P. 162 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

iswlkg 16101*

Generalization of iswlk 16095: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. W -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )

Theorem

wlkf 16102

The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)

|- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> F e. Word dom I )

Theorem

wlkfg 16103

The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)

|- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> F e. Word dom I )

Theorem

wlkcl 16104

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

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

Theorem

wlkclg 16105

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

|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` F ) e. NN0 )

Theorem

wlkp 16106

The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)

|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )

Theorem

wlkpg 16107

The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> P : ( 0 ... (♯ ` F ) ) --> V )

Theorem

wlkpwrdg 16108

The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> P e. Word V )

Theorem

wlklenvp1 16109

The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)

|- ( F (Walks ` G ) P -> (♯ ` P ) = ( (♯ ` F ) + 1 ) )

Theorem

wlklenvp1g 16110

The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)

|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` P ) = ( (♯ ` F ) + 1 ) )

Theorem

wlkm 16111*

The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.)

|- ( F (Walks ` G ) P -> E. x x e. P )

Theorem

wlkvtxm 16112*

A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)

|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> E. x x e. V )

Theorem

wlklenvm1 16113

The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)

|- ( F (Walks ` G ) P -> (♯ ` F ) = ( (♯ ` P ) - 1 ) )

Theorem

wlklenvm1g 16114

The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)

|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` F ) = ( (♯ ` P ) - 1 ) )

Theorem

ifpsnprss 16115

Lemma for wlkvtxeledgg 16116: Two adjacent (not necessarily different) vertices A and B in a walk are incident with an edge E. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)

|- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )

Theorem

wlkvtxeledgg 16116*

Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

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

Theorem

wlkvtxiedg 16117*

The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

|- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Theorem

wlkvtxiedgg 16118*

The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

|- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Theorem

relwlk 16119

The set (Walks ` G ) of all walks on G is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)

|- Rel (Walks ` G )

Theorem

wlkop 16120

A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)

|- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )

Theorem

wlkelvv 16121

A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)

|- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )

Theorem

wlkcprim 16122

A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)

|- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )

Theorem

wlk2f 16123*

If there is a walk W there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

|- ( W e. (Walks ` G ) -> E. f E. p f (Walks ` G ) p )

Theorem

wlkcompim 16124*

Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- F = ( 1st ` W )   &    |- P = ( 2nd ` W )   =>    |- ( W e. (Walks ` G ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )

Theorem

wlkelwrd 16125

The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- F = ( 1st ` W )   &    |- P = ( 2nd ` W )   =>    |- ( W e. (Walks ` G ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) )

Theorem

wlkeq 16126*

Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)

|- ( ( A e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )

Theorem

edginwlkd 16127

The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) (Revised by Jim Kingdon, 2-Feb-2026.)

|- I = (iEdg ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> F e. Word dom I )   &    |- ( ph -> K e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> ( I ` ( F ` K ) ) e. E )

Theorem

upgredginwlk 16128

The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.)

|- I = (iEdg ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ F e. Word dom I ) -> ( K e. ( 0..^ (♯ ` F ) ) -> ( I ` ( F ` K ) ) e. E ) )

Theorem

iedginwlk 16129

The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.)

|- I = (iEdg ` G )   =>    |- ( ( Fun I /\ F (Walks ` G ) P /\ X e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` X ) ) e. ran I )

Theorem

wlkl1loop 16130

A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)

|- ( ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )

Theorem

wlk1walkdom 16131*

A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.)

|- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> A. k e. ( 1..^ (♯ ` F ) ) 1o ~<_ ( ( I ` ( F ` ( k - 1 ) ) ) i^i ( I ` ( F ` k ) ) ) )

Theorem

upgriswlkdc 16132*

Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )

Theorem

upgrwlkedg 16133*

The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )

Theorem

upgrwlkcompim 16134*

Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- F = ( 1st ` W )   &    |- P = ( 2nd ` W )   =>    |- ( ( G e. UPGraph /\ W e. (Walks ` G ) ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )

Theorem

wlkvtxedg 16135*

The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)

|- E = (Edg ` G )   =>    |- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Theorem

upgrwlkvtxedg 16136*

The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)

|- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E )

Theorem

uspgr2wlkeq 16137*

Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)

|- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )

Theorem

uspgr2wlkeq2 16138

Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)

|- ( ( ( G e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )

Theorem

uspgr2wlkeqi 16139

Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.)

|- ( ( G e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A = B )

Theorem

umgrwlknloop 16140*

In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.)

|- ( ( G e. UMGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Theorem

wlkv0 16141

If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)

|- ( ( (Vtx ` G ) = (/) /\ W e. (Walks ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) )

Theorem

g0wlk0 16142

There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)

|- ( (Vtx ` G ) = (/) -> (Walks ` G ) = (/) )

Theorem

0wlk0 16143

There is no walk for the empty set, i.e. in a null graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)

|- (Walks ` (/) ) = (/)

