P. 153 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

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

Theorem

2sqlem1 15201*

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 15202*

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 15203

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 15204

Lemma for 2sqlem5 15206. (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 15205

Lemma for 2sqlem5 15206. (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 15206

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 15207*

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 15208*

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 15209*

Lemma for 2sqlem8 15210. (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 15210*

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 15211*

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 15212*

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 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 15213

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 15304 (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 5917, 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 5586 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-pap 7308, df-ap 8601		Yes	apadd1 8627, apne 8642
g	with "is a set" condition			No	1stvalg 6195, brtposg 6307, setsmsbasg 14647
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3533, negm 9680, ctm 7168, basmex 12677
seq3, sum3	recursive sequence	df-seqfrec 10519		Yes	seq3-1 10533, fsum3 11530
tap	tight apartness	df-tap 7310		Yes	df-tap 7310

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7310.

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

|- ph   =>    |- ph

12.1.2  Definitional examples

Theorem

ex-or 15214

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

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

Theorem

ex-an 15215

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

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

Theorem

1kp2ke3k 15216

Example for df-dec 9449, 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 9449 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 15217

Example for df-fl 10339. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 15218

Example for df-ceil 10340. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-exp 15219

Example for df-exp 10610. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-fac 15220

Example for df-fac 10797. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-bc 15221

Example for df-bc 10819. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 15222

Example for df-dvds 11931: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 15223

Example for df-gcd 12080. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

13.1  Mathboxes for user contributions

13.1.1  Mathbox guidelines

Theorem

mathbox 15224

(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

13.2  Mathbox for BJ

13.2.1  Propositional calculus

Theorem

bj-nnsn 15225

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 15226

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

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

Theorem

bj-nnim 15227

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 15228

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 15229

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

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

Theorem

bj-imnimnn 15230

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 15229 as its last step. (Contributed by BJ, 27-Oct-2024.)

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

13.2.1.1  Stable formulas

Some of the following theorems, like bj-sttru 15232 or bj-stfal 15234 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 15231

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

|- ( ph -> STAB ph )

Theorem

bj-sttru 15232

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

|- STAB T.

Theorem

bj-fast 15233

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

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 15234

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

|- STAB F.

Theorem

bj-nnst 15235

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 15482 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 ph

Theorem

bj-nnbist 15236

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 15249). (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stst 15237

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 15238

A conjunction with a stable consequent is stable. See stabnot 834 for negation , bj-stan 15239 for conjunction , and bj-stal 15241 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stan 15239

The conjunction of two stable formulas is stable. See bj-stim 15238 for implication, stabnot 834 for negation, and bj-stal 15241 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stand 15240

The conjunction of two stable formulas is stable. Deduction form of bj-stan 15239. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 15239 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 15241

The universal quantification of a stable formula is stable. See bj-stim 15238 for implication, stabnot 834 for negation, and bj-stan 15239 for conjunction. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-pm2.18st 15242

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

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

Theorem

bj-con1st 15243

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

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

13.2.1.2  Decidable formulas

Theorem

bj-trdc 15244

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

|- ( ph -> DECID ph )

Theorem

bj-dctru 15245

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

|- DECID T.

Theorem

bj-fadc 15246

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

|- ( -. ph -> DECID ph )

Theorem

bj-dcfal 15247

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

|- DECID F.

Theorem

bj-dcstab 15248

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

|- (DECID ph -> STAB ph )

Theorem

bj-nnbidc 15249

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 15236. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-nndcALT 15250

Alternate proof of nndc 852. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

bj-dcdc 15251

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 15252

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 15253

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 )

13.2.2  Predicate calculus

Theorem

bj-ex 15254*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1609 and 19.9ht 1652 or 19.23ht 1508). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 15255

Closed form of hbal 1488 (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 15256

Closed form of nfal 1587 (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 15257

Deduction form of spim 1749. (Contributed by BJ, 17-Oct-2019.)

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

Theorem

2spim 15258*

Double substitution, as in spim 1749. (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 15259*

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 15260*

Version of ch2var 15259 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 15261

Lemma for bj-vtoclgf 15268. (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 15262

Lemma for bj-vtoclgf 15268. (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 15263

A strengthening of sbiedh 1798 (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 15264

A strengthening of sbieh 1801 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

Theorem

bj-sbime 15265

A strengthening of sbie 1802 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

13.2.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 15266

Shorter proof of el2oss1o 6496 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

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

13.2.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 15267

Weakening two hypotheses of vtoclgf 2818. (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 15268

Weakening two hypotheses of vtoclgf 2818. (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 15269

Lemma for elabgf 2902. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgft1 15270

One implication of elabgf 2902, 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 15271

One implication of elabgf 2902. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgf2 15272

One implication of elabgf 2902. (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 15273*

One implication of elabf 2903. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabf2 15274*

One implication of elabf 2903. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab1 15275*

One implication of elab 2904. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab2a 15276*

One implication of elab 2904. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabg2 15277*

One implication of elabg 2906. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

bj-rspgt 15278

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2861 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 15279

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2861 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 15280*

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 15281

Version of intss1 3885 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 15282

Version of intss1 3885 using a class abstraction and implicit substitution. Closed form of intmin3 3897. (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 15283

Version of elssuni 3863 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 15284

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 ) )

13.2.5  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 15286).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 15333).

Syntax

wdcin 15285

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 15286*

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 15287

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 15288*

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 15289

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 15290

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9675. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

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

Theorem

sumdc2 15291*

Alternate proof of sumdc 11501, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11501). (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 )

13.2.6  Disjoint union

Theorem

djucllem 15292*

Lemma for djulcl 7110 and djurcl 7111. (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 ) )

Theorem

djulclALT 15293

Shortening of djulcl 7110 using djucllem 15292. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Theorem

djurclALT 15294

Shortening of djurcl 7111 using djucllem 15292. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

13.2.7  Miscellaneous

Theorem

funmptd 15295

The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5291, then prove funmptd 15295 from it, and then prove funmpt 5292 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ph -> Fun F )

Theorem

fnmptd 15296*

The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)

|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ph -> F Fn A )

Theorem

if0ab 15297*

Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3609, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 15298 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

|- if ( ph , A , (/) ) = { x e. A | ph }

Theorem

fmelpw1o 15298

With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 852, which translate to 1o and (/) respectively by iftrue 3562 and iffalse 3565, giving pwtrufal 15488).

As proved in if0ab 15297, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.)

|- if ( ph , 1o , (/) ) e. ~P 1o

Theorem

bj-charfun 15299*

Properties of the characteristic function on the class X of the class A. (Contributed by BJ, 15-Aug-2024.)

|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   =>    |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Theorem

bj-charfundc 15300*

Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)

|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )