Step |
Hyp |
Ref |
Expression |
1 |
|
wwlksnwwlksnon.v |
|- V = ( Vtx ` G ) |
2 |
|
wwlknbp1 |
|- ( W e. ( N WWalksN G ) -> ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) |
3 |
1
|
eqcomi |
|- ( Vtx ` G ) = V |
4 |
3
|
wrdeqi |
|- Word ( Vtx ` G ) = Word V |
5 |
4
|
eleq2i |
|- ( W e. Word ( Vtx ` G ) <-> W e. Word V ) |
6 |
5
|
biimpi |
|- ( W e. Word ( Vtx ` G ) -> W e. Word V ) |
7 |
6
|
3ad2ant2 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> W e. Word V ) |
8 |
|
nn0p1nn |
|- ( N e. NN0 -> ( N + 1 ) e. NN ) |
9 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ ( N + 1 ) ) <-> ( N + 1 ) e. NN ) |
10 |
8 9
|
sylibr |
|- ( N e. NN0 -> 0 e. ( 0 ..^ ( N + 1 ) ) ) |
11 |
10
|
3ad2ant1 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> 0 e. ( 0 ..^ ( N + 1 ) ) ) |
12 |
|
oveq2 |
|- ( ( # ` W ) = ( N + 1 ) -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ ( N + 1 ) ) ) |
13 |
12
|
eleq2d |
|- ( ( # ` W ) = ( N + 1 ) -> ( 0 e. ( 0 ..^ ( # ` W ) ) <-> 0 e. ( 0 ..^ ( N + 1 ) ) ) ) |
14 |
13
|
3ad2ant3 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( 0 e. ( 0 ..^ ( # ` W ) ) <-> 0 e. ( 0 ..^ ( N + 1 ) ) ) ) |
15 |
11 14
|
mpbird |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) |
16 |
15
|
adantl |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) |
17 |
|
wrdsymbcl |
|- ( ( W e. Word V /\ 0 e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` 0 ) e. V ) |
18 |
7 16 17
|
syl2an2 |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` 0 ) e. V ) |
19 |
|
fzonn0p1 |
|- ( N e. NN0 -> N e. ( 0 ..^ ( N + 1 ) ) ) |
20 |
19
|
3ad2ant1 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> N e. ( 0 ..^ ( N + 1 ) ) ) |
21 |
12
|
eleq2d |
|- ( ( # ` W ) = ( N + 1 ) -> ( N e. ( 0 ..^ ( # ` W ) ) <-> N e. ( 0 ..^ ( N + 1 ) ) ) ) |
22 |
21
|
3ad2ant3 |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( N e. ( 0 ..^ ( # ` W ) ) <-> N e. ( 0 ..^ ( N + 1 ) ) ) ) |
23 |
20 22
|
mpbird |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> N e. ( 0 ..^ ( # ` W ) ) ) |
24 |
|
wrdsymbcl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` N ) e. V ) |
25 |
7 23 24
|
syl2anc |
|- ( ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( W ` N ) e. V ) |
26 |
25
|
adantl |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` N ) e. V ) |
27 |
|
simpl |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> W e. ( N WWalksN G ) ) |
28 |
|
eqidd |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` 0 ) = ( W ` 0 ) ) |
29 |
|
eqidd |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> ( W ` N ) = ( W ` N ) ) |
30 |
|
eqeq2 |
|- ( a = ( W ` 0 ) -> ( ( W ` 0 ) = a <-> ( W ` 0 ) = ( W ` 0 ) ) ) |
31 |
30
|
3anbi2d |
|- ( a = ( W ` 0 ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = b ) ) ) |
32 |
|
eqeq2 |
|- ( b = ( W ` N ) -> ( ( W ` N ) = b <-> ( W ` N ) = ( W ` N ) ) ) |
33 |
32
|
3anbi3d |
|- ( b = ( W ` N ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = ( W ` N ) ) ) ) |
34 |
31 33
|
rspc2ev |
|- ( ( ( W ` 0 ) e. V /\ ( W ` N ) e. V /\ ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = ( W ` 0 ) /\ ( W ` N ) = ( W ` N ) ) ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) |
35 |
18 26 27 28 29 34
|
syl113anc |
|- ( ( W e. ( N WWalksN G ) /\ ( N e. NN0 /\ W e. Word ( Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) |
36 |
2 35
|
mpdan |
|- ( W e. ( N WWalksN G ) -> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) |
37 |
|
simp1 |
|- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) ) |
38 |
37
|
a1i |
|- ( ( a e. V /\ b e. V ) -> ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) ) ) |
39 |
38
|
rexlimivv |
|- ( E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) -> W e. ( N WWalksN G ) ) |
40 |
36 39
|
impbii |
|- ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) |
41 |
|
wwlknon |
|- ( W e. ( a ( N WWalksNOn G ) b ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) ) |
42 |
41
|
bicomi |
|- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> W e. ( a ( N WWalksNOn G ) b ) ) |
43 |
42
|
2rexbii |
|- ( E. a e. V E. b e. V ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = a /\ ( W ` N ) = b ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) |
44 |
40 43
|
bitri |
|- ( W e. ( N WWalksN G ) <-> E. a e. V E. b e. V W e. ( a ( N WWalksNOn G ) b ) ) |