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Theorem List for Intuitionistic Logic Explorer - 14401-14500 *Has distinct variable group(s)

Type

Label

Description

Statement

10.2 Basic number theory

10.2.1 Quadratic residues and the Legendre symbol

If the congruence has a solution we say that is a quadratic residue . If the congruence has no solution we say that is a quadratic nonresidue , see definition in [ApostolNT] p. 178. The Legendre symbol is defined in a way that its value is if is a quadratic residue and if is a quadratic nonresidue (and if divides ).

Originally, the Legendre symbol was defined for odd primes only (and arbitrary integers ) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs 14402 for the "Legendre symbol" is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for , but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol".

Syntax

clgs 14401

Extend class notation with the Legendre symbol function.

Definition

df-lgs 14402*

Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

zabsle1 14403

is the set of all integers with absolute value at most

. (Contributed by AV, 13-Jul-2021.)

Theorem

lgslem1 14404

When

is coprime to the prime

is equivalent

, and so adding

makes it equivalent to

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

$\$

$/\$

Theorem

lgslem2 14405

The set

of all integers with absolute value at most

contains

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgslem3 14406*

The set

of all integers with absolute value at most

is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgslem4 14407*

Lemma for lgsfcl2 14410. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)

$/\$

$\$

Theorem

lgsval 14408*

Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsfvalg 14409*

Value of the function

which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)

$/\$

Theorem

lgsfcl2 14410*

The function

is closed in integers with absolute value less than

(namely

, see zabsle1 14403). (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgscllem 14411*

The Legendre symbol is an element of

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsfcl 14412*

Closure of the function

which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsfle1 14413*

The function

has magnitude less or equal to

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsval2lem 14414*

Lemma for lgsval2 14420. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsval4lem 14415*

Lemma for lgsval4 14424. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgscl2 14416*

The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgs0 14417

The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorem

lgscl 14418

The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsle1 14419

The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 14418 this implies that it takes values in

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsval2 14420

The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime

). (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgs2 14421

The Legendre symbol at

. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorem

lgsval3 14422

The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

$\$

Theorem

lgsvalmod 14423

The Legendre symbol is equivalent to

. This theorem is also called "Euler's criterion", see theorem 9.2 in [ApostolNT] p. 180, or a representation of Euler's criterion using the Legendre symbol, (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

$\$

Theorem

lgsval4 14424*

Restate lgsval 14408 for nonzero

, where the function

has been abbreviated into a self-referential expression taking the value of

on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsfcl3 14425*

Closure of the function

which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsval4a 14426*

Same as lgsval4 14424 for positive

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgscl1 14427

The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.)

$/\$

Theorem

lgsneg 14428

The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsneg1 14429

The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsmod 14430

The Legendre (Jacobi) symbol is preserved under reduction

when

is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdilem 14431

Lemma for lgsdi 14441 and lgsdir 14439: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2lem1 14432

Lemma for lgsdir2 14437. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2lem2 14433

Lemma for lgsdir2 14437. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2lem3 14434

Lemma for lgsdir2 14437. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2lem4 14435

Lemma for lgsdir2 14437. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2lem5 14436

Lemma for lgsdir2 14437. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdir2 14437

The Legendre symbol is completely multiplicative at

. (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdirprm 14438

The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.)

$/\$

Theorem

lgsdir 14439

The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that

and

are odd positive integers). (Contributed by Mario Carneiro, 4-Feb-2015.)

$/\$

Theorem

lgsdilem2 14440*

Lemma for lgsdi 14441. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorem

lgsdi 14441

The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that

and

are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)

$/\$

Theorem

lgsne0 14442

The Legendre symbol is nonzero (and hence equal to

) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

$/\$

Theorem

lgsabs1 14443

The Legendre symbol is nonzero (and hence equal to

) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

$/\$

Theorem

lgssq 14444

The Legendre symbol at a square is equal to

. Together with lgsmod 14430 this implies that the Legendre symbol takes value

at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.)

$/\$

Theorem

lgssq2 14445

The Legendre symbol at a square is equal to

. (Contributed by Mario Carneiro, 5-Feb-2015.)

$/\$

Theorem

lgsprme0 14446

The Legendre symbol at any prime (even at 2) is

iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)

$/\$

Theorem

1lgs 14447

The Legendre symbol at

. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.)

