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Date | Label | Description |
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Theorem | ||
23-Jul-2024 | dceqnconst 13423 | Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 13422 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.) |
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15-Jul-2024 | fprodseq 11384 | The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.) |
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14-Jul-2024 | rexbid2 2443 | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
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14-Jul-2024 | ralbid2 2442 | Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
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12-Jul-2024 | 2irrexpqap 13103 |
There exist real numbers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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12-Jul-2024 | 2logb9irrap 13102 | Example for logbgcd1irrap 13095. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.) |
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11-Jul-2024 | logbgcd1irraplemexp 13093 |
Lemma for logbgcd1irrap 13095. Apartness of ![]() ![]() ![]() ![]() ![]() ![]() |
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11-Jul-2024 | reapef 12907 | Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.) |
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10-Jul-2024 | apcxp2 13066 | Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.) |
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9-Jul-2024 | logbgcd1irraplemap 13094 | Lemma for logbgcd1irrap 13095. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.) |
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9-Jul-2024 | apexp1 10496 | Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.) |
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5-Jul-2024 | logrpap0 13006 | The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.) |
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3-Jul-2024 | rplogbval 13070 | Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.) |
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3-Jul-2024 | logrpap0d 13007 | Deduction form of logrpap0 13006. (Contributed by Jim Kingdon, 3-Jul-2024.) |
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3-Jul-2024 | logrpap0b 13005 | The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.) |
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28-Jun-2024 | 2o01f 13364 |
Mapping zero and one between ![]() ![]() |
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28-Jun-2024 | 012of 13363 |
Mapping zero and one between ![]() ![]() |
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27-Jun-2024 | iooreen 13427 | An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.) |
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27-Jun-2024 | iooref1o 13426 | A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.) |
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25-Jun-2024 | neapmkvlem 13424 | Lemma for neapmkv 13425. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | ismkvnn 13420 | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | ismkvnnlem 13419 | Lemma for ismkvnn 13420. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | enmkvlem 7043 | Lemma for enmkv 7044. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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24-Jun-2024 | neapmkv 13425 | If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.) |
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24-Jun-2024 | enmkv 7044 |
Being Markov is invariant with respect to equinumerosity. For example,
this means that we can express the Markov's Principle as either
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21-Jun-2024 | redcwlpolemeq1 13421 | Lemma for redcwlpo 13422. A biconditionalized version of trilpolemeq1 13408. (Contributed by Jim Kingdon, 21-Jun-2024.) |
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20-Jun-2024 | redcwlpo 13422 |
Decidability of real number equality implies the Weak Limited Principle
of Omniscience (WLPO). We expect that we'd need some form of countable
choice to prove the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 13421). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10055 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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20-Jun-2024 | iswomninn 13418 |
Weak omniscience stated in terms of natural numbers. Similar to
iswomnimap 7048 but it will sometimes be more convenient to
use ![]() ![]() ![]() ![]() |
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20-Jun-2024 | iswomninnlem 13417 | Lemma for iswomnimap 7048. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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20-Jun-2024 | enwomni 7051 |
Weak omniscience is invariant with respect to equinumerosity. For
example, this means that we can express the Weak Limited Principle of
Omniscience as either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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20-Jun-2024 | enwomnilem 7050 | Lemma for enwomni 7051. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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19-Jun-2024 | rpabscxpbnd 13067 | Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.) |
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16-Jun-2024 | rpcxpsqrt 13050 |
The exponential function with exponent ![]() ![]() ![]() ![]() |
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13-Jun-2024 | rpcxpadd 13034 | Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.) |
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12-Jun-2024 | cxpap0 13033 | Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcncxpcl 13031 | Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcxp0 13027 | Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | cxpexpnn 13025 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | cxpexprp 13024 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcxpef 13023 | Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | df-rpcxp 12988 | Define the power function on complex numbers. Because df-relog 12987 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.) |
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10-Jun-2024 | trirec0xor 13413 |
Version of trirec0 13412 with exclusive-or.
The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.) |
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10-Jun-2024 | trirec0 13412 |
Every real number having a reciprocal or equaling zero is equivalent to
real number trichotomy.
This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 13411). (Contributed by Jim Kingdon, 10-Jun-2024.) |
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9-Jun-2024 | omniwomnimkv 7049 |
A set is omniscient if and only if it is weakly omniscient and Markov.
The case ![]() ![]() ![]() ![]() ![]() |
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9-Jun-2024 | iswomnimap 7048 | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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9-Jun-2024 | iswomni 7047 | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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9-Jun-2024 | df-womni 7046 |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function ![]() ![]()
In particular, The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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29-May-2024 | pw1nct 13371 | A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.) |
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28-May-2024 | sssneq 13370 | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
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24-May-2024 | dvmptcjx 12894 | Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.) |
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22-May-2024 | efltlemlt 12903 | Lemma for eflt 12904. The converse of efltim 11441 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.) |
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21-May-2024 | eflt 12904 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.) |
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19-May-2024 | apdifflemr 13415 | Lemma for apdiff 13416. (Contributed by Jim Kingdon, 19-May-2024.) |
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18-May-2024 | apdifflemf 13414 |
Lemma for apdiff 13416. Being apart from the point halfway between
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17-May-2024 | apdiff 13416 | The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.) |
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15-May-2024 | reeff1oleme 12901 | Lemma for reeff1o 12902. (Contributed by Jim Kingdon, 15-May-2024.) |
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14-May-2024 | df-relog 12987 | Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.) |
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7-May-2024 | ioocosf1o 12983 | The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.) |
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7-May-2024 | cos0pilt1 12981 |
Cosine is between minus one and one on the open interval between zero and
![]() |
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6-May-2024 | cos11 12982 |
Cosine is one-to-one over the closed interval from ![]() ![]() |
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5-May-2024 | omiunct 11993 | The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 11989 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.) |
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5-May-2024 | ctiunctal 11990 |
Variation of ctiunct 11989 which allows ![]() ![]() |
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3-May-2024 | cc4n 7103 |
Countable choice with a simpler restriction on how every set in the
countable collection needs to be inhabited. That is, compared with
cc4 7102, the hypotheses only require an A(n) for each
value of ![]() ![]() ![]() ![]() ![]() |
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3-May-2024 | cc4f 7101 |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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1-May-2024 | cc4 7102 |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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29-Apr-2024 | cc3 7100 | Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.) |
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27-Apr-2024 | cc2 7099 | Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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27-Apr-2024 | cc2lem 7098 | Lemma for cc2 7099. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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27-Apr-2024 | cc1 7097 | Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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19-Apr-2024 | omctfn 11992 | Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.) |
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13-Apr-2024 | prodmodclem2 11378 | Lemma for prodmodc 11379. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.) |
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11-Apr-2024 | prodmodclem2a 11377 | Lemma for prodmodc 11379. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
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11-Apr-2024 | prodmodclem3 11376 | Lemma for prodmodc 11379. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
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10-Apr-2024 | jcnd 642 | Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.) |
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4-Apr-2024 | prodrbdclem 11372 | Lemma for prodrbdc 11375. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.) |
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24-Mar-2024 | prodfdivap 11348 | The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
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24-Mar-2024 | prodfrecap 11347 | The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
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23-Mar-2024 | prodfap0 11346 | The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.) |
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22-Mar-2024 | prod3fmul 11342 | The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.) |
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21-Mar-2024 | df-proddc 11352 |
Define the product of a series with an index set of integers ![]() |
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19-Mar-2024 | cos02pilt1 12980 |
Cosine is less than one between zero and ![]() ![]() ![]() |
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19-Mar-2024 | cosq34lt1 12979 | Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.) |
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14-Mar-2024 | coseq0q4123 12963 |
Location of the zeroes of cosine in
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14-Mar-2024 | cosq23lt0 12962 | The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
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9-Mar-2024 | pilem3 12912 | Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.) |
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9-Mar-2024 | exmidonfin 7067 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6774 and nnon 4531. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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9-Mar-2024 | exmidonfinlem 7066 | Lemma for exmidonfin 7067. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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8-Mar-2024 | sin0pilem2 12911 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
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8-Mar-2024 | sin0pilem1 12910 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
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7-Mar-2024 | cosz12 12909 | Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.) |
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6-Mar-2024 | cos12dec 11510 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
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25-Feb-2024 | mul2lt0pn 9581 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | mul2lt0np 9580 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | lt0ap0 8434 | A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | negap0d 8417 | The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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24-Feb-2024 | lt0ap0d 8435 | A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.) |
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20-Feb-2024 | ivthdec 12830 | The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.) |
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20-Feb-2024 | ivthinclemex 12828 | Lemma for ivthinc 12829. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.) |
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19-Feb-2024 | ivthinclemuopn 12824 | Lemma for ivthinc 12829. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.) |
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19-Feb-2024 | dedekindicc 12819 | A Dedekind cut identifies a unique real number. Similar to df-inp 7298 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.) |
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Copyright terms: Public domain | W3C HTML validation [external] |