Theorem

wlk0prc 16144

There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)

|- ( ( S e/ _V /\ (Vtx ` S ) = (Vtx ` G ) ) -> (Walks ` G ) = (/) )

Theorem

wlklenvclwlk 16145

The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)

|- ( W e. Word (Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> (♯ ` F ) = (♯ ` W ) ) )

Theorem

2wlklem 16146*

Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

|- ( A. k e. { 0 , 1 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )

Theorem

upgr2wlkdc 16147*

Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ F ~~ 2o ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )

Theorem

wlkreslem 16148

Lemma for wlkres 16149. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (Walks ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> (Vtx ` S ) = V )   =>    |- ( ph -> S e. _V )

Theorem

wlkres 16149

The restriction <. H , Q >. of a walk <. F , P >. to an initial segment of the walk (of length N) forms a walk 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, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

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

12.3.2  Trails

Syntax

ctrls 16150

Extend class notation with trails (within a graph).

class Trails

Definition

df-trls 16151*

Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

|- Trails = ( g e. _V |-> { <. f , p >. | ( f (Walks ` g ) p /\ Fun `' f ) } )

Theorem

reltrls 16152

The set (Trails ` G ) of all trails on G is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.)

|- Rel (Trails ` G )

Theorem

trlsfvalg 16153*

The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)

|- ( G e. V -> (Trails ` G ) = { <. f , p >. | ( f (Walks ` G ) p /\ Fun `' f ) } )

Theorem

trlsv 16154

The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)

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

Theorem

istrl 16155

Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)

|- ( F (Trails ` G ) P <-> ( F (Walks ` G ) P /\ Fun `' F ) )

Theorem

trliswlk 16156

A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.)

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

Theorem

trlf1 16157

The enumeration F of a trail <. F , P >. is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.)

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

Theorem

trlreslem 16158

Lemma for trlres 16159. (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 (Trails ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- H = ( F prefix N )   =>    |- ( ph -> H : ( 0..^ (♯ ` H ) ) -1-1-onto-> dom ( I |` ( F " ( 0..^ N ) ) ) )

Theorem

trlres 16159

The restriction <. H , Q >. of a trail <. F , P >. to an initial segment of the trail (of length N) forms a trail on the subgraph S consisting of the edges in the initial segment. (Contributed 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 (Trails ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- H = ( F prefix N )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- Q = ( P |` ( 0 ... N ) )   =>    |- ( ph -> H (Trails ` S ) Q )

12.3.3  Closed walks as words

12.3.3.1  Closed walks as words

Syntax

cclwwlk 16160

Extend class notation with closed walks (in an undirected graph) as word over the set of vertices.

class ClWWalks

Definition

df-clwwlk 16161*

Define the set of all closed walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined elsewhere. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

|- ClWWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )

Theorem

clwwlkg 16162*

The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( G e. W -> (ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } )

Theorem

isclwwlk 16163*

Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( W e. (ClWWalks ` G ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) )

Theorem

clwwlkbp 16164

Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)

|- V = (Vtx ` G )   =>    |- ( W e. (ClWWalks ` G ) -> ( G e. _V /\ W e. Word V /\ W =/= (/) ) )

Theorem

clwwlkgt0 16165

There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)

|- ( W e. (ClWWalks ` G ) -> 0 < (♯ ` W ) )

Theorem

clwwlksswrd 16166

Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.)

|- (ClWWalks ` G ) C_ Word (Vtx ` G )

Theorem

clwwlkex 16167

Existence of the set of closed walks (represented by words). (Contributed by Jim Kingdon, 21-Feb-2026.)

|- ( G e. V -> (ClWWalks ` G ) e. _V )

Theorem

clwwlk1loop 16168

A closed walk of length 1 is a loop. (Contributed by AV, 24-Apr-2021.)

|- ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = 1 ) -> { ( W ` 0 ) , ( W ` 0 ) } e. (Edg ` G ) )

Theorem

clwwlkccatlem 16169*

Lemma for clwwlkccat 16170: index j is shifted up by (♯ ` A ), and the case i = ( (♯ ` A ) - 1 ) is covered by the "bridge" { (lastS ` A ) , ( B ` 0 ) } = { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ). (Contributed by AV, 23-Apr-2022.)

|- ( ( ( ( A e. Word (Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` A ) , ( A ` 0 ) } e. (Edg ` G ) ) /\ ( ( B e. Word (Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0..^ ( (♯ ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` B ) , ( B ` 0 ) } e. (Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0..^ ( (♯ ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. (Edg ` G ) )

Theorem

clwwlkccat 16170

The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 23-Apr-2022.)

|- ( ( A e. (ClWWalks ` G ) /\ B e. (ClWWalks ` G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )

Theorem

umgrclwwlkge2 16171

A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.)

|- ( G e. UMGraph -> ( P e. (ClWWalks ` G ) -> 2 <_ (♯ ` P ) ) )

12.3.3.2  Closed walks of a fixed length as words

Syntax

cclwwlkn 16172

Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices.

class ClWWalksN

Definition

df-clwwlkn 16173*

Define the set of all closed walks of a fixed length n as words over the set of vertices in a graph g. If 0 < n, such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) . For n = 0, the set is empty, see clwwlkn0 16177. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)

|- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } )

Theorem

clwwlkng 16174*

The set of closed walks of a fixed length N as words over the set of vertices in a graph G. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)

|- ( ( N e. NN0 /\ G e. V ) -> ( N ClWWalksN G ) = { w e. (ClWWalks ` G ) | (♯ ` w ) = N } )

Theorem

isclwwlkng 16175

A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)

|- ( N e. NN0 -> ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) ) )

Theorem

isclwwlkni 16176

A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Jim Kingdon, 22-Feb-2026.)

|- ( W e. ( N ClWWalksN G ) -> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) )

Theorem

clwwlkn0 16177

There is no closed walk of length 0 (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)

|- ( 0 ClWWalksN G ) = (/)

Theorem

clwwlkclwwlkn 16178

A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

|- ( W e. ( N ClWWalksN G ) -> W e. (ClWWalks ` G ) )

Theorem

clwwlksclwwlkn 16179

The closed walks of a fixed length as words are closed walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 12-Apr-2021.)

|- ( N ClWWalksN G ) C_ (ClWWalks ` G )

Theorem

clwwlknlen 16180

The length of a word representing a closed walk of a fixed length is this fixed length. (Contributed by AV, 22-Mar-2022.)

|- ( W e. ( N ClWWalksN G ) -> (♯ ` W ) = N )

Theorem

clwwlknnn 16181

The length of a closed walk of a fixed length as word is a positive integer. (Contributed by AV, 22-Mar-2022.)

|- ( W e. ( N ClWWalksN G ) -> N e. NN )

Theorem

clwwlknwrd 16182

A closed walk of a fixed length as word is a word over the vertices. (Contributed by AV, 30-Apr-2021.)

|- V = (Vtx ` G )   =>    |- ( W e. ( N ClWWalksN G ) -> W e. Word V )

Theorem

clwwlknbp 16183

Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

|- V = (Vtx ` G )   =>    |- ( W e. ( N ClWWalksN G ) -> ( W e. Word V /\ (♯ ` W ) = N ) )

Theorem

isclwwlknx 16184*

Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( 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 ) ) )

Theorem

clwwlknp 16185*

Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( W e. ( N ClWWalksN G ) -> ( ( W e. Word V /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) )

Theorem

clwwlkn1 16186

A closed walk of length 1 represented as word is a word consisting of 1 symbol representing a vertex connected to itself by (at least) one edge, that is, a loop. (Contributed by AV, 24-Apr-2021.) (Revised by AV, 11-Feb-2022.)

|- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )

Theorem

loopclwwlkn1b 16187

The singleton word consisting of a vertex V represents a closed walk of length 1 iff there is a loop at vertex V. (Contributed by AV, 11-Feb-2022.)

|- ( V e. (Vtx ` G ) -> ( { V } e. (Edg ` G ) <-> <" V "> e. ( 1 ClWWalksN G ) ) )

Theorem

clwwlkn1loopb 16188*

A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.)

|- ( W e. ( 1 ClWWalksN G ) <-> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) )

Theorem

clwwlkn2 16189

A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)

|- ( W e. ( 2 ClWWalksN G ) <-> ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Theorem

clwwlkext2edg 16190

If a word concatenated with a vertex represents a closed walk (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( W ++ <" Z "> ) e. ( N ClWWalksN G ) ) -> ( { (lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) )

Theorem

clwwlknccat 16191

The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 24-Apr-2022.)

|- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )

Theorem

umgr2cwwk2dif 16192

If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)

|- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> ( W ` 1 ) =/= ( W ` 0 ) )

Theorem

umgr2cwwkdifex 16193*

If a word represents a closed walk of length at least 2 in a undirected simple graph, there must be a symbol different from the first symbol of the word. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)

|- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> E. i e. ( 0..^ N ) ( W ` i ) =/= ( W ` 0 ) )

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 16194

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 16284 (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 6009, 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 5664 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 7450, df-ap 8745		Yes	apadd1 8771, apne 8786
g	with "is a set" condition			No	1stvalg 6297, brtposg 6411, setsmsbasg 15174
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3578, negm 9827, ctm 7292, basmex 13113
seq3, sum3	recursive sequence	df-seqfrec 10687		Yes	seq3-1 10701, fsum3 11919
tap	tight apartness	df-tap 7452		Yes	df-tap 7452

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7452.

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

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 16195

Example for ax-io 714. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

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

Theorem

ex-an 16196

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

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

Theorem

1kp2ke3k 16197

Example for df-dec 9595, 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 9595 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 16198

Example for df-fl 10507. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 16199

Example for df-ceil 10508. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-exp 16200

Example for df-exp 10778. (Contributed by AV, 4-Sep-2021.)

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