl 12410 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremnndivdivides 12411 Strong form of divides2 12408 for natural numbers. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremmoddvds 12412 Two ways to say . (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdvds0lem 12413 A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds1lem 12414* A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2lem 12415* A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvds 12416 An integer divides itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theorem1dvds 12417 1 divides any integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds0 12418 Any integer divides 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegdvdsb 12419 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsnegb 12420 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremabsdvdsb 12421 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsabsb 12422 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theorem0dvds 12423 Only 0 is divisible by 0 . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul1 12424 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul2 12425 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvdsexp 12426 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremmuldvds1 12427 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremmuldvds2 12428 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmul 12429 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulc 12430 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmulr 12431 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulcr 12432 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2ln 12433 If an integer divides each of two other integers, it divides any linear combination of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2add 12434 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2sub 12435 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdstr 12436 The divides relation is transitive. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmultr1 12437 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremdvdsmultr2 12438 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremordvdsmul 12439 If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremdvdssub2 12440 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsadd 12441 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsaddr 12442 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssub 12443 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssubr 12444 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdsadd2b 12445 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremfsumdvds 12446* If every term in a sum is divisible by , then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremdvdslelem 12447 Lemma for dvdsle 12448. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsle 12448 The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsleabs 12449 The divisors of a nonzero integer are bounded by its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theoremdvdseq 12450 If two integers divide each other, they must be equal, up to a difference in sign. (Contributed by Mario Carneiro, 30-May-2014.)

Theoremdvds1 12451 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremalzdvds 12452* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsext 12453* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremfzm1ndvds 12454 No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremfzo0dvdseq 12455 Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremfzocongeq 12456 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^

Theoremdvdsfac 12457 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremdvdsexp 12458 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdsmod 12459 Any number whose mod base is divisible by a divisor of the base is also divisible by . This means that primes will also be relatively prime to the base when reduced for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremodd2np1lem 12460* Lemma for odd2np1 12461. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 12461* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremoddm1even 12462 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoddp1even 12463 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.)

Theoremoexpneg 12464 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Theorem3dvds 12465* A rule for divisibility by 3 of a number written in base 10. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.)

6.1.4  The division algorithm

Theoremdivalglem0 12466 Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem1 12467 Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem2 12468* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem4 12469* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem5 12470* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem6 12471 Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem7 12472 Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem8 12473* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem9 12474* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalglem10 12475* Lemma for divalg 12476. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalg 12476* The division algorithm (theorem). Dividing an integer by a nonzero integer produces a (unique) quotient and a unique remainder . The proof does not use , or . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgb 12477* Express the division algorithm as stated in divalg 12476 in terms of . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdivalg2 12478* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivalgmod 12479* The result of the operator satisfies the requirements for the remainder in the division algorithm for a positive divisor (compare divalg2 12478 and divalgb 12477). This demonstration theorem justifies the use of to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdssub 12480 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsadd 12481 Corollary of the division algorithm. If an integer greater than divides , then it does not divide any of , ... . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsp1 12482 Special case of ndvdsadd 12481. If an integer greater than divides , it does not divide . (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremndvdsi 12483 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)

6.1.5  Bit sequences

Syntaxcbits 12484 Define the binary bits of an integer.
bits

Syntaxcsmu 12486 Define the sequence multiplication on bit sequences.
smul

Definitiondf-bits 12487* Define the binary bits of an integer. The expression bits means that the -th bit of is 1 (and its negation means the bit is 0). (Contributed by Mario Carneiro, 4-Sep-2016.)
bits

Theorembitsfval 12488* Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsval 12489 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsval2 12490 Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsss 12491 The set of bits of an integer is a subset of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsf 12492 The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0 12493 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0e 12494 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembits0o 12495 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits

Theorembitsp1 12496 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsp1e 12497 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsp1o 12498 The -th bit of is the -th bit of . (Contributed by Mario Carneiro, 5-Sep-2016.)
bits bits

Theorembitsfzolem 12499* Lemma for bitsfzo 12500. (Contributed by Mario Carneiro, 5-Sep-2016.)
bits ..^              ..^

Theorembitsfzo 12500 The bits of a number are all less than iff the number is nonnegative and less than . (Contributed by Mario Carneiro, 5-Sep-2016.)
..^ bits ..^

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