P. 271 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1wlkd 27001	In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )   &    |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F (Walks ` G ) P )
Theorem	1trld 27002	In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )   &    |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F (Trails ` G ) P )
Theorem	1pthd 27003	In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )   &    |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F (Paths ` G ) P )
Theorem	1pthond 27004	In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )   &    |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F ( X (PathsOn ` G ) Y ) P )
Theorem	upgr1wlkdlem1 27005	Lemma 1 for upgr1wlkd 27007. (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   =>    |- ( ( ph /\ X = Y ) -> ( (iEdg ` G ) ` J ) = { X } )
Theorem	upgr1wlkdlem2 27006	Lemma 2 for upgr1wlkd 27007. (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   =>    |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( (iEdg ` G ) ` J ) )
Theorem	upgr1wlkd 27007	In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> F (Walks ` G ) P )
Theorem	upgr1trld 27008	In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> F (Trails ` G ) P )
Theorem	upgr1pthd 27009	In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> F (Paths ` G ) P )
Theorem	upgr1pthond 27010	In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
|- P = <" X Y ">   &    |- F = <" J ">   &    |- ( ph -> X e. (Vtx ` G ) )   &    |- ( ph -> Y e. (Vtx ` G ) )   &    |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> F ( X (PathsOn ` G ) Y ) P )
Theorem	lppthon 27011	A loop (which is an edge at index J) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> <" J "> ( A (PathsOn ` G ) A ) <" A A "> )
Theorem	lp1cycl 27012	A loop (which is an edge at index J) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> <" J "> (Cycles ` G ) <" A A "> )
Theorem	1pthon2v 27013*	For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ E. e e. E { A , B } C_ e ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )
Theorem	1pthon2ve 27014*	For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UHGraph /\ ( A e. V /\ B e. V ) /\ { A , B } e. E ) -> E. f E. p f ( A (PathsOn ` G ) B ) p )
Theorem	wlk2v2elem1 27015	Lemma 1 for wlk2v2e 27017: F is a length 2 word of over { 0 }, the domain of the singleton word I. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
|- I = <" { X , Y } ">   &    |- F = <" 0 0 ">   =>    |- F e. Word dom I
Theorem	wlk2v2elem2 27016*	Lemma 2 for wlk2v2e 27017: The values of I after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
|- I = <" { X , Y } ">   &    |- F = <" 0 0 ">   &    |- X e. _V   &    |- Y e. _V   &    |- P = <" X Y X ">   =>    |- A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) }
Theorem	wlk2v2e 27017	In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that G is a simple graph (without loops) only if X =/= Y. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
|- I = <" { X , Y } ">   &    |- F = <" 0 0 ">   &    |- X e. _V   &    |- Y e. _V   &    |- P = <" X Y X ">   &    |- G = <. { X , Y } , I >.   =>    |- F (Walks ` G ) P
Theorem	ntrl2v2e 27018	A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see wlk2v2e 27017, but not a trail. Notice that G is a simple graph (without loops) only if X =/= Y. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- I = <" { X , Y } ">   &    |- F = <" 0 0 ">   &    |- X e. _V   &    |- Y e. _V   &    |- P = <" X Y X ">   &    |- G = <. { X , Y } , I >.   =>    |- -. F (Trails ` G ) P
Theorem	3wlkdlem1 27019	Lemma 1 for 3wlkd 27030. (Contributed by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   =>    |- ( # ` P ) = ( ( # ` F ) + 1 )
Theorem	3wlkdlem2 27020	Lemma 2 for 3wlkd 27030. (Contributed by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   =>    |- ( 0..^ ( # ` F ) ) = { 0 , 1 , 2 }
Theorem	3wlkdlem3 27021	Lemma 3 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   =>    |- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
Theorem	3wlkdlem4 27022*	Lemma 4 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   =>    |- ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )
Theorem	3wlkdlem5 27023*	Lemma 5 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   =>    |- ( ph -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
Theorem	3pthdlem1 27024*	Lemma 1 for 3pthd 27034. (Contributed by AV, 9-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   =>    |- ( ph -> A. k e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) )
Theorem	3wlkdlem6 27025	Lemma 6 for 3wlkd 27030. (Contributed by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   =>    |- ( ph -> ( A e. ( I ` J ) /\ B e. ( I ` K ) /\ C e. ( I ` L ) ) )
Theorem	3wlkdlem7 27026	Lemma 7 for 3wlkd 27030. (Contributed by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   =>    |- ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) )
Theorem	3wlkdlem8 27027	Lemma 8 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   =>    |- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K /\ ( F ` 2 ) = L ) )
Theorem	3wlkdlem9 27028	Lemma 9 for 3wlkd 27030. (Contributed by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   =>    |- ( ph -> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) )
Theorem	3wlkdlem10 27029*	Lemma 10 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   =>    |- ( ph -> A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
Theorem	3wlkd 27030	Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F (Walks ` G ) P )
Theorem	3wlkond 27031	A walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ph -> F ( A (WalksOn ` G ) D ) P )
Theorem	3trld 27032	Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   =>    |- ( ph -> F (Trails ` G ) P )
Theorem	3trlond 27033	A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   =>    |- ( ph -> F ( A (TrailsOn ` G ) D ) P )
Theorem	3pthd 27034	A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   =>    |- ( ph -> F (Paths ` G ) P )
Theorem	3pthond 27035	A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   =>    |- ( ph -> F ( A (PathsOn ` G ) D ) P )
Theorem	3spthd 27036	A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   &    |- ( ph -> A =/= D )   =>    |- ( ph -> F (SPaths ` G ) P )
Theorem	3spthond 27037	A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   &    |- ( ph -> A =/= D )   =>    |- ( ph -> F ( A (SPathsOn ` G ) D ) P )
Theorem	3cycld 27038	Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   &    |- ( ph -> A = D )   =>    |- ( ph -> F (Cycles ` G ) P )
Theorem	3cyclpd 27039	Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
|- P = <" A B C D ">   &    |- F = <" J K L ">   &    |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )   &    |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )   &    |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )   &    |- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )   &    |- ( ph -> A = D )   =>    |- ( ph -> ( F (Cycles ` G ) P /\ ( # ` F ) = 3 /\ ( P ` 0 ) = A ) )
Theorem	upgr3v3e3cycl 27040*	If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
|- E = (Edg ` G )   &    |- V = (Vtx ` G )   =>    |- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) )
Theorem	uhgr3cyclexlem 27041	Lemma for uhgr3cyclex 27042. (Contributed by AV, 12-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( ( ( A e. V /\ B e. V ) /\ A =/= B ) /\ ( ( J e. dom I /\ { B , C } = ( I ` J ) ) /\ ( K e. dom I /\ { C , A } = ( I ` K ) ) ) ) -> J =/= K )
Theorem	uhgr3cyclex 27042*	If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UHGraph /\ ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) /\ ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )
Theorem	umgr3cyclex 27043*	If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) /\ ( { A , B } e. E /\ { B , C } e. E /\ { C , A } e. E ) ) -> E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 /\ ( p ` 0 ) = A ) )
Theorem	umgr3v3e3cycl 27044*	If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( G e. UMGraph -> ( E. f E. p ( f (Cycles ` G ) p /\ ( # ` f ) = 3 ) <-> E. a e. V E. b e. V E. c e. V ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) ) )
Theorem	upgr4cycl4dv4e 27045*	If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ F (Cycles ` G ) P /\ ( # ` F ) = 4 ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) )
16.3.12  Connected graphs
Syntax	cconngr 27046	Extend class notation with connected graphs.
