P. 169 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

clwwlknonex2lem2 16801*

Lemma 2 for clwwlknonex2 16802: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ { X , Y } e. E ) -> A. i e. ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )

Theorem

clwwlknonex2 16802

Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )

Theorem

clwwlknonex2e 16803

Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk on vertex X. (Contributed by AV, 17-Apr-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X (ClWWalksNOn ` G ) N ) )

Theorem

clwwlknun 16804*

The set of closed walks of fixed length N in a simple graph G is the union of the closed walks of the fixed length N on each of the vertices of graph G. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

|- V = (Vtx ` G )   =>    |- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

12.4  Eulerian paths and the Konigsberg Bridge problem

12.4.1  Eulerian paths

According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."

Syntax

ceupth 16805

Extend class notation with Eulerian paths.

class EulerPaths

Definition

df-eupth 16806*

Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- EulerPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) } )

Theorem

releupth 16807

The set (EulerPaths ` G ) of all Eulerian paths on G is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- Rel (EulerPaths ` G )

Theorem

eupthsg 16808*

The Eulerian paths on the graph G. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( G e. V -> (EulerPaths ` G ) = { <. f , p >. | ( f (Trails ` G ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom I ) } )

Theorem

eupthv 16809

The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)

|- ( F (EulerPaths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )

Theorem

iseupth 16810

The property " <. F , P >. is an Eulerian path on the graph G". An Eulerian path is defined as bijection F from the edges to a set 0 ... ( N - 1 ) and a function P : ( 0 ... N ) --> V into the vertices such that for each 0 <_ k < N, F ( k ) is an edge from P ( k ) to P ( k + 1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P <-> ( F (Trails ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -onto-> dom I ) )

Theorem

iseupthf1o 16811

The property " <. F , P >. is an Eulerian path on the graph G". An Eulerian path is defined as bijection F from the edges to a set 0 ... ( N - 1 ) and a function P : ( 0 ... N ) --> V into the vertices such that for each 0 <_ k < N, F ( k ) is an edge from P ( k ) to P ( k + 1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P <-> ( F (Walks ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I ) )

Theorem

eupthi 16812

Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> ( F (Walks ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I ) )

Theorem

eupthf1o 16813

The F function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I )

Theorem

eupthfi 16814

Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> dom I e. Fin )

Theorem

eupthseg 16815

The N-th edge in an eulerian path is the edge having P ( N ) and P ( N + 1 ) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( ( F (EulerPaths ` G ) P /\ N e. ( 0..^ (♯ ` F ) ) ) -> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) )

Theorem

eupthcl 16816

An Eulerian path has length ♯ ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> (♯ ` F ) e. NN0 )

Theorem

eupthistrl 16817

An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> F (Trails ` G ) P )

Theorem

eupthiswlk 16818

An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)

|- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )

Theorem

eupthpf 16819

The P function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )

Theorem

eupthres 16820

The restriction <. H , Q >. of an Eulerian path <. F , P >. to an initial segment of the path (of length N) forms an Eulerian path on the subgraph S consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- H = ( F prefix N )   &    |- Q = ( P |` ( 0 ... N ) )   &    |- (Vtx ` S ) = V   =>    |- ( ph -> H (EulerPaths ` S ) Q )

Theorem

eupth2lem1 16821

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

|- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Theorem

eupth2lem2dc 16822

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

|- ( ph -> B e. X )   &    |- ( ph -> DECID A = B )   &    |- ( ph -> B =/= C )   &    |- ( ph -> B = U )   =>    |- ( ph -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Theorem

trlsegvdeglem1 16823

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   =>    |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )

Theorem

trlsegvdeglem2 16824

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> Fun (iEdg ` X ) )

Theorem

trlsegvdeglem3 16825

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> Fun (iEdg ` Y ) )

Theorem

trlsegvdeglem4 16826

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Theorem

trlsegvdeglem5 16827

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )

Theorem

trlsegvdeglem6 16828

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` X ) e. Fin )

Theorem

trlsegvdeglem7 16829

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

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` Y ) e. Fin )

Theorem

trlsegvdegfi 16830

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 Z of a trail <. F , P >. in a pseudograph G which is composed of subtrails X and Y, where Y consists of a single edge, the vertex degree of any vertex U within Z is the sum of the vertex degree of U within X and the vertex degree of U within Y. 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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Theorem

eupth2lem3lem1fi 16831

Lemma for eupth2lem3fi 16839. (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )

Theorem

eupth2lem3lem2fi 16832

Lemma for eupth2lem3fi 16839. (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. NN0 )

Theorem

eupth2lem3lem3fi 16833*

Lemma for eupth2lem3fi 16839. If a loop { ( P ` N ) , ( P ` ( N + 1 ) ) } 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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )   =>    |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem6fi 16834*

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 ( P ` 0 ) and ( P ` N ): 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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem5 16835

Lemma for eupth2fi 16842. (Contributed by AV, 25-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )

Theorem

eupth2lem3lem4fi 16836*

Lemma for eupth2lem3fi 16839. 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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )   &    |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )   =>    |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem7fi 16837*

Lemma for eupth2lem3fi 16839: Combining trlsegvdegfi 16830, eupth2lem3lem3fi 16833, eupth2lem3lem4fi 16836 and eupth2lem3lem6fi 16834. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ph -> ( -. 2 || ( (VtxDeg ` Z ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupthvdres 16838

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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. W )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- H = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >.   =>    |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Theorem

eupth2lem3fi 16839*

Lemma for eupth2fi 16842. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   &    |- H = <. V , ( I |` ( F " ( 0..^ N ) ) ) >.   &    |- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> ( N + 1 ) <_ (♯ ` F ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` H ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   =>    |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lembfi 16840*

Lemma for eupth2fi 16842 (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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )

Theorem

eupth2lemsfi 16841*

Lemma for eupth2fi 16842 (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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )

Theorem

eupth2fi 16842*

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.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

Theorem

eulerpathprum 16843*

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.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ F (EulerPaths ` G ) P /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

Theorem

eulerpathum 16844*

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.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ E. j j e. (EulerPaths ` G ) /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

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 G is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: G = <. ( 0 ... 3 ) , <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } "> >., see konigsbergumgr 16850. konigsberg 16856 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theorem

konigsbergvtx 16845

The set of vertices of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (Vtx ` G ) = ( 0 ... 3 )

Theorem

konigsbergiedg 16846

The indexed edges of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (iEdg ` G ) = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">

Theorem

konigsbergiedgwen 16847*

The indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- E e. Word { x e. ~P V | x ~~ 2o }

Theorem

konigsbergssiedgwpren 16848*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | x ~~ 2o } )

Theorem

konigsbergssiedgwen 16849*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )

Theorem

konigsbergumgr 16850

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

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- G e. UMGraph

Theorem

konigsberglem1 16851

Lemma 1 for konigsberg 16856: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 0 ) = 3

Theorem

konigsberglem2 16852

Lemma 2 for konigsberg 16856: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 1 ) = 3

Theorem

konigsberglem3 16853

Lemma 3 for konigsberg 16856: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 3 ) = 3

Theorem

konigsberglem4 16854*

Lemma 4 for konigsberg 16856: Vertices 0 , 1 , 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- { 0 , 1 , 3 } C_ { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) }

Theorem

konigsberglem5 16855*

Lemma 5 for konigsberg 16856: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- 2 < (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )

Theorem

konigsberg 16856

The Königsberg Bridge problem. If G is the Königsberg graph, i.e. a graph on four vertices 0 , 1 , 2 , 3, with edges { 0 , 1 } , { 0 , 2 } , { 0 , 3 } , { 1 , 2 } , { 1 , 2 } , { 2 , 3 } , { 2 , 3 }, then vertices 0 , 1 , 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpathum 16844 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.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (EulerPaths ` G ) = (/)

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 16857

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 16960 (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 8913	#	Yes	apadd1 8939, apne 8954
g	with "is a set" condition			No	1stvalg 6376, brtposg 6525, setsmsbasg 15632
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3614, negm 10025, ctm 7450, basmex 13463
pap	apartness predicate	df-pap 7609	Ap	Yes	papeq1 7610, papirr 7612
seq3, sum3	recursive sequence	df-seqfrec 10899		Yes	seq3-1 10913, fsum3 12172
tap	tight apartness	df-tap 7616	TAp	Yes	df-tap 7616

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7616.

