Step |
Hyp |
Ref |
Expression |
1 |
|
conngrv2edg.v |
|- V = ( Vtx ` G ) |
2 |
|
conngrv2edg.i |
|- I = ( iEdg ` G ) |
3 |
1
|
fvexi |
|- V e. _V |
4 |
|
simp3 |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> 1 < ( # ` V ) ) |
5 |
|
simp2 |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> N e. V ) |
6 |
|
hashgt12el2 |
|- ( ( V e. _V /\ 1 < ( # ` V ) /\ N e. V ) -> E. v e. V N =/= v ) |
7 |
3 4 5 6
|
mp3an2i |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> E. v e. V N =/= v ) |
8 |
1
|
isconngr |
|- ( G e. ConnGraph -> ( G e. ConnGraph <-> A. a e. V A. b e. V E. f E. p f ( a ( PathsOn ` G ) b ) p ) ) |
9 |
|
oveq1 |
|- ( a = N -> ( a ( PathsOn ` G ) b ) = ( N ( PathsOn ` G ) b ) ) |
10 |
9
|
breqd |
|- ( a = N -> ( f ( a ( PathsOn ` G ) b ) p <-> f ( N ( PathsOn ` G ) b ) p ) ) |
11 |
10
|
2exbidv |
|- ( a = N -> ( E. f E. p f ( a ( PathsOn ` G ) b ) p <-> E. f E. p f ( N ( PathsOn ` G ) b ) p ) ) |
12 |
|
oveq2 |
|- ( b = v -> ( N ( PathsOn ` G ) b ) = ( N ( PathsOn ` G ) v ) ) |
13 |
12
|
breqd |
|- ( b = v -> ( f ( N ( PathsOn ` G ) b ) p <-> f ( N ( PathsOn ` G ) v ) p ) ) |
14 |
13
|
2exbidv |
|- ( b = v -> ( E. f E. p f ( N ( PathsOn ` G ) b ) p <-> E. f E. p f ( N ( PathsOn ` G ) v ) p ) ) |
15 |
11 14
|
rspc2v |
|- ( ( N e. V /\ v e. V ) -> ( A. a e. V A. b e. V E. f E. p f ( a ( PathsOn ` G ) b ) p -> E. f E. p f ( N ( PathsOn ` G ) v ) p ) ) |
16 |
15
|
ad2ant2r |
|- ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> ( A. a e. V A. b e. V E. f E. p f ( a ( PathsOn ` G ) b ) p -> E. f E. p f ( N ( PathsOn ` G ) v ) p ) ) |
17 |
|
pthontrlon |
|- ( f ( N ( PathsOn ` G ) v ) p -> f ( N ( TrailsOn ` G ) v ) p ) |
18 |
|
trlsonwlkon |
|- ( f ( N ( TrailsOn ` G ) v ) p -> f ( N ( WalksOn ` G ) v ) p ) |
19 |
|
simpl |
|- ( ( f ( N ( WalksOn ` G ) v ) p /\ ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) ) -> f ( N ( WalksOn ` G ) v ) p ) |
20 |
|
simprr |
|- ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> N =/= v ) |
21 |
|
wlkon2n0 |
|- ( ( f ( N ( WalksOn ` G ) v ) p /\ N =/= v ) -> ( # ` f ) =/= 0 ) |
22 |
20 21
|
sylan2 |
|- ( ( f ( N ( WalksOn ` G ) v ) p /\ ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) ) -> ( # ` f ) =/= 0 ) |
23 |
19 22
|
jca |
|- ( ( f ( N ( WalksOn ` G ) v ) p /\ ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) ) -> ( f ( N ( WalksOn ` G ) v ) p /\ ( # ` f ) =/= 0 ) ) |
24 |
23
|
ex |
|- ( f ( N ( WalksOn ` G ) v ) p -> ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> ( f ( N ( WalksOn ` G ) v ) p /\ ( # ` f ) =/= 0 ) ) ) |
25 |
17 18 24
|
3syl |
|- ( f ( N ( PathsOn ` G ) v ) p -> ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> ( f ( N ( WalksOn ` G ) v ) p /\ ( # ` f ) =/= 0 ) ) ) |
26 |
2
|
wlkonl1iedg |
|- ( ( f ( N ( WalksOn ` G ) v ) p /\ ( # ` f ) =/= 0 ) -> E. e e. ran I N e. e ) |
27 |
25 26
|
syl6com |
|- ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> ( f ( N ( PathsOn ` G ) v ) p -> E. e e. ran I N e. e ) ) |
28 |
27
|
exlimdvv |
|- ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> ( E. f E. p f ( N ( PathsOn ` G ) v ) p -> E. e e. ran I N e. e ) ) |
29 |
16 28
|
syldc |
|- ( A. a e. V A. b e. V E. f E. p f ( a ( PathsOn ` G ) b ) p -> ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> E. e e. ran I N e. e ) ) |
30 |
8 29
|
syl6bi |
|- ( G e. ConnGraph -> ( G e. ConnGraph -> ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> E. e e. ran I N e. e ) ) ) |
31 |
30
|
pm2.43i |
|- ( G e. ConnGraph -> ( ( ( N e. V /\ 1 < ( # ` V ) ) /\ ( v e. V /\ N =/= v ) ) -> E. e e. ran I N e. e ) ) |
32 |
31
|
expd |
|- ( G e. ConnGraph -> ( ( N e. V /\ 1 < ( # ` V ) ) -> ( ( v e. V /\ N =/= v ) -> E. e e. ran I N e. e ) ) ) |
33 |
32
|
3impib |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> ( ( v e. V /\ N =/= v ) -> E. e e. ran I N e. e ) ) |
34 |
33
|
expd |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> ( v e. V -> ( N =/= v -> E. e e. ran I N e. e ) ) ) |
35 |
34
|
rexlimdv |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> ( E. v e. V N =/= v -> E. e e. ran I N e. e ) ) |
36 |
7 35
|
mpd |
|- ( ( G e. ConnGraph /\ N e. V /\ 1 < ( # ` V ) ) -> E. e e. ran I N e. e ) |