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Theorem List for Intuitionistic Logic Explorer - 15201-15300 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

lgs1 15201

The Legendre symbol at

. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)

Theorem

lgsmodeq 15202

The Legendre (Jacobi) symbol is preserved under reduction

when

is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)

$/\$

Theorem

lgsmulsqcoprm 15203

The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)

$/\$

Theorem

lgsdirnn0 15204

Variation on lgsdir 15192 valid for all

but only for positive

. (The exact location of the failure of this law is for

in which case

but

.) (Contributed by Mario Carneiro, 28-Apr-2016.)

$/\$

Theorem

lgsdinn0 15205

Variation on lgsdi 15194 valid for all

but only for positive

. (The exact location of the failure of this law is for

, and some

in which case

but

when

is not a quadratic residue mod

.) (Contributed by Mario Carneiro, 28-Apr-2016.)

$/\$

11.3.3 Gauss' Lemma

Gauss' Lemma is valid for any integer not dividing the given prime number. In the following, only the special case for 2 (not dividing any odd prime) is proven, see gausslemma2d 15226. The general case is still to prove.

Theorem

gausslemma2dlem0a 15206

Auxiliary lemma 1 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0b 15207

Auxiliary lemma 2 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0c 15208

Auxiliary lemma 3 for gausslemma2d 15226. (Contributed by AV, 13-Jul-2021.)

$\$

Theorem

gausslemma2dlem0d 15209

Auxiliary lemma 4 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0e 15210

Auxiliary lemma 5 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0f 15211

Auxiliary lemma 6 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0g 15212

Auxiliary lemma 7 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0h 15213

Auxiliary lemma 8 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem0i 15214

Auxiliary lemma 9 for gausslemma2d 15226. (Contributed by AV, 14-Jul-2021.)

$\$

Theorem

gausslemma2dlem1a 15215*

Lemma for gausslemma2dlem1 15218. (Contributed by AV, 1-Jul-2021.)

$\$

Theorem

gausslemma2dlem1cl 15216

Lemma for gausslemma2dlem1 15218. Closure of the body of the definition of

. (Contributed by Jim Kingdon, 10-Aug-2025.)

$\$

Theorem

gausslemma2dlem1f1o 15217*

Lemma for gausslemma2dlem1 15218. (Contributed by Jim Kingdon, 9-Aug-2025.)

$\$

Theorem

gausslemma2dlem1 15218*

Lemma 1 for gausslemma2d 15226. (Contributed by AV, 5-Jul-2021.)

$\$

Theorem

gausslemma2dlem2 15219*

Lemma 2 for gausslemma2d 15226. (Contributed by AV, 4-Jul-2021.)

$\$

Theorem

gausslemma2dlem3 15220*

Lemma 3 for gausslemma2d 15226. (Contributed by AV, 4-Jul-2021.)

$\$

Theorem

gausslemma2dlem4 15221*

Lemma 4 for gausslemma2d 15226. (Contributed by AV, 16-Jun-2021.)

$\$

Theorem

gausslemma2dlem5a 15222*

Lemma for gausslemma2dlem5 15223. (Contributed by AV, 8-Jul-2021.)

$\$

Theorem

gausslemma2dlem5 15223*

Lemma 5 for gausslemma2d 15226. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem6 15224*

Lemma 6 for gausslemma2d 15226. (Contributed by AV, 16-Jun-2021.)

$\$

Theorem

gausslemma2dlem7 15225*

Lemma 7 for gausslemma2d 15226. (Contributed by AV, 13-Jul-2021.)

$\$

Theorem

gausslemma2d 15226*

Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer

: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

$\$

11.3.4 Quadratic reciprocity

Theorem

lgseisenlem1 15227*

Lemma for lgseisen 15231. If

and

, then for any even

is also an even integer

. To simplify these statements, we divide all the even numbers by

, so that it becomes the statement that

is an integer between

and

. (Contributed by Mario Carneiro, 17-Jun-2015.)

$\$

Theorem

lgseisenlem2 15228*

Lemma for lgseisen 15231. The function

is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)

$\$

Theorem

lgseisenlem3 15229*

Lemma for lgseisen 15231. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.)

$\$

ℤ/nℤ

mulGrp

RHom

Theorem

lgseisenlem4 15230*

Lemma for lgseisen 15231. (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.)

$\$

ℤ/nℤ

mulGrp

RHom

Theorem

lgseisen 15231*

Eisenstein's lemma, an expression for

when

are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

$\$

Theorem

lgsquadlemsfi 15232*

Lemma for lgsquad 15237.

is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)

$\$

$/\$

Theorem

lgsquadlemofi 15233*

Lemma for lgsquad 15237. There are finitely many members of

with odd first part. (Contributed by Jim Kingdon, 16-Sep-2025.)

$\$

$/\$

Theorem

lgsquadlem1 15234*

Lemma for lgsquad 15237. Count the members of

with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)

$\$

$/\$

♯

Theorem

lgsquadlem2 15235*

Lemma for lgsquad 15237. Count the members of

with even coordinates, and combine with lgsquadlem1 15234 to get the total count of lattice points in

(up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)

$\$

$/\$

♯

Theorem

lgsquadlem3 15236*

Lemma for lgsquad 15237. (Contributed by Mario Carneiro, 18-Jun-2015.)

$\$

$/\$

Theorem

lgsquad 15237

The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If

and

are distinct odd primes, then the product of the Legendre symbols

and

is the parity of

. This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.)

$\$

$/\$

$\$

$/\$

Theorem

lgsquad2lem1 15238

Lemma for lgsquad2 15240. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

lgsquad2lem2 15239*

Lemma for lgsquad2 15240. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

$\$

$/\$

Theorem

lgsquad2 15240

Extend lgsquad 15237 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

lgsquad3 15241

Extend lgsquad2 15240 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

Theorem

m1lgs 15242

The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime

iff

(mod

). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)

$\$

Theorem

2lgslem1a1 15243*

Lemma 1 for 2lgslem1a 15245. (Contributed by AV, 16-Jun-2021.)

$/\$

Theorem

2lgslem1a2 15244

Lemma 2 for 2lgslem1a 15245. (Contributed by AV, 18-Jun-2021.)

$/\$

Theorem

2lgslem1a 15245*

Lemma 1 for 2lgslem1 15248. (Contributed by AV, 18-Jun-2021.)

$/\$

Theorem

2lgslem1b 15246*

Lemma 2 for 2lgslem1 15248. (Contributed by AV, 18-Jun-2021.)

Theorem

2lgslem1c 15247

Lemma 3 for 2lgslem1 15248. (Contributed by AV, 19-Jun-2021.)

$/\$

Theorem

2lgslem1 15248*

Lemma 1 for 2lgs 15261. (Contributed by AV, 19-Jun-2021.)

$/\$

♯

$/\$

Theorem

2lgslem2 15249

Lemma 2 for 2lgs 15261. (Contributed by AV, 20-Jun-2021.)

$/\$

Theorem

2lgslem3a 15250

Lemma for 2lgslem3a1 15254. (Contributed by AV, 14-Jul-2021.)

$/\$

Theorem

2lgslem3b 15251

Lemma for 2lgslem3b1 15255. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3c 15252

Lemma for 2lgslem3c1 15256. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3d 15253

Lemma for 2lgslem3d1 15257. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3a1 15254

Lemma 1 for 2lgslem3 15258. (Contributed by AV, 15-Jul-2021.)

$/\$

Theorem

2lgslem3b1 15255

Lemma 2 for 2lgslem3 15258. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3c1 15256

Lemma 3 for 2lgslem3 15258. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3d1 15257

Lemma 4 for 2lgslem3 15258. (Contributed by AV, 15-Jul-2021.)

$/\$

Theorem

2lgslem3 15258

Lemma 3 for 2lgs 15261. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgs2 15259

The Legendre symbol for

. (Contributed by AV, 20-Jun-2021.)

Theorem

2lgslem4 15260

Lemma 4 for 2lgs 15261: special case of 2lgs 15261 for

. (Contributed by AV, 20-Jun-2021.)

Theorem

2lgs 15261

The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime

iff

(mod

), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of

(lgs2 15174) to some degree, by demanding that reciprocity extend to the case

. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.)

Theorem

2lgsoddprmlem1 15262

Lemma 1 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)

$/\$

Theorem

2lgsoddprmlem2 15263

Lemma 2 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)

$/\$

Theorem

2lgsoddprmlem3a 15264

Lemma 1 for 2lgsoddprmlem3 15268. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3b 15265

Lemma 2 for 2lgsoddprmlem3 15268. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3c 15266

Lemma 3 for 2lgsoddprmlem3 15268. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3d 15267

Lemma 4 for 2lgsoddprmlem3 15268. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3 15268

Lemma 3 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)

$/\$

Theorem

2lgsoddprmlem4 15269

Lemma 4 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)

$/\$

Theorem

2lgsoddprm 15270

The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for

at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight (

= -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.)

$\$

11.3.5 All primes 4n+1 are the sum of two squares

Theorem

2sqlem1 15271*

Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

2sqlem2 15272*

Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

mul2sq 15273

Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

Theorem

2sqlem3 15274

Lemma for 2sqlem5 15276. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem4 15275

Lemma for 2sqlem5 15276. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem5 15276

Lemma for 2sq . If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem6 15277*

Lemma for 2sq . If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem7 15278*

Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

Theorem

2sqlem8a 15279*

Lemma for 2sqlem8 15280. (Contributed by Mario Carneiro, 4-Jun-2016.)

$/\$

Theorem

2sqlem8 15280*

Lemma for 2sq . (Contributed by Mario Carneiro, 20-Jun-2015.)

$/\$

Theorem

2sqlem9 15281*

Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

Theorem

2sqlem10 15282*

Lemma for 2sq . Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

PART 12 GUIDES AND MISCELLANEA

12.1 Guides (conventions, explanations, and examples)

12.1.1 Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
Theorems of propositional calculus - [Heyting].
Theorems of pure predicate calculus - Metamath Proof Explorer.
Theorems of equality and substitution - Metamath Proof Explorer.
Axioms of set theory - [Crosilla].
Development of set theory - Chapter 10 of [HoTT].
Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
Theorems about real numbers - [Geuvers].

Theorem

conventions 15283

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 15374 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5918, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5587 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
iset.mm versus set.mm names
Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Syntax?	Example(s)
ap	apart	df-pap 7310, df-ap 8603	Yes	apadd1 8629, apne 8644
g	with "is a set" condition		No	1stvalg 6197, brtposg 6309, setsmsbasg 14658
m	inhabited (from "member")		No	r19.2m 3534, negm 9683, ctm 7170, basmex 12680
seq3, sum3	recursive sequence	df-seqfrec 10522	Yes	seq3-1 10536, fsum3 11533
tap	tight apartness	df-tap 7312	Yes	df-tap 7312

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7312.

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

12.1.2 Definitional examples

Theorem

ex-or 15284

Example for ax-io 710. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

$\/$

Theorem

ex-an 15285

Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

$/\$

Theorem

1kp2ke3k 15286

Example for df-dec 9452, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with , commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9452 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

;;;

Theorem

ex-fl 15287

Example for df-fl 10342. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

$/\$

Theorem

ex-ceil 15288

Example for df-ceil 10343. (Contributed by AV, 4-Sep-2021.)

⌈

$/\$

⌈

Theorem

ex-exp 15289

Example for df-exp 10613. (Contributed by AV, 4-Sep-2021.)

;

$/\$

Theorem

ex-fac 15290

Example for df-fac 10800. (Contributed by AV, 4-Sep-2021.)

;;

Theorem

ex-bc 15291

Example for df-bc 10822. (Contributed by AV, 4-Sep-2021.)

;

Theorem

ex-dvds 15292

Example for df-dvds 11934: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

Theorem

ex-gcd 15293

Example for df-gcd 12083. (Contributed by AV, 5-Sep-2021.)

PART 13 SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

13.1 Mathboxes for user contributions

13.1.1 Mathbox guidelines

Theorem

mathbox 15294

(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

13.2 Mathbox for BJ

13.2.1 Propositional calculus

Theorem

bj-nnsn 15295

As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

Theorem

bj-nnor 15296

Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)

$\/$

Theorem

bj-nnim 15297

The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)

Theorem

bj-nnan 15298

The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

$/\$

Theorem

bj-nnclavius 15299

Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)

Theorem

bj-imnimnn 15300

If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 15299 as its last step. (Contributed by BJ, 27-Oct-2024.)

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