Step |
Hyp |
Ref |
Expression |
1 |
|
isclwlke.v |
|- V = ( Vtx ` G ) |
2 |
|
isclwlke.i |
|- I = ( iEdg ` G ) |
3 |
|
clwlkcomp.1 |
|- F = ( 1st ` W ) |
4 |
|
clwlkcomp.2 |
|- P = ( 2nd ` W ) |
5 |
1 2 3 4
|
clwlkcompim |
|- ( W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) |
6 |
5
|
adantl |
|- ( ( G e. UPGraph /\ W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) |
7 |
|
simprl |
|- ( ( ( G e. UPGraph /\ W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) ) |
8 |
|
clwlkwlk |
|- ( W e. ( ClWalks ` G ) -> W e. ( Walks ` G ) ) |
9 |
1 2 3 4
|
upgrwlkcompim |
|- ( ( 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 ) ) } ) ) |
10 |
9
|
simp3d |
|- ( ( G e. UPGraph /\ W e. ( Walks ` G ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
11 |
8 10
|
sylan2 |
|- ( ( G e. UPGraph /\ W e. ( ClWalks ` G ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
12 |
11
|
adantr |
|- ( ( ( G e. UPGraph /\ W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
13 |
|
simprrr |
|- ( ( ( G e. UPGraph /\ W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) -> ( P ` 0 ) = ( P ` ( # ` F ) ) ) |
14 |
7 12 13
|
3jca |
|- ( ( ( G e. UPGraph /\ W e. ( ClWalks ` 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 ) ) ) /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) ) -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
15 |
6 14
|
mpdan |
|- ( ( G e. UPGraph /\ W e. ( ClWalks ` G ) ) -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |