Step | Hyp | Ref
| Expression |
1 | | fstwrdne 14452 |
. . . . . . . . . 10
β’ ((π β Word π β§ π β β
) β (πβ0) β π) |
2 | 1 | s1cld 14500 |
. . . . . . . . 9
β’ ((π β Word π β§ π β β
) β β¨β(πβ0)ββ© β
Word π) |
3 | | ccatlen 14472 |
. . . . . . . . 9
β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π) β (β―β(π ++ β¨β(πβ0)ββ©)) =
((β―βπ) +
(β―ββ¨β(πβ0)ββ©))) |
4 | 2, 3 | syldan 592 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β (β―β(π ++ β¨β(πβ0)ββ©)) =
((β―βπ) +
(β―ββ¨β(πβ0)ββ©))) |
5 | | s1len 14503 |
. . . . . . . . 9
β’
(β―ββ¨β(πβ0)ββ©) = 1 |
6 | 5 | oveq2i 7372 |
. . . . . . . 8
β’
((β―βπ) +
(β―ββ¨β(πβ0)ββ©)) =
((β―βπ) +
1) |
7 | 4, 6 | eqtrdi 2789 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β (β―β(π ++ β¨β(πβ0)ββ©)) =
((β―βπ) +
1)) |
8 | 7 | oveq1d 7376 |
. . . . . 6
β’ ((π β Word π β§ π β β
) β ((β―β(π ++ β¨β(πβ0)ββ©)) β
1) = (((β―βπ) +
1) β 1)) |
9 | | lencl 14430 |
. . . . . . . . 9
β’ (π β Word π β (β―βπ) β
β0) |
10 | 9 | nn0cnd 12483 |
. . . . . . . 8
β’ (π β Word π β (β―βπ) β β) |
11 | 10 | adantr 482 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β (β―βπ) β
β) |
12 | | 1cnd 11158 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β 1 β
β) |
13 | 11, 12, 12 | addsubd 11541 |
. . . . . 6
β’ ((π β Word π β§ π β β
) β (((β―βπ) + 1) β 1) =
(((β―βπ) β
1) + 1)) |
14 | 8, 13 | eqtrd 2773 |
. . . . 5
β’ ((π β Word π β§ π β β
) β ((β―β(π ++ β¨β(πβ0)ββ©)) β
1) = (((β―βπ)
β 1) + 1)) |
15 | 14 | oveq2d 7377 |
. . . 4
β’ ((π β Word π β§ π β β
) β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)) =
(0..^(((β―βπ)
β 1) + 1))) |
16 | 15 | raleqdv 3312 |
. . 3
β’ ((π β Word π β§ π β β
) β (βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β βπ β
(0..^(((β―βπ)
β 1) + 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
17 | | lennncl 14431 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β (β―βπ) β
β) |
18 | | nnm1nn0 12462 |
. . . . . . . 8
β’
((β―βπ)
β β β ((β―βπ) β 1) β
β0) |
19 | 17, 18 | syl 17 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β ((β―βπ) β 1) β
β0) |
20 | | elnn0uz 12816 |
. . . . . . 7
β’
(((β―βπ)
β 1) β β0 β ((β―βπ) β 1) β
(β€β₯β0)) |
21 | 19, 20 | sylib 217 |
. . . . . 6
β’ ((π β Word π β§ π β β
) β ((β―βπ) β 1) β
(β€β₯β0)) |
22 | | fzosplitsn 13689 |
. . . . . 6
β’
(((β―βπ)
β 1) β (β€β₯β0) β
(0..^(((β―βπ)
β 1) + 1)) = ((0..^((β―βπ) β 1)) βͺ {((β―βπ) β 1)})) |
23 | 21, 22 | syl 17 |
. . . . 5
β’ ((π β Word π β§ π β β
) β
(0..^(((β―βπ)
β 1) + 1)) = ((0..^((β―βπ) β 1)) βͺ {((β―βπ) β 1)})) |
24 | 23 | raleqdv 3312 |
. . . 4
β’ ((π β Word π β§ π β β
) β (βπ β
(0..^(((β―βπ)
β 1) + 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β βπ β
((0..^((β―βπ)
β 1)) βͺ {((β―βπ) β 1)}){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
25 | | ralunb 4155 |
. . . 4
β’
(βπ β
((0..^((β―βπ)
β 1)) βͺ {((β―βπ) β 1)}){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β (βπ β
(0..^((β―βπ)
β 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β§ βπ β {((β―βπ) β 1)} {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
26 | 24, 25 | bitrdi 287 |
. . 3
β’ ((π β Word π β§ π β β
) β (βπ β
(0..^(((β―βπ)
β 1) + 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β (βπ β
(0..^((β―βπ)
β 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β§ βπ β {((β―βπ) β 1)} {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ)))) |
27 | | simpl 484 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β π β Word π) |
28 | 9 | nn0zd 12533 |
. . . . . . . . . 10
β’ (π β Word π β (β―βπ) β β€) |
29 | 28 | adantr 482 |
. . . . . . . . 9
β’ ((π β Word π β§ π β β
) β (β―βπ) β
β€) |
30 | | elfzom1elfzo 13649 |
. . . . . . . . 9
β’
(((β―βπ)
β β€ β§ π
β (0..^((β―βπ) β 1))) β π β (0..^(β―βπ))) |
31 | 29, 30 | sylan 581 |
. . . . . . . 8
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β π β
(0..^(β―βπ))) |
32 | | ccats1val1 14523 |
. . . . . . . 8
β’ ((π β Word π β§ π β (0..^(β―βπ))) β ((π ++ β¨β(πβ0)ββ©)βπ) = (πβπ)) |
33 | 27, 31, 32 | syl2an2r 684 |
. . . . . . 7
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β ((π ++ β¨β(πβ0)ββ©)βπ) = (πβπ)) |
34 | | elfzom1elp1fzo 13648 |
. . . . . . . . 9
β’
(((β―βπ)
β β€ β§ π
β (0..^((β―βπ) β 1))) β (π + 1) β (0..^(β―βπ))) |
35 | 29, 34 | sylan 581 |
. . . . . . . 8
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β (π + 1) β
(0..^(β―βπ))) |
36 | | ccats1val1 14523 |
. . . . . . . 8
β’ ((π β Word π β§ (π + 1) β (0..^(β―βπ))) β ((π ++ β¨β(πβ0)ββ©)β(π + 1)) = (πβ(π + 1))) |
37 | 27, 35, 36 | syl2an2r 684 |
. . . . . . 7
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β ((π ++ β¨β(πβ0)ββ©)β(π + 1)) = (πβ(π + 1))) |
38 | 33, 37 | preq12d 4706 |
. . . . . 6
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} = {(πβπ), (πβ(π + 1))}) |
39 | 38 | eleq1d 2819 |
. . . . 5
β’ (((π β Word π β§ π β β
) β§ π β (0..^((β―βπ) β 1))) β ({((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β {(πβπ), (πβ(π + 1))} β (EdgβπΊ))) |
40 | 39 | ralbidva 3169 |
. . . 4
β’ ((π β Word π β§ π β β
) β (βπ β
(0..^((β―βπ)
β 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ))) |
41 | | ovex 7394 |
. . . . . 6
β’
((β―βπ)
β 1) β V |
42 | | fveq2 6846 |
. . . . . . . 8
β’ (π = ((β―βπ) β 1) β ((π ++ β¨β(πβ0)ββ©)βπ) = ((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1))) |
43 | | fvoveq1 7384 |
. . . . . . . 8
β’ (π = ((β―βπ) β 1) β ((π ++ β¨β(πβ0)ββ©)β(π + 1)) = ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))) |
44 | 42, 43 | preq12d 4706 |
. . . . . . 7
β’ (π = ((β―βπ) β 1) β {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} = {((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)), ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))}) |
45 | 44 | eleq1d 2819 |
. . . . . 6
β’ (π = ((β―βπ) β 1) β ({((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β {((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)), ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))} β
(EdgβπΊ))) |
46 | 41, 45 | ralsn 4646 |
. . . . 5
β’
(βπ β
{((β―βπ) β
1)} {((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β {((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)), ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))} β
(EdgβπΊ)) |
47 | | fzo0end 13673 |
. . . . . . . . . 10
β’
((β―βπ)
β β β ((β―βπ) β 1) β
(0..^(β―βπ))) |
48 | 17, 47 | syl 17 |
. . . . . . . . 9
β’ ((π β Word π β§ π β β
) β ((β―βπ) β 1) β
(0..^(β―βπ))) |
49 | | ccats1val1 14523 |
. . . . . . . . 9
β’ ((π β Word π β§ ((β―βπ) β 1) β
(0..^(β―βπ)))
β ((π ++
β¨β(πβ0)ββ©)β((β―βπ) β 1)) = (πβ((β―βπ) β 1))) |
50 | 48, 49 | syldan 592 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)) = (πβ((β―βπ) β 1))) |
51 | | lsw 14461 |
. . . . . . . . 9
β’ (π β Word π β (lastSβπ) = (πβ((β―βπ) β 1))) |
52 | 51 | adantr 482 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β (lastSβπ) = (πβ((β―βπ) β 1))) |
53 | 50, 52 | eqtr4d 2776 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)) = (lastSβπ)) |
54 | | npcan1 11588 |
. . . . . . . . . . 11
β’
((β―βπ)
β β β (((β―βπ) β 1) + 1) = (β―βπ)) |
55 | 10, 54 | syl 17 |
. . . . . . . . . 10
β’ (π β Word π β (((β―βπ) β 1) + 1) = (β―βπ)) |
56 | 55 | adantr 482 |
. . . . . . . . 9
β’ ((π β Word π β§ π β β
) β (((β―βπ) β 1) + 1) =
(β―βπ)) |
57 | 56 | fveq2d 6850 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1)) = ((π ++ β¨β(πβ0)ββ©)β(β―βπ))) |
58 | | eqidd 2734 |
. . . . . . . . 9
β’ ((π β Word π β§ π β β
) β (β―βπ) = (β―βπ)) |
59 | | ccats1val2 14524 |
. . . . . . . . 9
β’ ((π β Word π β§ (πβ0) β π β§ (β―βπ) = (β―βπ)) β ((π ++ β¨β(πβ0)ββ©)β(β―βπ)) = (πβ0)) |
60 | 27, 1, 58, 59 | syl3anc 1372 |
. . . . . . . 8
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©)β(β―βπ)) = (πβ0)) |
61 | 57, 60 | eqtrd 2773 |
. . . . . . 7
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1)) = (πβ0)) |
62 | 53, 61 | preq12d 4706 |
. . . . . 6
β’ ((π β Word π β§ π β β
) β {((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)), ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))} =
{(lastSβπ), (πβ0)}) |
63 | 62 | eleq1d 2819 |
. . . . 5
β’ ((π β Word π β§ π β β
) β ({((π ++ β¨β(πβ0)ββ©)β((β―βπ) β 1)), ((π ++ β¨β(πβ0)ββ©)β(((β―βπ) β 1) + 1))} β
(EdgβπΊ) β
{(lastSβπ), (πβ0)} β (EdgβπΊ))) |
64 | 46, 63 | bitrid 283 |
. . . 4
β’ ((π β Word π β§ π β β
) β (βπ β {((β―βπ) β 1)} {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β {(lastSβπ), (πβ0)} β (EdgβπΊ))) |
65 | 40, 64 | anbi12d 632 |
. . 3
β’ ((π β Word π β§ π β β
) β ((βπ β
(0..^((β―βπ)
β 1)){((π ++
β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β§ βπ β {((β―βπ) β 1)} {((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ)) β (βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)))) |
66 | 16, 26, 65 | 3bitrd 305 |
. 2
β’ ((π β Word π β§ π β β
) β (βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ) β (βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)))) |
67 | 27, 2 | jca 513 |
. . . . 5
β’ ((π β Word π β§ π β β
) β (π β Word π β§ β¨β(πβ0)ββ© β Word π)) |
68 | | ccat0 14473 |
. . . . . . . 8
β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π) β ((π ++ β¨β(πβ0)ββ©) = β
β
(π = β
β§
β¨β(πβ0)ββ© =
β
))) |
69 | | simpl 484 |
. . . . . . . 8
β’ ((π = β
β§
β¨β(πβ0)ββ© = β
) β
π =
β
) |
70 | 68, 69 | syl6bi 253 |
. . . . . . 7
β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π) β ((π ++ β¨β(πβ0)ββ©) = β
β
π =
β
)) |
71 | 70 | necon3d 2961 |
. . . . . 6
β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π) β (π β β
β (π ++ β¨β(πβ0)ββ©) β
β
)) |
72 | 71 | adantld 492 |
. . . . 5
β’ ((π β Word π β§ β¨β(πβ0)ββ© β Word π) β ((π β Word π β§ π β β
) β (π ++ β¨β(πβ0)ββ©) β
β
)) |
73 | 67, 72 | mpcom 38 |
. . . 4
β’ ((π β Word π β§ π β β
) β (π ++ β¨β(πβ0)ββ©) β
β
) |
74 | | wrdv 14426 |
. . . . . . 7
β’ (π β Word π β π β Word V) |
75 | | s1cli 14502 |
. . . . . . 7
β’
β¨β(πβ0)ββ© β Word
V |
76 | | ccatalpha 14490 |
. . . . . . 7
β’ ((π β Word V β§
β¨β(πβ0)ββ© β Word V) β
((π ++ β¨β(πβ0)ββ©) β
Word π β (π β Word π β§ β¨β(πβ0)ββ© β Word π))) |
77 | 74, 75, 76 | sylancl 587 |
. . . . . 6
β’ (π β Word π β ((π ++ β¨β(πβ0)ββ©) β Word π β (π β Word π β§ β¨β(πβ0)ββ© β Word π))) |
78 | 77 | adantr 482 |
. . . . 5
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©) β Word π β (π β Word π β§ β¨β(πβ0)ββ© β Word π))) |
79 | 27, 2, 78 | mpbir2and 712 |
. . . 4
β’ ((π β Word π β§ π β β
) β (π ++ β¨β(πβ0)ββ©) β Word π) |
80 | 73, 79 | jca 513 |
. . 3
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©) β β
β§
(π ++ β¨β(πβ0)ββ©) β
Word π)) |
81 | | clwwlkwwlksb.v |
. . . . . 6
β’ π = (VtxβπΊ) |
82 | | eqid 2733 |
. . . . . 6
β’
(EdgβπΊ) =
(EdgβπΊ) |
83 | 81, 82 | iswwlks 28830 |
. . . . 5
β’ ((π ++ β¨β(πβ0)ββ©) β
(WWalksβπΊ) β
((π ++ β¨β(πβ0)ββ©) β
β
β§ (π ++
β¨β(πβ0)ββ©) β Word π β§ βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
84 | | df-3an 1090 |
. . . . 5
β’ (((π ++ β¨β(πβ0)ββ©) β
β
β§ (π ++
β¨β(πβ0)ββ©) β Word π β§ βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ)) β (((π ++ β¨β(πβ0)ββ©) β β
β§
(π ++ β¨β(πβ0)ββ©) β
Word π) β§ βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
85 | 83, 84 | bitri 275 |
. . . 4
β’ ((π ++ β¨β(πβ0)ββ©) β
(WWalksβπΊ) β
(((π ++ β¨β(πβ0)ββ©) β
β
β§ (π ++
β¨β(πβ0)ββ©) β Word π) β§ βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
86 | 85 | a1i 11 |
. . 3
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©) β
(WWalksβπΊ) β
(((π ++ β¨β(πβ0)ββ©) β
β
β§ (π ++
β¨β(πβ0)ββ©) β Word π) β§ βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ)))) |
87 | 80, 86 | mpbirand 706 |
. 2
β’ ((π β Word π β§ π β β
) β ((π ++ β¨β(πβ0)ββ©) β
(WWalksβπΊ) β
βπ β
(0..^((β―β(π ++
β¨β(πβ0)ββ©)) β 1)){((π ++ β¨β(πβ0)ββ©)βπ), ((π ++ β¨β(πβ0)ββ©)β(π + 1))} β (EdgβπΊ))) |
88 | 81, 82 | isclwwlk 28977 |
. . . 4
β’ (π β (ClWWalksβπΊ) β ((π β Word π β§ π β β
) β§ βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ))) |
89 | | 3anass 1096 |
. . . 4
β’ (((π β Word π β§ π β β
) β§ βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)) β ((π β Word π β§ π β β
) β§ (βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)))) |
90 | 88, 89 | bitri 275 |
. . 3
β’ (π β (ClWWalksβπΊ) β ((π β Word π β§ π β β
) β§ (βπ β
(0..^((β―βπ)
β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)))) |
91 | 90 | baib 537 |
. 2
β’ ((π β Word π β§ π β β
) β (π β (ClWWalksβπΊ) β (βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)))) |
92 | 66, 87, 91 | 3bitr4rd 312 |
1
β’ ((π β Word π β§ π β β
) β (π β (ClWWalksβπΊ) β (π ++ β¨β(πβ0)ββ©) β
(WWalksβπΊ))) |