Home Intuitionistic Logic ExplorerTheorem List (p. 96 of 137) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminfsupneg 9501* If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9500. (Contributed by Jim Kingdon, 15-Jan-2022.)

Theoremsupminfex 9502* A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
inf

Theoremeluznn0 9503 Membership in a nonnegative upper set of integers implies membership in . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremeluznn 9504 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)

Theoremeluz2b1 9505 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2gt1 9506 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)

Theoremeluz2b2 9507 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b3 9508 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremuz2m1nn 9509 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)

Theorem1nuz2 9510 1 is not in . (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremelnn1uz2 9511 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremuz2mulcl 9512 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremindstr2 9513* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremeluzdc 9514 Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
DECID

Theoremublbneg 9515* The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9500. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremeqreznegel 9516* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegm 9517* The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)

Theoremlbzbi 9518* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnn01to3 9519 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)

Theoremnn0ge2m1nnALT 9520 Alternate proof of nn0ge2m1nn 9144: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9439, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 9144. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

4.4.12  Rational numbers (as a subset of complex numbers)

Syntaxcq 9521 Extend class notation to include the class of rationals.

Definitiondf-q 9522 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9524 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

Theoremdivfnzn 9523 Division restricted to is a function. Given excluded middle, it would be easy to prove this for . The key difference is that an element of is apart from zero, whereas being an element of implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)

Theoremelq 9524* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)

Theoremqmulz 9525* If is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)

Theoremznq 9526 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)

Theoremqre 9527 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)

Theoremzq 9528 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)

Theoremzssq 9529 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssq 9530 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremnnssq 9531 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremqssre 9532 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

Theoremqsscn 9533 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremqex 9534 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnq 9535 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)

Theoremqcn 9536 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)

Theoremqnegcl 9538 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)

Theoremqmulcl 9539 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)

Theoremqsubcl 9540 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremqapne 9541 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
#

Theoremqltlen 9542 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8501 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremqlttri2 9543 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)

Theoremqreccl 9544 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqdivcl 9545 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqrevaddcl 9546 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremnnrecq 9547 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)

Theoremirradd 9548 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)

Theoremirrmul 9549 The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.)

Theoremelpq 9550* A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)

Theoremelpqb 9551* A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)

4.4.13  Complex numbers as pairs of reals

Theoremcnref1o 9552* There is a natural one-to-one mapping from to , where we map to . In our construction of the complex numbers, this is in fact our definition of (see df-c 7732), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)

4.5  Order sets

4.5.1  Positive reals (as a subset of complex numbers)

Syntaxcrp 9553 Extend class notation to include the class of positive reals.

Definitiondf-rp 9554 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremelrp 9555 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)

Theoremelrpii 9556 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)

Theorem1rp 9557 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)

Theorem2rp 9558 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theorem3rp 9559 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremrpre 9560 A positive real is a real. (Contributed by NM, 27-Oct-2007.)

Theoremrpxr 9561 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremrpcn 9562 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)

Theoremnnrp 9563 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)

Theoremrpssre 9564 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)

Theoremrpgt0 9565 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)

Theoremrpge0 9566 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)

Theoremrpregt0 9567 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrprege0 9568 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpne0 9569 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)

Theoremrpap0 9570 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremrprene0 9571 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)

Theoremrpreap0 9572 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremrpcnne0 9573 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)

Theoremrpcnap0 9574 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremralrp 9575 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Theoremrexrp 9576 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrpaddcl 9577 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpmulcl 9578 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpdivcl 9579 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)

Theoremrpreccl 9580 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)

Theoremrphalfcl 9581 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpgecl 9582 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalflt 9583 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrerpdivcl 9584 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)

Theoremge0p1rp 9585 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremrpnegap 9586 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
#

Theoremnegelrp 9587 Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)

Theoremnegelrpd 9588 The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Theorem0nrp 9589 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremltsubrp 9590 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)

Theoremltaddrp 9591 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)

Theoremdifrp 9592 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 9593 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 9594 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzgt1rpn0n1 9595 An integer greater than 1 is a positive real number not equal to 0 or 1. Useful for working with integer logarithm bases (which is a common case, e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux, 26-Sep-2017.) (Proof shortened by AV, 9-Jul-2022.)

Theoremrpred 9596 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 9597 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 9598 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 9599 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 9600 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13623
 Copyright terms: Public domain < Previous  Next >