Step |
Hyp |
Ref |
Expression |
1 |
|
wwlknbp1 |
|- ( W e. ( N WWalksN G ) -> ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) |
2 |
|
simp2 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> W e. Word ( Vtx ` G ) ) |
3 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
4 |
|
fzval3 |
|- ( N e. ZZ -> ( 0 ... N ) = ( 0 ..^ ( N + 1 ) ) ) |
5 |
3 4
|
syl |
|- ( N e. NN0 -> ( 0 ... N ) = ( 0 ..^ ( N + 1 ) ) ) |
6 |
5
|
3ad2ant1 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( 0 ... N ) = ( 0 ..^ ( N + 1 ) ) ) |
7 |
6
|
eleq2d |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( i e. ( 0 ... N ) <-> i e. ( 0 ..^ ( N + 1 ) ) ) ) |
8 |
7
|
biimpa |
|- ( ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ i e. ( 0 ... N ) ) -> i e. ( 0 ..^ ( N + 1 ) ) ) |
9 |
|
oveq2 |
|- ( ( # ` W ) = ( N + 1 ) -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ ( N + 1 ) ) ) |
10 |
9
|
eleq2d |
|- ( ( # ` W ) = ( N + 1 ) -> ( i e. ( 0 ..^ ( # ` W ) ) <-> i e. ( 0 ..^ ( N + 1 ) ) ) ) |
11 |
10
|
3ad2ant3 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( i e. ( 0 ..^ ( # ` W ) ) <-> i e. ( 0 ..^ ( N + 1 ) ) ) ) |
12 |
11
|
adantr |
|- ( ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ i e. ( 0 ... N ) ) -> ( i e. ( 0 ..^ ( # ` W ) ) <-> i e. ( 0 ..^ ( N + 1 ) ) ) ) |
13 |
8 12
|
mpbird |
|- ( ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ i e. ( 0 ... N ) ) -> i e. ( 0 ..^ ( # ` W ) ) ) |
14 |
|
wrdsymbcl |
|- ( ( W e. Word ( Vtx ` G ) /\ i e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` i ) e. ( Vtx ` G ) ) |
15 |
2 13 14
|
syl2an2r |
|- ( ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ i e. ( 0 ... N ) ) -> ( W ` i ) e. ( Vtx ` G ) ) |
16 |
15
|
ralrimiva |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> A. i e. ( 0 ... N ) ( W ` i ) e. ( Vtx ` G ) ) |
17 |
1 16
|
syl |
|- ( W e. ( N WWalksN G ) -> A. i e. ( 0 ... N ) ( W ` i ) e. ( Vtx ` G ) ) |