Theorem

lgs1 14448

The Legendre symbol at

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

Theorem

lgsmodeq 14449

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 14450

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 14451

Variation on lgsdir 14439 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 14452

Variation on lgsdi 14441 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.)

$/\$

10.2.2 Quadratic reciprocity

Theorem

lgseisenlem1 14453*

Lemma for Eisenstein's lemma. 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 14454*

Lemma for Eisenstein's lemma. The function

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

$\$

Theorem

m1lgs 14455

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

2lgsoddprmlem1 14456

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

$/\$

Theorem

2lgsoddprmlem2 14457

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

$/\$

Theorem

2lgsoddprmlem3a 14458

Lemma 1 for 2lgsoddprmlem3 14462. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3b 14459

Lemma 2 for 2lgsoddprmlem3 14462. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3c 14460

Lemma 3 for 2lgsoddprmlem3 14462. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3d 14461

Lemma 4 for 2lgsoddprmlem3 14462. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3 14462

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

$/\$

Theorem

2lgsoddprmlem4 14463

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

$/\$

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

Theorem

2sqlem1 14464*

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

Theorem

2sqlem2 14465*

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

Theorem

mul2sq 14466

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 14467

Lemma for 2sqlem5 14469. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem4 14468

Lemma for 2sqlem5 14469. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem5 14469

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 14470*

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 14471*

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

$/\$

Theorem

2sqlem8a 14472*

Lemma for 2sqlem8 14473. (Contributed by Mario Carneiro, 4-Jun-2016.)

$/\$

Theorem

2sqlem8 14473*

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

$/\$

Theorem

2sqlem9 14474*

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

$/\$

Theorem

2sqlem10 14475*

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 11 GUIDES AND MISCELLANEA

11.1 Guides (conventions, explanations, and examples)

11.1.1 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 14476

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 14567 (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 5874, 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 5548 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 7247, df-ap 8539	Yes	apadd1 8565, apne 8580
g	with "is a set" condition		No	1stvalg 6143, brtposg 6255, setsmsbasg 13982
seq3, sum3	recursive sequence	df-seqfrec 10446	Yes	seq3-1 10460, fsum3 11395
tap	tight apartness	df-tap 7249	Yes	df-tap 7249

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7249.

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

11.1.2 Definitional examples

Theorem

ex-or 14477

Example for ax-io 709. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

$\/$

Theorem

ex-an 14478

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

$/\$

Theorem

1kp2ke3k 14479

Example for df-dec 9385, 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 9385 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 14480

Example for df-fl 10270. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

$/\$

Theorem

ex-ceil 14481

Example for df-ceil 10271. (Contributed by AV, 4-Sep-2021.)

⌈

$/\$

⌈

Theorem

ex-exp 14482

Example for df-exp 10520. (Contributed by AV, 4-Sep-2021.)

;

$/\$

Theorem

ex-fac 14483

Example for df-fac 10706. (Contributed by AV, 4-Sep-2021.)

;;

Theorem

ex-bc 14484

Example for df-bc 10728. (Contributed by AV, 4-Sep-2021.)

;

Theorem

ex-dvds 14485

Example for df-dvds 11795: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

Theorem

ex-gcd 14486

Example for df-gcd 11944. (Contributed by AV, 5-Sep-2021.)

PART 12 SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

12.1 Mathboxes for user contributions

12.1.1 Mathbox guidelines

Theorem

mathbox 14487

(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.)

12.2 Mathbox for BJ

12.2.1 Propositional calculus

Theorem

bj-nnsn 14488

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 14489

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

$\/$

Theorem

bj-nnim 14490

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

Theorem

bj-nnan 14491

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

$/\$

Theorem

bj-nnclavius 14492

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

Theorem

bj-imnimnn 14493

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 14492 as its last step. (Contributed by BJ, 27-Oct-2024.)

12.2.1.1 Stable formulas

Some of the following theorems, like bj-sttru 14495 or bj-stfal 14497 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

Theorem

bj-trst 14494

A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-sttru 14495

The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)

STAB

Theorem

bj-fast 14496

A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-stfal 14497

The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)

STAB

Theorem

bj-nnst 14498

Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 14745 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in

-intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in

-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

STAB

Theorem

bj-nnbist 14499

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if

is a classical tautology, then

is an intuitionistic tautology. Therefore, if

is a classical tautology, then

is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 14512). (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-stst 14500

Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)

STAB STAB

STAB

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