class ConnGraph
Definition	df-conngr 27047*	Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 26990 and dfconngr1 27048. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- ConnGraph = { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. v E. f E. p f ( k (PathsOn ` g ) n ) p }
Theorem	dfconngr1 27048*	Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- ConnGraph = { g | [. (Vtx ` g ) / v ]. A. k e. v A. n e. ( v \ { k } ) E. f E. p f ( k (PathsOn ` g ) n ) p }
Theorem	isconngr 27049*	The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- V = (Vtx ` G )   =>    |- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. V E. f E. p f ( k (PathsOn ` G ) n ) p ) )
Theorem	isconngr1 27050*	The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- V = (Vtx ` G )   =>    |- ( G e. W -> ( G e. ConnGraph <-> A. k e. V A. n e. ( V \ { k } ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
Theorem	cusconngr 27051	A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- ( ( G e. UHGraph /\ G e. ComplGraph ) -> G e. ConnGraph )
Theorem	0conngr 27052	A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- (/) e. ConnGraph
Theorem	0vconngr 27053	A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> G e. ConnGraph )
Theorem	1conngr 27054	A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
|- ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph )
Theorem	conngrv2edg 27055*	A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 27159. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> E. e e. ran I N e. e )
Theorem	vdn0conngrumgrv2 27056	A vertex in a connected multigraph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
|- V = (Vtx ` G )   =>    |- ( ( ( G e. ConnGraph /\ G e. UMGraph ) /\ ( N e. V /\ 1 < ( # ` V ) ) ) -> ( (VtxDeg ` G ) ` N ) =/= 0 )
16.4  Eulerian paths and the Konigsberg Bridge problem
16.4.1  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." Correspondingly, an Eulerian path is defined as "a trail containing all edges" (see definition in [Bollobas] p. 16) in df-eupth 27058 resp. iseupth 27061. (EulerPaths ` G ) is the set of all Eulerian paths in graph G, see eupths 27060. An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), or, with other words, a circuit which is an Eulerian path. The function mapping a graph to the set of its Eulerian paths is defined as EulerPaths in df-eupth 27058, whereas there is no explicit definition for Eulerian circuits (yet): The statement " <. F , P >. is an Eulerian circuit" is formally expressed by ( F (EulerPaths ` G ) P /\ F (Circuits ` G ) P ). Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 27077). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 27105. An Eulerian path does not have to be a path in the meaning of definition df-pths 26612, because it may traverse some vertices more than once. Therefore, "Eulerian trail" would be a more appropriate name. The main result of this section is (one direction of) Euler's Theorem: "A non-trivial connected graph has an Euler[ian] circuit iff each vertex has even degree." (see part 1 of theorem 12 in [Bollobas] p. 16 and theorem 1.8.1 in [Diestel] p. 22) or, expressed with Eulerian paths: "A connected graph has an Euler[ian] trail from a vertex x to a vertex y (not equal with x) iff x and y are the only vertices of odd degree." (see part 2 of theorem 12 in [Bollobas] p. 17). In eulerpath 27101, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 27102 shows that a pseudograph with an Eulerian circuit has only vertices of even degree.
Syntax	ceupth 27057	Extend class notation with Eulerian paths.
class EulerPaths
Definition	df-eupth 27058*	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 27059	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	eupths 27060*	The Eulerian paths on the graph G. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)
|- I = (iEdg ` G )   =>    |- (EulerPaths ` G ) = { <. f , p >. | ( f (Trails ` G ) p /\ f : ( 0..^ ( # ` f ) ) -onto-> dom I ) }
Theorem	iseupth 27061	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 27062	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 27063	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 27064	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 27065	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 27066	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	upgriseupth 27067*	The property " <. F , P >. is an Eulerian path on the pseudograph G". (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 )   &    |- V = (Vtx ` G )   =>    |- ( G e. UPGraph -> ( F (EulerPaths ` G ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
Theorem	upgreupthi 27068*	Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- I = (iEdg ` G )   &    |- V = (Vtx ` G )   =>    |- ( ( G e. UPGraph /\ F (EulerPaths ` G ) P ) -> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
Theorem	upgreupthseg 27069	The N-th edge in an eulerian path is the edge from P ( N ) to P ( N + 1 ). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. UPGraph /\ F (EulerPaths ` G ) P /\ N e. ( 0..^ ( # ` F ) ) ) -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
Theorem	eupthcl 27070	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 27071	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 27072	An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)
|- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
Theorem	eupthpf 27073	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	eupth0 27074	There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( A e. V /\ I = (/) ) -> (/) (EulerPaths ` G ) { <. 0 , A >. } )
Theorem	eupthres 27075	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.) (Proof shortened by AV, 30-Oct-2021.)
|- 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 |` ( 0..^ N ) )   &    |- Q = ( P |` ( 0 ... N ) )   &    |- (Vtx ` S ) = V   =>    |- ( ph -> H (EulerPaths ` S ) Q )
Theorem	eupthp1 27076	Append one path segment to an Eulerian path <. F , P >. to become an Eulerian path <. H , Q >. of the supergraph S obtained by adding the new edge to the graph G. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> B e. _V )   &    |- ( ph -> C e. V )   &    |- ( ph -> -. B e. dom I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- N = ( # ` F )   &    |- ( ph -> E e. (Edg ` G ) )   &    |- ( ph -> { ( P ` N ) , C } C_ E )   &    |- (iEdg ` S ) = ( I u. { <. B , E >. } )   &    |- H = ( F u. { <. N , B >. } )   &    |- Q = ( P u. { <. ( N + 1 ) , C >. } )   &    |- (Vtx ` S ) = V   &    |- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )   =>    |- ( ph -> H (EulerPaths ` S ) Q )
Theorem	eupth2eucrct 27077	Append one path segment to an Eulerian path <. F , P >. which may not be an (Eulerian) circuit to become an Eulerian circuit <. H , Q >. of the supergraph S obtained by adding the new edge to the graph G. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> B e. _V )   &    |- ( ph -> C e. V )   &    |- ( ph -> -. B e. dom I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- N = ( # ` F )   &    |- ( ph -> E e. (Edg ` G ) )   &    |- ( ph -> { ( P ` N ) , C } C_ E )   &    |- (iEdg ` S ) = ( I u. { <. B , E >. } )   &    |- H = ( F u. { <. N , B >. } )   &    |- Q = ( P u. { <. ( N + 1 ) , C >. } )   &    |- (Vtx ` S ) = V   &    |- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )   &    |- ( ph -> C = ( P ` 0 ) )   =>    |- ( ph -> ( H (EulerPaths ` S ) Q /\ H (Circuits ` S ) Q ) )
Theorem	eupth2lem1 27078	Lemma for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.)
|- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
Theorem	eupth2lem2 27079	Lemma for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.)
|- B e. _V   =>    |- ( ( B =/= C /\ B = U ) -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )
Theorem	trlsegvdeglem1 27080	Lemma for trlsegvdeg 27087. (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 27081	Lemma for trlsegvdeg 27087. (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 27082	Lemma for trlsegvdeg 27087. (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 27083	Lemma for trlsegvdeg 27087. (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 27084	Lemma for trlsegvdeg 27087. (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 27085	Lemma for trlsegvdeg 27087. (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 27086	Lemma for trlsegvdeg 27087. (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	trlsegvdeg 27087	Formerly part of proof of eupth2lem3 27096: If a trail in a graph G induces a subgraph Z with the vertices V of G and the edges being the edges of the walk, and a subgraph X with the vertices V of G and the edges being the edges of the walk except the last one, and a subgraph Y with the vertices V of G and one edges being the last edge of the walk, then the vertex degree of any vertex U of G 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 -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
Theorem	eupth2lem3lem1 27088	Lemma for eupth2lem3 27096. (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 -> ( (VtxDeg ` X ) ` U ) e. NN0 )
Theorem	eupth2lem3lem2 27089	Lemma for eupth2lem3 27096. (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 -> ( (VtxDeg ` Y ) ` U ) e. NN0 )
Theorem	eupth2lem3lem3 27090*	Lemma for eupth2lem3 27096, formerly part of proof of eupth2lem3 27096: 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 -> { 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	eupth2lem3lem4 27091*	Lemma for eupth2lem3 27096, formerly part of proof of eupth2lem3 27096: 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 -> { 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	eupth2lem3lem5 27092*	Lemma for eupth2 27099. (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	eupth2lem3lem6 27093*	Formerly part of proof of eupth2lem3 27096: 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 -> { 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	eupth2lem3lem7 27094*	Lemma for eupth2lem3 27096: Combining trlsegvdeg 27087, eupth2lem3lem3 27090, eupth2lem3lem4 27091 and eupth2lem3lem6 27093. (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 -> { 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 27095	Formerly part of proof of eupth2 27099: 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	eupth2lem3 27096*	Lemma for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- 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	eupth2lemb 27097*	Lemma for eupth2 27099 (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). Formerly part of proof for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )
Theorem	eupth2lems 27098*	Lemma for eupth2 27099 (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. Formerly part of proof for eupth2 27099. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   =>    |- ( ( 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	eupth2 27099*	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. UPGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( # ` F ) ) , (/) , { ( P ` 0 ) , ( P ` ( # ` F ) ) } ) )
Theorem	eulerpathpr 27100*	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. UPGraph /\ F (EulerPaths ` G ) P ) -> ( # ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )