P. 138 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

lgssq2 13701

The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.)

|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L ( N ^ 2 ) ) = 1 )

Theorem

lgsprme0 13702

The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)

|- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )

Theorem

1lgs 13703

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

|- ( N e. ZZ -> ( 1 /L N ) = 1 )

Theorem

lgs1 13704

The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)

|- ( A e. ZZ -> ( A /L 1 ) = 1 )

Theorem

lgsmodeq 13705

The Legendre (Jacobi) symbol is preserved under reduction mod n when n is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)

|- ( ( A e. ZZ /\ B e. ZZ /\ ( N e. NN /\ -. 2 || N ) ) -> ( ( A mod N ) = ( B mod N ) -> ( A /L N ) = ( B /L N ) ) )

Theorem

lgsmulsqcoprm 13706

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.)

|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( B /L N ) )

Theorem

lgsdirnn0 13707

Variation on lgsdir 13695 valid for all A , B but only for positive N. (The exact location of the failure of this law is for A = 0, B < 0, N = -u 1 in which case ( 0 /L -u 1 ) = 1 but ( B /L -u 1 ) = -u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)

|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Theorem

lgsdinn0 13708

Variation on lgsdi 13697 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

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

Theorem

2sqlem1 13709*

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

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )

Theorem

2sqlem2 13710*

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

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( A e. S <-> E. x e. ZZ E. y e. ZZ A = ( ( x ^ 2 ) + ( y ^ 2 ) ) )

Theorem

mul2sq 13711

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.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

Theorem

2sqlem3 13712

Lemma for 2sqlem5 13714. (Contributed by Mario Carneiro, 20-Jun-2015.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )   &    |- ( ph -> P || ( ( C x. B ) + ( A x. D ) ) )   =>    |- ( ph -> N e. S )

Theorem

2sqlem4 13713

Lemma for 2sqlem5 13714. (Contributed by Mario Carneiro, 20-Jun-2015.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )   =>    |- ( ph -> N e. S )

Theorem

2sqlem5 13714

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.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( N x. P ) e. S )   &    |- ( ph -> P e. S )   =>    |- ( ph -> N e. S )

Theorem

2sqlem6 13715*

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.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> A. p e. Prime ( p || B -> p e. S ) )   &    |- ( ph -> ( A x. B ) e. S )   =>    |- ( ph -> A e. S )

Theorem

2sqlem7 13716*

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

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   =>    |- Y C_ ( S i^i NN )

Theorem

2sqlem8a 13717*

Lemma for 2sqlem8 13718. (Contributed by Mario Carneiro, 4-Jun-2016.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> ( A gcd B ) = 1 )   &    |- ( ph -> N = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- D = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( C gcd D ) e. NN )

Theorem

2sqlem8 13718*

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

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> ( A gcd B ) = 1 )   &    |- ( ph -> N = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- D = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- E = ( C / ( C gcd D ) )   &    |- F = ( D / ( C gcd D ) )   =>    |- ( ph -> M e. S )

Theorem

2sqlem9 13719*

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

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. Y )   =>    |- ( ph -> M e. S )

Theorem

2sqlem10 13720*

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.)

|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   =>    |- ( ( A e. Y /\ B e. NN /\ B || A ) -> B e. S )

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 13721

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 13812 (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 5850, 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 5526 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	Expression	Syntax?	Example(s)
ap	apart	df-ap 8494		Yes	apadd1 8520, apne 8535
g	with "is a set" condition			No	1stvalg 6119, brtposg 6231, setsmsbasg 13238
seq3, sum3	recursive sequence	df-seqfrec 10395		Yes	seq3-1 10409, fsum3 11343

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 11343.

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

|- ph   =>    |- ph

11.1.2  Definitional examples

Theorem

ex-or 13722

Example for ax-io 704. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 3 \/ 4 = 4 )

Theorem

ex-an 13723

Example for ax-ia1 105. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 2 /\ 3 = 3 )

Theorem

1kp2ke3k 13724

Example for df-dec 9337, 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 ( 2 + 1 ) = 3, 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 9337 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.)

|- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Theorem

ex-fl 13725

Example for df-fl 10219. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

|- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Theorem

ex-ceil 13726

Example for df-ceil 10220. (Contributed by AV, 4-Sep-2021.)

|- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )

Theorem

ex-exp 13727

Example for df-exp 10469. (Contributed by AV, 4-Sep-2021.)

|- ( ( 5 ^ 2 ) = ; 2 5 /\ ( -u 3 ^ -u 2 ) = ( 1 / 9 ) )

Theorem

ex-fac 13728

Example for df-fac 10653. (Contributed by AV, 4-Sep-2021.)

|- ( ! ` 5 ) = ;; 1 2 0

Theorem

ex-bc 13729

Example for df-bc 10675. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 13730

Example for df-dvds 11743: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 13731

Example for df-gcd 11891. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 12  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

12.1  Mathboxes for user contributions

12.1.1  Mathbox guidelines

Theorem

mathbox 13732

(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.)

|- ph   =>    |- ph

12.2  Mathbox for BJ

12.2.1  Propositional calculus

Theorem

bj-nnsn 13733

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

|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )

Theorem

bj-nnor 13734

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

|- ( -. -. ( ph \/ ps ) <-> ( -. ph -> -. -. ps ) )

Theorem

bj-nnim 13735

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

|- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )

Theorem

bj-nnan 13736

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

|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )

Theorem

bj-nnclavius 13737

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

|- ( ( -. ph -> ph ) -> -. -. ph )

Theorem

bj-imnimnn 13738

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 13737 as its last step. (Contributed by BJ, 27-Oct-2024.)

|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps

12.2.1.1  Stable formulas

Some of the following theorems, like bj-sttru 13740 or bj-stfal 13742 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 13739

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

|- ( ph -> STAB ph )

Theorem

bj-sttru 13740

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

|- STAB T.

Theorem

bj-fast 13741

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

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 13742

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

|- STAB F.

Theorem

bj-nnst 13743

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 13990 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 105, ax-ia2 106, ax-ia3 107 are via sylibr 133, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

|- -. -. STAB ph

Theorem

bj-nnbist 13744

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if ph is a classical tautology, then -. -. ph is an intuitionistic tautology. Therefore, if ph is a classical tautology, then ph is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 13757). (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ph -> (STAB ph <-> ph ) )

Theorem

bj-stst 13745

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 ph <-> STAB ph )

Theorem

bj-stim 13746

A conjunction with a stable consequent is stable. See stabnot 828 for negation , bj-stan 13747 for conjunction , and bj-stal 13749 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

|- (STAB ps -> STAB ( ph -> ps ) )

Theorem

bj-stan 13747

The conjunction of two stable formulas is stable. See bj-stim 13746 for implication, stabnot 828 for negation, and bj-stal 13749 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )

Theorem

bj-stand 13748

The conjunction of two stable formulas is stable. Deduction form of bj-stan 13747. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 13747 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

|- ( ph -> STAB ps )   &    |- ( ph -> STAB ch )   =>    |- ( ph -> STAB ( ps /\ ch ) )

Theorem

bj-stal 13749

The universal quantification of a stable formula is stable. See bj-stim 13746 for implication, stabnot 828 for negation, and bj-stan 13747 for conjunction. (Contributed by BJ, 24-Nov-2023.)

|- ( A. xSTAB ph -> STAB A. x ph )

Theorem

bj-pm2.18st 13750

Clavius law for stable formulas. See pm2.18dc 850. (Contributed by BJ, 4-Dec-2023.)

|- (STAB ph -> ( ( -. ph -> ph ) -> ph ) )

Theorem

bj-con1st 13751

Contraposition when the antecedent is a negated stable proposition. See con1dc 851. (Contributed by BJ, 11-Nov-2024.)

|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )

12.2.1.2  Decidable formulas

Theorem

bj-trdc 13752

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

|- ( ph -> DECID ph )

Theorem

bj-dctru 13753

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

|- DECID T.

Theorem

bj-fadc 13754

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

|- ( -. ph -> DECID ph )

Theorem

bj-dcfal 13755

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

|- DECID F.

Theorem

bj-dcstab 13756

A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

|- (DECID ph -> STAB ph )

Theorem

bj-nnbidc 13757

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 13744. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ph -> (DECID ph <-> ph ) )

Theorem

bj-nndcALT 13758

Alternate proof of nndc 846. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

bj-dcdc 13759

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

|- (DECID DECID ph <-> DECID ph )

Theorem

bj-stdc 13760

Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)

|- (STAB DECID ph <-> DECID ph )

Theorem

bj-dcst 13761

Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)

|- (DECID STAB ph <-> STAB ph )

12.2.2  Predicate calculus

Theorem

bj-ex 13762*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1591 and 19.9ht 1634 or 19.23ht 1490). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 13763

Closed form of hbal 1470 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

|- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) )

Theorem

bj-nfalt 13764

Closed form of nfal 1569 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)

|- ( A. x F/ y ph -> F/ y A. x ph )

Theorem

spimd 13765

Deduction form of spim 1731. (Contributed by BJ, 17-Oct-2019.)

|- ( ph -> F/ x ch )   &    |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> ch ) )

Theorem

2spim 13766*

Double substitution, as in spim 1731. (Contributed by BJ, 17-Oct-2019.)

|- F/ x ch   &    |- F/ z ch   &    |- ( ( x = y /\ z = t ) -> ( ps -> ch ) )   =>    |- ( A. z A. x ps -> ch )

Theorem

ch2var 13767*

Implicit substitution of y for x and t for z into a theorem. (Contributed by BJ, 17-Oct-2019.)

|- F/ x ps   &    |- F/ z ps   &    |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps

Theorem

ch2varv 13768*

Version of ch2var 13767 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

|- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps

Theorem

bj-exlimmp 13769

Lemma for bj-vtoclgf 13776. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

|- F/ x ps   &    |- ( ch -> ph )   =>    |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )

Theorem

bj-exlimmpi 13770

Lemma for bj-vtoclgf 13776. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

|- F/ x ps   &    |- ( ch -> ph )   &    |- ( ch -> ( ph -> ps ) )   =>    |- ( E. x ch -> ps )

Theorem

bj-sbimedh 13771

A strengthening of sbiedh 1780 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps -> ch ) )

Theorem

bj-sbimeh 13772

A strengthening of sbieh 1783 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )

Theorem

bj-sbime 13773

A strengthening of sbie 1784 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )

12.2.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 13774

Shorter proof of el2oss1o 6420 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. 2o -> A C_ 1o )

12.2.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 13775

Weakening two hypotheses of vtoclgf 2788. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )

Theorem

bj-vtoclgf 13776

Weakening two hypotheses of vtoclgf 2788. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ps )

Theorem

elabgf0 13777

Lemma for elabgf 2872. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( A e. { x | ph } <-> ph ) )

Theorem

elabgft1 13778

One implication of elabgf 2872, in closed form. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )

Theorem

elabgf1 13779

One implication of elabgf 2872. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elabgf2 13780

One implication of elabgf 2872. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Theorem

elabf1 13781*

One implication of elabf 2873. (Contributed by BJ, 21-Nov-2019.)

|- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elabf2 13782*

One implication of elabf 2873. (Contributed by BJ, 21-Nov-2019.)

|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )

Theorem

elab1 13783*

One implication of elab 2874. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elab2a 13784*

One implication of elab 2874. (Contributed by BJ, 21-Nov-2019.)

|- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )

Theorem

elabg2 13785*

One implication of elabg 2876. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> A e. { x | ph } ) )

Theorem

bj-rspgt 13786

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2831 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x e. B ph -> ( A e. B -> ps ) ) )

Theorem

bj-rspg 13787

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2831 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )

Theorem

cbvrald 13788*

Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Theorem

bj-intabssel 13789

Version of intss1 3844 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Theorem

bj-intabssel1 13790

Version of intss1 3844 using a class abstraction and implicit substitution. Closed form of intmin3 3856. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )

Theorem

bj-elssuniab 13791

Version of elssuni 3822 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Theorem

bj-sseq 13792

If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)

|- ( ph -> ( ps <-> A C_ B ) )   &    |- ( ph -> ( ch <-> B C_ A ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

12.2.5  Decidability of classes

The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that " A is decidable in B " if A. x e. BDECID x e. A (see df-dcin 13794).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 13841).

Syntax

wdcin 13793

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 13794*

Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)

|- ( A DECID_in B <-> A. x e. B DECID x e. A )

Theorem

decidi 13795

Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

|- ( A DECID_in B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )

Theorem

decidr 13796*

Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )   =>    |- ( ph -> A DECID_in B )

Theorem

decidin 13797

If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> A C_ B )   &    |- ( ph -> A DECID_in B )   &    |- ( ph -> B DECID_in C )   =>    |- ( ph -> A DECID_in C )

Theorem

uzdcinzz 13798

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9562. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

|- ( M e. ZZ -> ( ZZ>= ` M ) DECID_in ZZ )

Theorem

sumdc2 13799*

Alternate proof of sumdc 11314, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11314). (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. x e. ( ZZ>= ` M )DECID x e. A )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> DECID N e. A )

12.2.6  Disjoint union

Theorem

djucllem 13800*

Lemma for djulcl 7026 and djurcl 7027. (Contributed by BJ, 4-Jul-2022.)

|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( A e. B -> ( ( F |` B ) ` A ) e. ( { X } X. B ) )