Step |
Hyp |
Ref |
Expression |
1 |
|
simplll |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> A e. Word ( Vtx ` G ) ) |
2 |
|
simplr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> B e. Word ( Vtx ` G ) ) |
3 |
|
lencl |
|- ( A e. Word ( Vtx ` G ) -> ( # ` A ) e. NN0 ) |
4 |
3
|
nn0zd |
|- ( A e. Word ( Vtx ` G ) -> ( # ` A ) e. ZZ ) |
5 |
|
fzossrbm1 |
|- ( ( # ` A ) e. ZZ -> ( 0 ..^ ( ( # ` A ) - 1 ) ) C_ ( 0 ..^ ( # ` A ) ) ) |
6 |
4 5
|
syl |
|- ( A e. Word ( Vtx ` G ) -> ( 0 ..^ ( ( # ` A ) - 1 ) ) C_ ( 0 ..^ ( # ` A ) ) ) |
7 |
6
|
ad2antrr |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) -> ( 0 ..^ ( ( # ` A ) - 1 ) ) C_ ( 0 ..^ ( # ` A ) ) ) |
8 |
7
|
sselda |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> i e. ( 0 ..^ ( # ` A ) ) ) |
9 |
|
ccatval1 |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ i e. ( 0 ..^ ( # ` A ) ) ) -> ( ( A ++ B ) ` i ) = ( A ` i ) ) |
10 |
1 2 8 9
|
syl3anc |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( ( A ++ B ) ` i ) = ( A ` i ) ) |
11 |
4
|
ad2antrr |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) -> ( # ` A ) e. ZZ ) |
12 |
|
elfzom1elp1fzo |
|- ( ( ( # ` A ) e. ZZ /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( i + 1 ) e. ( 0 ..^ ( # ` A ) ) ) |
13 |
11 12
|
sylan |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( i + 1 ) e. ( 0 ..^ ( # ` A ) ) ) |
14 |
|
ccatval1 |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ ( i + 1 ) e. ( 0 ..^ ( # ` A ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( A ` ( i + 1 ) ) ) |
15 |
1 2 13 14
|
syl3anc |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( A ` ( i + 1 ) ) ) |
16 |
10 15
|
preq12d |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( A ` i ) , ( A ` ( i + 1 ) ) } ) |
17 |
16
|
eleq1d |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
18 |
17
|
biimprd |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) /\ i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) ) -> ( { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
19 |
18
|
ralimdva |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ B e. Word ( Vtx ` G ) ) -> ( A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
20 |
19
|
impancom |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) ) -> ( B e. Word ( Vtx ` G ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
21 |
20
|
3adant3 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( B e. Word ( Vtx ` G ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
22 |
21
|
com12 |
|- ( B e. Word ( Vtx ` G ) -> ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
23 |
22
|
adantr |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
24 |
23
|
3ad2ant1 |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
25 |
24
|
impcom |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
26 |
25
|
3adant3 |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
27 |
|
simprl |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> A e. Word ( Vtx ` G ) ) |
28 |
|
simpll |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> B e. Word ( Vtx ` G ) ) |
29 |
|
simprr |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> A =/= (/) ) |
30 |
|
ccatval1lsw |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) = ( lastS ` A ) ) |
31 |
27 28 29 30
|
syl3anc |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) = ( lastS ` A ) ) |
32 |
31
|
adantr |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) = ( lastS ` A ) ) |
33 |
3
|
nn0cnd |
|- ( A e. Word ( Vtx ` G ) -> ( # ` A ) e. CC ) |
34 |
|
npcan1 |
|- ( ( # ` A ) e. CC -> ( ( ( # ` A ) - 1 ) + 1 ) = ( # ` A ) ) |
35 |
33 34
|
syl |
|- ( A e. Word ( Vtx ` G ) -> ( ( ( # ` A ) - 1 ) + 1 ) = ( # ` A ) ) |
36 |
35
|
ad2antrl |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( ( # ` A ) - 1 ) + 1 ) = ( # ` A ) ) |
37 |
36
|
fveq2d |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) = ( ( A ++ B ) ` ( # ` A ) ) ) |
38 |
|
simplr |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> B =/= (/) ) |
39 |
|
ccatval21sw |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( A ++ B ) ` ( # ` A ) ) = ( B ` 0 ) ) |
40 |
27 28 38 39
|
syl3anc |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( # ` A ) ) = ( B ` 0 ) ) |
41 |
37 40
|
eqtrd |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) = ( B ` 0 ) ) |
42 |
41
|
adantr |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) = ( B ` 0 ) ) |
43 |
|
simpr |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ` 0 ) = ( B ` 0 ) ) |
44 |
42 43
|
eqtr4d |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) = ( A ` 0 ) ) |
45 |
32 44
|
preq12d |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } = { ( lastS ` A ) , ( A ` 0 ) } ) |
46 |
45
|
eleq1d |
|- ( ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) <-> { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) ) |
47 |
46
|
exbiri |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
48 |
47
|
com23 |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) ) -> ( { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
49 |
48
|
expimpd |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
50 |
49
|
3ad2ant1 |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
51 |
50
|
com12 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
52 |
51
|
3adant2 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) ) |
53 |
52
|
3imp |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) |
54 |
|
ralunb |
|- ( A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. { ( ( # ` A ) - 1 ) } { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
55 |
|
ovex |
|- ( ( # ` A ) - 1 ) e. _V |
56 |
|
fveq2 |
|- ( i = ( ( # ` A ) - 1 ) -> ( ( A ++ B ) ` i ) = ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) ) |
57 |
|
fvoveq1 |
|- ( i = ( ( # ` A ) - 1 ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) ) |
58 |
56 57
|
preq12d |
|- ( i = ( ( # ` A ) - 1 ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } ) |
59 |
58
|
eleq1d |
|- ( i = ( ( # ` A ) - 1 ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) |
60 |
55 59
|
ralsn |
|- ( A. i e. { ( ( # ` A ) - 1 ) } { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) |
61 |
60
|
anbi2i |
|- ( ( A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. { ( ( # ` A ) - 1 ) } { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) <-> ( A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) |
62 |
54 61
|
bitri |
|- ( A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( ( A ++ B ) ` ( ( # ` A ) - 1 ) ) , ( ( A ++ B ) ` ( ( ( # ` A ) - 1 ) + 1 ) ) } e. ( Edg ` G ) ) ) |
63 |
26 53 62
|
sylanbrc |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
64 |
|
0z |
|- 0 e. ZZ |
65 |
|
lennncl |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( # ` A ) e. NN ) |
66 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
67 |
66
|
fveq2i |
|- ( ZZ>= ` ( 0 + 1 ) ) = ( ZZ>= ` 1 ) |
68 |
67
|
eleq2i |
|- ( ( # ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) <-> ( # ` A ) e. ( ZZ>= ` 1 ) ) |
69 |
|
elnnuz |
|- ( ( # ` A ) e. NN <-> ( # ` A ) e. ( ZZ>= ` 1 ) ) |
70 |
68 69
|
bitr4i |
|- ( ( # ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) <-> ( # ` A ) e. NN ) |
71 |
65 70
|
sylibr |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( # ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) ) |
72 |
|
fzosplitsnm1 |
|- ( ( 0 e. ZZ /\ ( # ` A ) e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( 0 ..^ ( # ` A ) ) = ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) ) |
73 |
64 71 72
|
sylancr |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( 0 ..^ ( # ` A ) ) = ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) ) |
74 |
73
|
raleqdv |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( A. i e. ( 0 ..^ ( # ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
75 |
74
|
3ad2ant1 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( A. i e. ( 0 ..^ ( # ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
76 |
75
|
3ad2ant1 |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A. i e. ( 0 ..^ ( # ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( ( 0 ..^ ( ( # ` A ) - 1 ) ) u. { ( ( # ` A ) - 1 ) } ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
77 |
63 76
|
mpbird |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0 ..^ ( # ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
78 |
|
lencl |
|- ( B e. Word ( Vtx ` G ) -> ( # ` B ) e. NN0 ) |
79 |
78
|
nn0zd |
|- ( B e. Word ( Vtx ` G ) -> ( # ` B ) e. ZZ ) |
80 |
|
peano2zm |
|- ( ( # ` B ) e. ZZ -> ( ( # ` B ) - 1 ) e. ZZ ) |
81 |
79 80
|
syl |
|- ( B e. Word ( Vtx ` G ) -> ( ( # ` B ) - 1 ) e. ZZ ) |
82 |
81
|
ad2antrl |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` B ) - 1 ) e. ZZ ) |
83 |
82
|
adantr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( # ` B ) - 1 ) e. ZZ ) |
84 |
83
|
anim1ci |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) /\ ( ( # ` B ) - 1 ) e. ZZ ) ) |
85 |
|
fzosubel3 |
|- ( ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) /\ ( ( # ` B ) - 1 ) e. ZZ ) -> ( i - ( # ` A ) ) e. ( 0 ..^ ( ( # ` B ) - 1 ) ) ) |
86 |
|
fveq2 |
|- ( j = ( i - ( # ` A ) ) -> ( B ` j ) = ( B ` ( i - ( # ` A ) ) ) ) |
87 |
|
fvoveq1 |
|- ( j = ( i - ( # ` A ) ) -> ( B ` ( j + 1 ) ) = ( B ` ( ( i - ( # ` A ) ) + 1 ) ) ) |
88 |
86 87
|
preq12d |
|- ( j = ( i - ( # ` A ) ) -> { ( B ` j ) , ( B ` ( j + 1 ) ) } = { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } ) |
89 |
88
|
eleq1d |
|- ( j = ( i - ( # ` A ) ) -> ( { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) <-> { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } e. ( Edg ` G ) ) ) |
90 |
89
|
rspcv |
|- ( ( i - ( # ` A ) ) e. ( 0 ..^ ( ( # ` B ) - 1 ) ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } e. ( Edg ` G ) ) ) |
91 |
84 85 90
|
3syl |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } e. ( Edg ` G ) ) ) |
92 |
|
simp-4l |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> A e. Word ( Vtx ` G ) ) |
93 |
|
simprl |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> B e. Word ( Vtx ` G ) ) |
94 |
93
|
ad2antrr |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> B e. Word ( Vtx ` G ) ) |
95 |
3
|
adantr |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( # ` A ) e. NN0 ) |
96 |
78
|
adantr |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( # ` B ) e. NN0 ) |
97 |
|
nn0addcl |
|- ( ( ( # ` A ) e. NN0 /\ ( # ` B ) e. NN0 ) -> ( ( # ` A ) + ( # ` B ) ) e. NN0 ) |
98 |
97
|
nn0zd |
|- ( ( ( # ` A ) e. NN0 /\ ( # ` B ) e. NN0 ) -> ( ( # ` A ) + ( # ` B ) ) e. ZZ ) |
99 |
95 96 98
|
syl2an |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) + ( # ` B ) ) e. ZZ ) |
100 |
|
1nn0 |
|- 1 e. NN0 |
101 |
|
eluzmn |
|- ( ( ( ( # ` A ) + ( # ` B ) ) e. ZZ /\ 1 e. NN0 ) -> ( ( # ` A ) + ( # ` B ) ) e. ( ZZ>= ` ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) |
102 |
99 100 101
|
sylancl |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) + ( # ` B ) ) e. ( ZZ>= ` ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) |
103 |
33
|
ad2antrr |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( # ` A ) e. CC ) |
104 |
78
|
nn0cnd |
|- ( B e. Word ( Vtx ` G ) -> ( # ` B ) e. CC ) |
105 |
104
|
ad2antrl |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( # ` B ) e. CC ) |
106 |
|
1cnd |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> 1 e. CC ) |
107 |
103 105 106
|
addsubassd |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( ( # ` A ) + ( # ` B ) ) - 1 ) = ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) |
108 |
107
|
fveq2d |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ZZ>= ` ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) = ( ZZ>= ` ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) |
109 |
102 108
|
eleqtrd |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) + ( # ` B ) ) e. ( ZZ>= ` ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) |
110 |
|
fzoss2 |
|- ( ( ( # ` A ) + ( # ` B ) ) e. ( ZZ>= ` ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) C_ ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
111 |
109 110
|
syl |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) C_ ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
112 |
111
|
adantr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) C_ ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
113 |
112
|
sselda |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
114 |
|
ccatval2 |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) -> ( ( A ++ B ) ` i ) = ( B ` ( i - ( # ` A ) ) ) ) |
115 |
92 94 113 114
|
syl3anc |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` i ) = ( B ` ( i - ( # ` A ) ) ) ) |
116 |
107
|
oveq2d |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) = ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) |
117 |
116
|
eleq2d |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( i e. ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) <-> i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
118 |
117
|
adantr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) <-> i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
119 |
|
eluzmn |
|- ( ( ( # ` A ) e. ZZ /\ 1 e. NN0 ) -> ( # ` A ) e. ( ZZ>= ` ( ( # ` A ) - 1 ) ) ) |
120 |
4 100 119
|
sylancl |
|- ( A e. Word ( Vtx ` G ) -> ( # ` A ) e. ( ZZ>= ` ( ( # ` A ) - 1 ) ) ) |
121 |
120
|
ad3antrrr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( # ` A ) e. ( ZZ>= ` ( ( # ` A ) - 1 ) ) ) |
122 |
|
fzoss1 |
|- ( ( # ` A ) e. ( ZZ>= ` ( ( # ` A ) - 1 ) ) -> ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) C_ ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) |
123 |
121 122
|
syl |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) C_ ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) |
124 |
123
|
sseld |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( ( # ` A ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) -> i e. ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) ) |
125 |
118 124
|
sylbird |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> i e. ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) ) |
126 |
125
|
imp |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> i e. ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) ) |
127 |
4
|
adantr |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( # ` A ) e. ZZ ) |
128 |
79
|
adantr |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( # ` B ) e. ZZ ) |
129 |
|
simpl |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. ZZ ) -> ( # ` A ) e. ZZ ) |
130 |
|
zaddcl |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. ZZ ) -> ( ( # ` A ) + ( # ` B ) ) e. ZZ ) |
131 |
129 130
|
jca |
|- ( ( ( # ` A ) e. ZZ /\ ( # ` B ) e. ZZ ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ ) ) |
132 |
127 128 131
|
syl2an |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ ) ) |
133 |
132
|
adantr |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ ) ) |
134 |
|
elfzoelz |
|- ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> i e. ZZ ) |
135 |
|
1zzd |
|- ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> 1 e. ZZ ) |
136 |
134 135
|
jca |
|- ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> ( i e. ZZ /\ 1 e. ZZ ) ) |
137 |
|
elfzomelpfzo |
|- ( ( ( ( # ` A ) e. ZZ /\ ( ( # ` A ) + ( # ` B ) ) e. ZZ ) /\ ( i e. ZZ /\ 1 e. ZZ ) ) -> ( i e. ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) <-> ( i + 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) ) |
138 |
133 136 137
|
syl2an |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( i e. ( ( ( # ` A ) - 1 ) ..^ ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) <-> ( i + 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) ) |
139 |
126 138
|
mpbid |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( i + 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) |
140 |
|
ccatval2 |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) /\ ( i + 1 ) e. ( ( # ` A ) ..^ ( ( # ` A ) + ( # ` B ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i + 1 ) - ( # ` A ) ) ) ) |
141 |
92 94 139 140
|
syl3anc |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i + 1 ) - ( # ` A ) ) ) ) |
142 |
134
|
zcnd |
|- ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> i e. CC ) |
143 |
142
|
adantl |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> i e. CC ) |
144 |
|
1cnd |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> 1 e. CC ) |
145 |
103
|
ad2antrr |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( # ` A ) e. CC ) |
146 |
143 144 145
|
addsubd |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( ( i + 1 ) - ( # ` A ) ) = ( ( i - ( # ` A ) ) + 1 ) ) |
147 |
146
|
fveq2d |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( B ` ( ( i + 1 ) - ( # ` A ) ) ) = ( B ` ( ( i - ( # ` A ) ) + 1 ) ) ) |
148 |
141 147
|
eqtrd |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( ( A ++ B ) ` ( i + 1 ) ) = ( B ` ( ( i - ( # ` A ) ) + 1 ) ) ) |
149 |
115 148
|
preq12d |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } = { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } ) |
150 |
149
|
eleq1d |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> { ( B ` ( i - ( # ` A ) ) ) , ( B ` ( ( i - ( # ` A ) ) + 1 ) ) } e. ( Edg ` G ) ) ) |
151 |
91 150
|
sylibrd |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
152 |
151
|
impancom |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) ) -> ( i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) -> { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
153 |
152
|
ralrimiv |
|- ( ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
154 |
153
|
exp31 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
155 |
154
|
expcom |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) ) |
156 |
155
|
com23 |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) ) |
157 |
156
|
com24 |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) -> ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) ) |
158 |
157
|
imp |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) ) -> ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
159 |
158
|
3adant3 |
|- ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
160 |
159
|
com12 |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
161 |
160
|
3ad2ant1 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) -> ( ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) -> ( ( A ` 0 ) = ( B ` 0 ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) ) |
162 |
161
|
3imp |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
163 |
|
ralunb |
|- ( A. i e. ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> ( A. i e. ( 0 ..^ ( # ` A ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) /\ A. i e. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
164 |
77 162 163
|
sylanbrc |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |
165 |
|
ccatlen |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) ) -> ( # ` ( A ++ B ) ) = ( ( # ` A ) + ( # ` B ) ) ) |
166 |
165
|
oveq1d |
|- ( ( A e. Word ( Vtx ` G ) /\ B e. Word ( Vtx ` G ) ) -> ( ( # ` ( A ++ B ) ) - 1 ) = ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) |
167 |
166
|
ad2ant2r |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` ( A ++ B ) ) - 1 ) = ( ( ( # ` A ) + ( # ` B ) ) - 1 ) ) |
168 |
167 107
|
eqtrd |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( ( # ` ( A ++ B ) ) - 1 ) = ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) |
169 |
168
|
oveq2d |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) = ( 0 ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) |
170 |
|
elnn0uz |
|- ( ( # ` A ) e. NN0 <-> ( # ` A ) e. ( ZZ>= ` 0 ) ) |
171 |
3 170
|
sylib |
|- ( A e. Word ( Vtx ` G ) -> ( # ` A ) e. ( ZZ>= ` 0 ) ) |
172 |
171
|
adantr |
|- ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) -> ( # ` A ) e. ( ZZ>= ` 0 ) ) |
173 |
|
lennncl |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( # ` B ) e. NN ) |
174 |
|
nnm1nn0 |
|- ( ( # ` B ) e. NN -> ( ( # ` B ) - 1 ) e. NN0 ) |
175 |
173 174
|
syl |
|- ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) -> ( ( # ` B ) - 1 ) e. NN0 ) |
176 |
|
fzoun |
|- ( ( ( # ` A ) e. ( ZZ>= ` 0 ) /\ ( ( # ` B ) - 1 ) e. NN0 ) -> ( 0 ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
177 |
172 175 176
|
syl2an |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( 0 ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
178 |
169 177
|
eqtrd |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) ) -> ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
179 |
178
|
3ad2antr1 |
|- ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) ) -> ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
180 |
179
|
3ad2antl1 |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) ) -> ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
181 |
180
|
3adant3 |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) = ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) ) |
182 |
181
|
raleqdv |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A. i e. ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) <-> A. i e. ( ( 0 ..^ ( # ` A ) ) u. ( ( # ` A ) ..^ ( ( # ` A ) + ( ( # ` B ) - 1 ) ) ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) ) |
183 |
164 182
|
mpbird |
|- ( ( ( ( A e. Word ( Vtx ` G ) /\ A =/= (/) ) /\ A. i e. ( 0 ..^ ( ( # ` A ) - 1 ) ) { ( A ` i ) , ( A ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` A ) , ( A ` 0 ) } e. ( Edg ` G ) ) /\ ( ( B e. Word ( Vtx ` G ) /\ B =/= (/) ) /\ A. j e. ( 0 ..^ ( ( # ` B ) - 1 ) ) { ( B ` j ) , ( B ` ( j + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` B ) , ( B ` 0 ) } e. ( Edg ` G ) ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) |