P. 154 of Theorem List - Intuitionistic Logic Explorer

Home	Intuitionistic Logic Explorer Theorem List (p. 154 of 158)	< 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 - 15301-15400 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

gausslemma2dlem1f1o 15301*

Lemma for gausslemma2dlem1 15302. (Contributed by Jim Kingdon, 9-Aug-2025.)

$\$

Theorem

gausslemma2dlem1 15302*

Lemma 1 for gausslemma2d 15310. (Contributed by AV, 5-Jul-2021.)

$\$

Theorem

gausslemma2dlem2 15303*

Lemma 2 for gausslemma2d 15310. (Contributed by AV, 4-Jul-2021.)

$\$

Theorem

gausslemma2dlem3 15304*

Lemma 3 for gausslemma2d 15310. (Contributed by AV, 4-Jul-2021.)

$\$

Theorem

gausslemma2dlem4 15305*

Lemma 4 for gausslemma2d 15310. (Contributed by AV, 16-Jun-2021.)

$\$

Theorem

gausslemma2dlem5a 15306*

Lemma for gausslemma2dlem5 15307. (Contributed by AV, 8-Jul-2021.)

$\$

Theorem

gausslemma2dlem5 15307*

Lemma 5 for gausslemma2d 15310. (Contributed by AV, 9-Jul-2021.)

$\$

Theorem

gausslemma2dlem6 15308*

Lemma 6 for gausslemma2d 15310. (Contributed by AV, 16-Jun-2021.)

$\$

Theorem

gausslemma2dlem7 15309*

Lemma 7 for gausslemma2d 15310. (Contributed by AV, 13-Jul-2021.)

$\$

Theorem

gausslemma2d 15310*

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.)

$\$

11.3.6 Quadratic reciprocity

Theorem

lgseisenlem1 15311*

Lemma for lgseisen 15315. 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 15312*

Lemma for lgseisen 15315. The function

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

$\$

Theorem

lgseisenlem3 15313*

Lemma for lgseisen 15315. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.)

$\$

ℤ/nℤ

mulGrp

RHom

Theorem

lgseisenlem4 15314*

Lemma for lgseisen 15315. (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.)

$\$

ℤ/nℤ

mulGrp

RHom

Theorem

lgseisen 15315*

Eisenstein's lemma, an expression for

when

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

$\$

Theorem

lgsquadlemsfi 15316*

Lemma for lgsquad 15321.

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

$\$

$/\$

Theorem

lgsquadlemofi 15317*

Lemma for lgsquad 15321. There are finitely many members of

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

$\$

$/\$

Theorem

lgsquadlem1 15318*

Lemma for lgsquad 15321. Count the members of

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

$\$

$/\$

♯

Theorem

lgsquadlem2 15319*

Lemma for lgsquad 15321. Count the members of

with even coordinates, and combine with lgsquadlem1 15318 to get the total count of lattice points in

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

$\$

$/\$

♯

Theorem

lgsquadlem3 15320*

Lemma for lgsquad 15321. (Contributed by Mario Carneiro, 18-Jun-2015.)

$\$

$/\$

Theorem

lgsquad 15321

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 15322

Lemma for lgsquad2 15324. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

lgsquad2lem2 15323*

Lemma for lgsquad2 15324. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

$\$

$/\$

Theorem

lgsquad2 15324

Extend lgsquad 15321 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem

lgsquad3 15325

Extend lgsquad2 15324 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

$/\$

Theorem

m1lgs 15326

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 15327*

Lemma 1 for 2lgslem1a 15329. (Contributed by AV, 16-Jun-2021.)

$/\$

Theorem

2lgslem1a2 15328

Lemma 2 for 2lgslem1a 15329. (Contributed by AV, 18-Jun-2021.)

$/\$

Theorem

2lgslem1a 15329*

Lemma 1 for 2lgslem1 15332. (Contributed by AV, 18-Jun-2021.)

$/\$

Theorem

2lgslem1b 15330*

Lemma 2 for 2lgslem1 15332. (Contributed by AV, 18-Jun-2021.)

Theorem

2lgslem1c 15331

Lemma 3 for 2lgslem1 15332. (Contributed by AV, 19-Jun-2021.)

$/\$

Theorem

2lgslem1 15332*

Lemma 1 for 2lgs 15345. (Contributed by AV, 19-Jun-2021.)

$/\$

♯

$/\$

Theorem

2lgslem2 15333

Lemma 2 for 2lgs 15345. (Contributed by AV, 20-Jun-2021.)

$/\$

Theorem

2lgslem3a 15334

Lemma for 2lgslem3a1 15338. (Contributed by AV, 14-Jul-2021.)

$/\$

Theorem

2lgslem3b 15335

Lemma for 2lgslem3b1 15339. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3c 15336

Lemma for 2lgslem3c1 15340. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3d 15337

Lemma for 2lgslem3d1 15341. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3a1 15338

Lemma 1 for 2lgslem3 15342. (Contributed by AV, 15-Jul-2021.)

$/\$

Theorem

2lgslem3b1 15339

Lemma 2 for 2lgslem3 15342. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3c1 15340

Lemma 3 for 2lgslem3 15342. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgslem3d1 15341

Lemma 4 for 2lgslem3 15342. (Contributed by AV, 15-Jul-2021.)

$/\$

Theorem

2lgslem3 15342

Lemma 3 for 2lgs 15345. (Contributed by AV, 16-Jul-2021.)

$/\$

Theorem

2lgs2 15343

The Legendre symbol for

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

Theorem

2lgslem4 15344

Lemma 4 for 2lgs 15345: special case of 2lgs 15345 for

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

Theorem

2lgs 15345

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 15258) 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 15346

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

$/\$

Theorem

2lgsoddprmlem2 15347

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

$/\$

Theorem

2lgsoddprmlem3a 15348

Lemma 1 for 2lgsoddprmlem3 15352. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3b 15349

Lemma 2 for 2lgsoddprmlem3 15352. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3c 15350

Lemma 3 for 2lgsoddprmlem3 15352. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3d 15351

Lemma 4 for 2lgsoddprmlem3 15352. (Contributed by AV, 20-Jul-2021.)

Theorem

2lgsoddprmlem3 15352

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

$/\$

Theorem

2lgsoddprmlem4 15353

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

$/\$

Theorem

2lgsoddprm 15354

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.)

$\$

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

Theorem

2sqlem1 15355*

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

Theorem

2sqlem2 15356*

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

Theorem

mul2sq 15357

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 15358

Lemma for 2sqlem5 15360. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem4 15359

Lemma for 2sqlem5 15360. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem

2sqlem5 15360

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 15361*

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 15362*

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

$/\$

Theorem

2sqlem8a 15363*

Lemma for 2sqlem8 15364. (Contributed by Mario Carneiro, 4-Jun-2016.)

$/\$

Theorem

2sqlem8 15364*

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

$/\$

Theorem

2sqlem9 15365*

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

$/\$

Theorem

2sqlem10 15366*

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 15367

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 15458 (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 5921, 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 5590 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 7315, df-ap 8609	Yes	apadd1 8635, apne 8650
g	with "is a set" condition		No	1stvalg 6200, brtposg 6312, setsmsbasg 14715
m	inhabited (from "member")		No	r19.2m 3537, negm 9689, ctm 7175, basmex 12737
seq3, sum3	recursive sequence	df-seqfrec 10540	Yes	seq3-1 10554, fsum3 11552
tap	tight apartness	df-tap 7317	Yes	df-tap 7317

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7317.

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

12.1.2 Definitional examples

Theorem

ex-or 15368

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

$\/$

Theorem

ex-an 15369

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

$/\$

Theorem

1kp2ke3k 15370

Example for df-dec 9458, 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 9458 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 15371

Example for df-fl 10360. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

$/\$

Theorem

ex-ceil 15372

Example for df-ceil 10361. (Contributed by AV, 4-Sep-2021.)

⌈

$/\$

⌈

Theorem

ex-exp 15373

Example for df-exp 10631. (Contributed by AV, 4-Sep-2021.)

;

$/\$

Theorem

ex-fac 15374

Example for df-fac 10818. (Contributed by AV, 4-Sep-2021.)

;;

Theorem

ex-bc 15375

Example for df-bc 10840. (Contributed by AV, 4-Sep-2021.)

;

Theorem

ex-dvds 15376

Example for df-dvds 11953: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

Theorem

ex-gcd 15377

Example for df-gcd 12121. (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 15378

(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 15379

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 15380

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

$\/$

Theorem

bj-nnim 15381

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

Theorem

bj-nnan 15382

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

$/\$

Theorem

bj-nnclavius 15383

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

Theorem

bj-imnimnn 15384

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 15383 as its last step. (Contributed by BJ, 27-Oct-2024.)

13.2.1.1 Stable formulas

Some of the following theorems, like bj-sttru 15386 or bj-stfal 15388 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 15385

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

STAB

Theorem

bj-sttru 15386

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

STAB

Theorem

bj-fast 15387

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

STAB

Theorem

bj-stfal 15388

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

STAB

Theorem

bj-nnst 15389

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 15636 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 15390

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 15403). (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-stst 15391

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

Theorem

bj-stim 15392

A conjunction with a stable consequent is stable. See stabnot 834 for negation , bj-stan 15393 for conjunction , and bj-stal 15395 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-stan 15393

The conjunction of two stable formulas is stable. See bj-stim 15392 for implication, stabnot 834 for negation, and bj-stal 15395 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

STAB

$/\$ STAB

STAB

$/\$

Theorem

bj-stand 15394

The conjunction of two stable formulas is stable. Deduction form of bj-stan 15393. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 15393 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

STAB

$/\$

Theorem

bj-stal 15395

The universal quantification of a stable formula is stable. See bj-stim 15392 for implication, stabnot 834 for negation, and bj-stan 15393 for conjunction. (Contributed by BJ, 24-Nov-2023.)

STAB

Theorem

bj-pm2.18st 15396

Clavius law for stable formulas. See pm2.18dc 856. (Contributed by BJ, 4-Dec-2023.)

STAB

Theorem

bj-con1st 15397

Contraposition when the antecedent is a negated stable proposition. See con1dc 857. (Contributed by BJ, 11-Nov-2024.)

STAB

13.2.1.2 Decidable formulas

Theorem

bj-trdc 15398

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

DECID

Theorem

bj-dctru 15399

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

DECID

Theorem

bj-fadc 15400

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

DECID

< Previous Next >

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 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15728

< Previous Next >