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

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 16858

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

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

Theorem

ex-an 16859

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

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

Theorem

1kp2ke3k 16860

Example for df-dec 9783, 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 9783 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 16861

Example for df-fl 10716. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 16862

Example for df-ceil 10717. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-exp 16863

Example for df-exp 10990. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-fac 16864

Example for df-fac 11179. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-bc 16865

Example for df-bc 11201. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 16866

Example for df-dvds 12573: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 16867

Example for df-gcd 12749. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1  Mathboxes for user contributions

14.1.1  Mathbox guidelines

Theorem

mathbox 16868

(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

14.2  Mathbox for Matthew House

Theorem

depindlem1 16869*

Lemma for depind 16872. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> ( F : NN0 --> _V /\ ( F ` 0 ) = A /\ A. n e. NN0 ( F ` ( n + 1 ) ) = ( ( H ` n ) ` ( F ` n ) ) ) )

Theorem

depindlem2 16870*

Lemma for depind 16872. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> F e. X_ n e. NN0 ( P ` n ) )

Theorem

depindlem3 16871*

Lemma for depind 16872. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> A. f e. X_ n e. NN0 ( P ` n ) ( ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) -> f = F ) )

Theorem

depind 16872*

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

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   =>    |- ( ph -> E! f e. X_ n e. NN0 ( P ` n ) ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) )

Theorem

lealltlt1 16873*

Alternative definition for <_ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( x < A -> x < B ) ) )

Theorem

lealltlt2 16874*

Alternative definition for <_ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( B < x -> A < x ) ) )

Theorem

dichmul0orlem1 16875

Lemma for dichmul0or 16882. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) = 0 )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> A = 0 )

Theorem

dichmul0orlem2 16876

Lemma for dichmul0or 16882. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) = 0 )   &    |- ( ph -> ( abs ` A ) <_ ( abs ` B ) )   =>    |- ( ph -> A = 0 )

Theorem

dichmul0orlem3 16877*

Lemma for dichmul0or 16882. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ph -> A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) = 0 )   =>    |- ( ph -> ( A = 0 \/ B = 0 ) )

Theorem

dichmul0orlem4 16878

Lemma for dichmul0or 16882. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ph -> A e. RR )   =>    |- ( ph -> ( ( ( abs ` A ) + A ) x. ( ( abs ` A ) - A ) ) = 0 )

Theorem

dichmul0orlem5 16879

Lemma for dichmul0or 16882. (Contributed by Matthew House, 29-Jun-2026.)

|- ( ph -> A e. RR )   &    |- ( ph -> ( ( abs ` A ) + A ) = 0 )   =>    |- ( ph -> A <_ 0 )

Theorem

dichmul0orlem6 16880

Lemma for dichmul0or 16882. (Contributed by Matthew House, 28-Jun-2026.)

|- ( ph -> A e. RR )   &    |- ( ph -> ( ( abs ` A ) - A ) = 0 )   =>    |- ( ph -> 0 <_ A )

Theorem

dichmul0orlem7 16881*

Lemma for dichmul0or 16882. (Contributed by Matthew House, 28-Jun-2026.)

|- ( ph -> A. x e. CC A. y e. CC ( ( x x. y ) = 0 -> ( x = 0 \/ y = 0 ) ) )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> ( A <_ 0 \/ 0 <_ A ) )

Theorem

dichmul0or 16882*

Real number dichotomy is equivalent to the zero product principle for complex numbers: if a product is zero, one of its factors must be zero. (Contributed by Matthew House, 29-Jun-2026.)

|- ( A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) <-> A. z e. CC A. w e. CC ( ( z x. w ) = 0 -> ( z = 0 \/ w = 0 ) ) )

14.3  Mathbox for BJ

14.3.1  Propositional calculus

Theorem

bj-nnsn 16883

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 16884

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

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

Theorem

bj-nnim 16885

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 16886

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 16887

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

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

Theorem

bj-imnimnn 16888

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 16887 as its last step. (Contributed by BJ, 27-Oct-2024.)

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

14.3.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16890 or bj-stfal 16892 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 16889

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

|- ( ph -> STAB ph )

Theorem

bj-sttru 16890

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

|- STAB T.

Theorem

bj-fast 16891

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

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 16892

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

|- STAB F.

Theorem

bj-nnst 16893

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 17138 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 16894

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 16907). (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stst 16895

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 16896

A conjunction with a stable consequent is stable. See stabnot 845 for negation , bj-stan 16897 for conjunction , and bj-stal 16899 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stan 16897

The conjunction of two stable formulas is stable. See bj-stim 16896 for implication, stabnot 845 for negation, and bj-stal 16899 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stand 16898

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16897. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16897 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 16899

The universal quantification of a stable formula is stable. See bj-stim 16896 for implication, stabnot 845 for negation, and bj-stan 16897 for conjunction. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-pm2.18st 16900

Clavius law for stable formulas. See pm2.18dc 867. (Contributed by BJ, 4-Dec-2023.)

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