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Theorem List for Intuitionistic Logic Explorer - 16701-16800 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

trlsegvdeglem1 16701

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

$/\$

Theorem

trlsegvdeglem2 16702

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

Theorem

trlsegvdeglem3 16703

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

Theorem

trlsegvdeglem4 16704

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

..^

Theorem

trlsegvdeglem5 16705

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

Theorem

trlsegvdeglem6 16706

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

Theorem

trlsegvdeglem7 16707

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

Theorem

trlsegvdegfi 16708

The effect on vertex degree of adding one edge to a trail. In the following, a subgraph induced by a segment of a trail is called a "subtrail": For any subtrail

of a trail

in a pseudograph

which is composed of subtrails

and

, where

consists of a single edge, the vertex degree of any vertex

within

is the sum of the vertex degree of

within

and the vertex degree of

within

. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UPGraph

VtxDeg

Theorem

eupth2lem3lem1fi 16709

Lemma for eupth2lem3fi 16717. (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UPGraph

VtxDeg

Theorem

eupth2lem3lem2fi 16710

Lemma for eupth2lem3fi 16717. (Contributed by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UPGraph

VtxDeg

Theorem

eupth2lem3lem3fi 16711*

Lemma for eupth2lem3fi 16717. If a loop

is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UPGraph

VtxDeg

if-

$/\$

VtxDeg

Theorem

eupth2lem3lem6fi 16712*

If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than

and

: if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UPGraph

VtxDeg

$/\$

VtxDeg

Theorem

eupth2lem3lem5 16713

Lemma for eupth2fi 16720. (Contributed by AV, 25-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

VtxDeg

Theorem

eupth2lem3lem4fi 16714*

Lemma for eupth2lem3fi 16717. If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UMGraph

VtxDeg

if-

$/\$

$\/$

VtxDeg

Theorem

eupth2lem3lem7fi 16715*

Lemma for eupth2lem3fi 16717: Combining trlsegvdegfi 16708, eupth2lem3lem3fi 16711, eupth2lem3lem4fi 16714 and eupth2lem3lem6fi 16712. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)

Vtx

iEdg

..^

♯

Trails

Vtx

iEdg

..^

iEdg

UMGraph

VtxDeg

Theorem

eupthvdres 16716

The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

iEdg

EulerPaths

..^

♯

VtxDeg

Theorem

eupth2lem3fi 16717*

Lemma for eupth2fi 16720. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

iEdg

UMGraph

EulerPaths

..^

♯

VtxDeg

Theorem

eupth2lembfi 16718*

Lemma for eupth2fi 16720 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

iEdg

UMGraph

EulerPaths

VtxDeg

..^

Theorem

eupth2lemsfi 16719*

Lemma for eupth2fi 16720 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

iEdg

UMGraph

EulerPaths

$/\$

♯

VtxDeg

..^

♯

VtxDeg

..^

Theorem

eupth2fi 16720*

The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

iEdg

UMGraph

EulerPaths

VtxDeg

♯

Theorem

eulerpathprum 16721*

A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

UMGraph $/\$

EulerPaths

$/\$

♯

VtxDeg

Theorem

eulerpathum 16722*

A multigraph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Vtx

UMGraph $/\$

EulerPaths

$/\$

♯

VtxDeg

12.4.2 The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: , see konigsbergumgr 16728. konigsberg 16734 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theorem

konigsbergvtx 16723

The set of vertices of the Königsberg graph

. (Contributed by AV, 28-Feb-2021.)

Vtx

Theorem

konigsbergiedg 16724

The indexed edges of the Königsberg graph

. (Contributed by AV, 28-Feb-2021.)

iEdg

Theorem

konigsbergiedgwen 16725*

The indexed edges of the Königsberg graph

is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

Word

Theorem

konigsbergssiedgwpren 16726*

Each subset of the indexed edges of the Königsberg graph

is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

Word

$/\$

Word

$/\$

Word

Theorem

konigsbergssiedgwen 16727*

Each subset of the indexed edges of the Königsberg graph

is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

Word

$/\$

Word

$/\$

Word

$\/$

Theorem

konigsbergumgr 16728

The Königsberg graph

is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)

UMGraph

Theorem

konigsberglem1 16729

Lemma 1 for konigsberg 16734: Vertex

has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

VtxDeg

Theorem

konigsberglem2 16730

Lemma 2 for konigsberg 16734: Vertex

has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

VtxDeg

Theorem

konigsberglem3 16731

Lemma 3 for konigsberg 16734: Vertex

has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

VtxDeg

Theorem

konigsberglem4 16732*

Lemma 4 for konigsberg 16734: Vertices

are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

VtxDeg

Theorem

konigsberglem5 16733*

Lemma 5 for konigsberg 16734: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

♯

VtxDeg

Theorem

konigsberg 16734

The Königsberg Bridge problem. If

is the Königsberg graph, i.e. a graph on four vertices

, with edges

, then vertices

each have degree three, and

has degree five, so there are four vertices of odd degree and thus by eulerpathum 16722 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)

EulerPaths

PART 13 GUIDES AND MISCELLANEA

13.1 Guides (conventions, explanations, and examples)

13.1.1 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 16735

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 16838 (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 6084, 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 5727 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 8910	#	Yes	apadd1 8936, apne 8951
g	with "is a set" condition			No	1stvalg 6376, brtposg 6525, setsmsbasg 15580
m	inhabited (from "member")			No	r19.2m 3614, negm 10015, ctm 7449, basmex 13412
pap	apartness predicate	df-pap 7608	Ap	Yes	papeq1 7609, papirr 7611
seq3, sum3	recursive sequence	df-seqfrec 10885		Yes	seq3-1 10899, fsum3 12154
tap	tight apartness	df-tap 7615	TAp	Yes	df-tap 7615

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7615.

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

13.1.2 Definitional examples

Theorem

ex-or 16736

Example for ax-io 721. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

$\/$

Theorem

ex-an 16737

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

$/\$

Theorem

1kp2ke3k 16738

Example for df-dec 9778, 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 9778 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 16739

Example for df-fl 10705. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

$/\$

Theorem

ex-ceil 16740

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

⌈

$/\$

⌈

Theorem

ex-exp 16741

Example for df-exp 10976. (Contributed by AV, 4-Sep-2021.)

;

$/\$

Theorem

ex-fac 16742

Example for df-fac 11164. (Contributed by AV, 4-Sep-2021.)

;;

Theorem

ex-bc 16743

Example for df-bc 11186. (Contributed by AV, 4-Sep-2021.)

;

Theorem

ex-dvds 16744

Example for df-dvds 12555: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

Theorem

ex-gcd 16745

Example for df-gcd 12731. (Contributed by AV, 5-Sep-2021.)

PART 14 SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1 Mathboxes for user contributions

14.1.1 Mathbox guidelines

Theorem

mathbox 16746

(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.)

14.2 Mathbox for Matthew House

Theorem

depindlem1 16747*

Lemma for depind 16750. (Contributed by Matthew House, 14-Apr-2026.)

$/\$

Theorem

depindlem2 16748*

Lemma for depind 16750. (Contributed by Matthew House, 14-Apr-2026.)

Theorem

depindlem3 16749*

Lemma for depind 16750. (Contributed by Matthew House, 14-Apr-2026.)

$/\$

Theorem

depind 16750*

Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)

$/\$

Theorem

lealltlt1 16751*

Alternative definition for

on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

$/\$

Theorem

lealltlt2 16752*

Alternative definition for

on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

$/\$

Theorem

dichmul0orlem1 16753