Step | Hyp | Ref
| Expression |
1 | | clwwlknnn 29286 |
. . 3
β’ ((π ++ β¨βπββ©) β (π ClWWalksN πΊ) β π β β) |
2 | | clwwlkext2edg.v |
. . . . 5
β’ π = (VtxβπΊ) |
3 | | clwwlkext2edg.e |
. . . . 5
β’ πΈ = (EdgβπΊ) |
4 | 2, 3 | isclwwlknx 29289 |
. . . 4
β’ (π β β β ((π ++ β¨βπββ©) β (π ClWWalksN πΊ) β (((π ++ β¨βπββ©) β Word π β§ βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β§ (β―β(π ++ β¨βπββ©)) = π))) |
5 | | ige2m2fzo 13695 |
. . . . . . . . . . . . . . 15
β’ (π β
(β€β₯β2) β (π β 2) β (0..^(π β 1))) |
6 | 5 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β (π β 2) β
(0..^(π β
1))) |
7 | 6 | adantr 482 |
. . . . . . . . . . . . 13
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (π β 2) β (0..^(π β 1))) |
8 | | oveq1 7416 |
. . . . . . . . . . . . . . . 16
β’
((β―β(π
++ β¨βπββ©)) = π β ((β―β(π ++ β¨βπββ©)) β 1) = (π β 1)) |
9 | 8 | oveq2d 7425 |
. . . . . . . . . . . . . . 15
β’
((β―β(π
++ β¨βπββ©)) = π β (0..^((β―β(π ++ β¨βπββ©)) β 1)) =
(0..^(π β
1))) |
10 | 9 | eleq2d 2820 |
. . . . . . . . . . . . . 14
β’
((β―β(π
++ β¨βπββ©)) = π β ((π β 2) β
(0..^((β―β(π ++
β¨βπββ©)) β 1)) β (π β 2) β (0..^(π β 1)))) |
11 | 10 | adantl 483 |
. . . . . . . . . . . . 13
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π β 2) β
(0..^((β―β(π ++
β¨βπββ©)) β 1)) β (π β 2) β (0..^(π β 1)))) |
12 | 7, 11 | mpbird 257 |
. . . . . . . . . . . 12
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (π β 2) β
(0..^((β―β(π ++
β¨βπββ©)) β
1))) |
13 | | fveq2 6892 |
. . . . . . . . . . . . . . 15
β’ (π = (π β 2) β ((π ++ β¨βπββ©)βπ) = ((π ++ β¨βπββ©)β(π β 2))) |
14 | | fvoveq1 7432 |
. . . . . . . . . . . . . . 15
β’ (π = (π β 2) β ((π ++ β¨βπββ©)β(π + 1)) = ((π ++ β¨βπββ©)β((π β 2) + 1))) |
15 | 13, 14 | preq12d 4746 |
. . . . . . . . . . . . . 14
β’ (π = (π β 2) β {((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} = {((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))}) |
16 | 15 | eleq1d 2819 |
. . . . . . . . . . . . 13
β’ (π = (π β 2) β ({((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β {((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))} β πΈ)) |
17 | 16 | rspcv 3609 |
. . . . . . . . . . . 12
β’ ((π β 2) β
(0..^((β―β(π ++
β¨βπββ©)) β 1)) β
(βπ β
(0..^((β―β(π ++
β¨βπββ©)) β 1)){((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β {((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))} β πΈ)) |
18 | 12, 17 | syl 17 |
. . . . . . . . . . 11
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β {((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))} β πΈ)) |
19 | | wrdlenccats1lenm1 14572 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β Word π β ((β―β(π ++ β¨βπββ©)) β 1) =
(β―βπ)) |
20 | 19 | eqcomd 2739 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β Word π β (β―βπ) = ((β―β(π ++ β¨βπββ©)) β 1)) |
21 | 20 | adantr 482 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β Word π β§ π β π) β (β―βπ) = ((β―β(π ++ β¨βπββ©)) β 1)) |
22 | 21, 8 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β Word π β§ π β π) β§ (β―β(π ++ β¨βπββ©)) = π) β (β―βπ) = (π β 1)) |
23 | 22 | ex 414 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β Word π β§ π β π) β ((β―β(π ++ β¨βπββ©)) = π β (β―βπ) = (π β 1))) |
24 | 23 | 3adant3 1133 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β (β―βπ) = (π β 1))) |
25 | | eluzelcn 12834 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β
(β€β₯β2) β π β β) |
26 | | 1cnd 11209 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β
(β€β₯β2) β 1 β β) |
27 | 25, 26, 26 | subsub4d 11602 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β
(β€β₯β2) β ((π β 1) β 1) = (π β (1 + 1))) |
28 | | 1p1e2 12337 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (1 + 1) =
2 |
29 | 28 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β
(β€β₯β2) β (1 + 1) = 2) |
30 | 29 | oveq2d 7425 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β
(β€β₯β2) β (π β (1 + 1)) = (π β 2)) |
31 | 27, 30 | eqtr2d 2774 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β
(β€β₯β2) β (π β 2) = ((π β 1) β 1)) |
32 | 31 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β (π β 2) =
((π β 1) β
1)) |
33 | | oveq1 7416 |
. . . . . . . . . . . . . . . . . . . 20
β’
((β―βπ) =
(π β 1) β
((β―βπ) β
1) = ((π β 1) β
1)) |
34 | 33 | eqcomd 2739 |
. . . . . . . . . . . . . . . . . . 19
β’
((β―βπ) =
(π β 1) β
((π β 1) β 1) =
((β―βπ) β
1)) |
35 | 32, 34 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
(π β 2) =
((β―βπ) β
1)) |
36 | 35 | ex 414 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―βπ) =
(π β 1) β (π β 2) =
((β―βπ) β
1))) |
37 | 24, 36 | syld 47 |
. . . . . . . . . . . . . . . 16
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β (π β 2) = ((β―βπ) β 1))) |
38 | 37 | imp 408 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (π β 2) = ((β―βπ) β 1)) |
39 | 38 | fveq2d 6896 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β(π β 2)) = ((π ++ β¨βπββ©)β((β―βπ) β 1))) |
40 | | simpl1 1192 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β π β Word π) |
41 | | s1cl 14552 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β π β β¨βπββ© β Word π) |
42 | 41 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β β¨βπββ© β Word π) |
43 | 42 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
β¨βπββ©
β Word π) |
44 | | eluz2 12828 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π β
(β€β₯β2) β (2 β β€ β§ π β β€ β§ 2 β€
π)) |
45 | | zre 12562 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β€ β π β
β) |
46 | | 1red 11215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β€
π) β 1 β
β) |
47 | | 2re 12286 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ 2 β
β |
48 | 47 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β€
π) β 2 β
β) |
49 | | simpl 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β€
π) β π β β) |
50 | | 1lt2 12383 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ 1 <
2 |
51 | 50 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β€
π) β 1 <
2) |
52 | | simpr 486 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β€
π) β 2 β€ π) |
53 | 46, 48, 49, 51, 52 | ltletrd 11374 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((π β β β§ 2 β€
π) β 1 < π) |
54 | | 1red 11215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (π β β β 1 β
β) |
55 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (π β β β π β
β) |
56 | 54, 55 | posdifd 11801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (π β β β (1 <
π β 0 < (π β 1))) |
57 | 56 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((π β β β§ 2 β€
π) β (1 < π β 0 < (π β 1))) |
58 | 53, 57 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ ((π β β β§ 2 β€
π) β 0 < (π β 1)) |
59 | 58 | ex 414 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β β (2 β€
π β 0 < (π β 1))) |
60 | 45, 59 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β β€ β (2 β€
π β 0 < (π β 1))) |
61 | 60 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (2 β
β€ β (π β
β€ β (2 β€ π
β 0 < (π β
1)))) |
62 | 61 | 3imp 1112 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((2
β β€ β§ π
β β€ β§ 2 β€ π) β 0 < (π β 1)) |
63 | 44, 62 | sylbi 216 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β
(β€β₯β2) β 0 < (π β 1)) |
64 | 63 | ad2antlr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β Word π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β 0
< (π β
1)) |
65 | | breq2 5153 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’
((β―βπ) =
(π β 1) β (0
< (β―βπ)
β 0 < (π β
1))) |
66 | 65 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β Word π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β (0
< (β―βπ)
β 0 < (π β
1))) |
67 | 64, 66 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β Word π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β 0
< (β―βπ)) |
68 | | hashneq0 14324 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β Word π β (0 < (β―βπ) β π β β
)) |
69 | 68 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π β Word π β§ π β (β€β₯β2))
β (0 < (β―βπ) β π β β
)) |
70 | 69 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β Word π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β (0
< (β―βπ)
β π β
β
)) |
71 | 67, 70 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β Word π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β π β β
) |
72 | 71 | 3adantl2 1168 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β π β β
) |
73 | 40, 43, 72 | 3jca 1129 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
(π β Word π β§ β¨βπββ© β Word π β§ π β β
)) |
74 | 73 | ex 414 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―βπ) =
(π β 1) β (π β Word π β§ β¨βπββ© β Word π β§ π β β
))) |
75 | 24, 74 | syld 47 |
. . . . . . . . . . . . . . . 16
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β (π β Word π β§ β¨βπββ© β Word π β§ π β β
))) |
76 | 75 | imp 408 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (π β Word π β§ β¨βπββ© β Word π β§ π β β
)) |
77 | | ccatval1lsw 14534 |
. . . . . . . . . . . . . . 15
β’ ((π β Word π β§ β¨βπββ© β Word π β§ π β β
) β ((π ++ β¨βπββ©)β((β―βπ) β 1)) =
(lastSβπ)) |
78 | 76, 77 | syl 17 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β((β―βπ) β 1)) =
(lastSβπ)) |
79 | 39, 78 | eqtrd 2773 |
. . . . . . . . . . . . 13
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β(π β 2)) = (lastSβπ)) |
80 | | 2m1e1 12338 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (2
β 1) = 1 |
81 | 80 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β
(β€β₯β2) β (2 β 1) = 1) |
82 | 81 | eqcomd 2739 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β
(β€β₯β2) β 1 = (2 β 1)) |
83 | 82 | oveq2d 7425 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β
(β€β₯β2) β (π β 1) = (π β (2 β 1))) |
84 | | 2cnd 12290 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β
(β€β₯β2) β 2 β β) |
85 | 25, 84, 26 | subsubd 11599 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β
(β€β₯β2) β (π β (2 β 1)) = ((π β 2) +
1)) |
86 | 83, 85 | eqtr2d 2774 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β
(β€β₯β2) β ((π β 2) + 1) = (π β 1)) |
87 | 86 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((π β 2) + 1)
= (π β
1)) |
88 | | eqeq2 2745 |
. . . . . . . . . . . . . . . . . 18
β’
((β―βπ) =
(π β 1) β
(((π β 2) + 1) =
(β―βπ) β
((π β 2) + 1) =
(π β
1))) |
89 | 87, 88 | syl5ibrcom 246 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―βπ) =
(π β 1) β
((π β 2) + 1) =
(β―βπ))) |
90 | 24, 89 | syld 47 |
. . . . . . . . . . . . . . . 16
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β ((π β 2) + 1) = (β―βπ))) |
91 | 90 | imp 408 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π β 2) + 1) = (β―βπ)) |
92 | 91 | fveq2d 6896 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β((π β 2) + 1)) = ((π ++ β¨βπββ©)β(β―βπ))) |
93 | | id 22 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π) β (π β Word π β§ π β π)) |
94 | 93 | 3adant3 1133 |
. . . . . . . . . . . . . . . 16
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β (π β Word π β§ π β π)) |
95 | 94 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (π β Word π β§ π β π)) |
96 | | ccatws1ls 14583 |
. . . . . . . . . . . . . . 15
β’ ((π β Word π β§ π β π) β ((π ++ β¨βπββ©)β(β―βπ)) = π) |
97 | 95, 96 | syl 17 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β(β―βπ)) = π) |
98 | 92, 97 | eqtrd 2773 |
. . . . . . . . . . . . 13
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ((π ++ β¨βπββ©)β((π β 2) + 1)) = π) |
99 | 79, 98 | preq12d 4746 |
. . . . . . . . . . . 12
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β {((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))} = {(lastSβπ), π}) |
100 | 99 | eleq1d 2819 |
. . . . . . . . . . 11
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β ({((π ++ β¨βπββ©)β(π β 2)), ((π ++ β¨βπββ©)β((π β 2) + 1))} β πΈ β {(lastSβπ), π} β πΈ)) |
101 | 18, 100 | sylibd 238 |
. . . . . . . . . 10
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―β(π ++
β¨βπββ©)) = π) β (βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β {(lastSβπ), π} β πΈ)) |
102 | 101 | ex 414 |
. . . . . . . . 9
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β (βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β {(lastSβπ), π} β πΈ))) |
103 | 102 | com13 88 |
. . . . . . . 8
β’
(βπ β
(0..^((β―β(π ++
β¨βπββ©)) β 1)){((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β ((β―β(π ++ β¨βπββ©)) = π β ((π β Word π β§ π β π β§ π β (β€β₯β2))
β {(lastSβπ),
π} β πΈ))) |
104 | 103 | 3ad2ant2 1135 |
. . . . . . 7
β’ (((π ++ β¨βπββ©) β Word π β§ βπ β
(0..^((β―β(π ++
β¨βπββ©)) β 1)){((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β ((β―β(π ++ β¨βπββ©)) = π β ((π β Word π β§ π β π β§ π β (β€β₯β2))
β {(lastSβπ),
π} β πΈ))) |
105 | 104 | imp31 419 |
. . . . . 6
β’
(((((π ++
β¨βπββ©) β Word π β§ βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β§ (β―β(π ++ β¨βπββ©)) = π) β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β {(lastSβπ),
π} β πΈ) |
106 | 94 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
(π β Word π β§ π β π)) |
107 | | lswccats1 14584 |
. . . . . . . . . . . . . . 15
β’ ((π β Word π β§ π β π) β (lastSβ(π ++ β¨βπββ©)) = π) |
108 | 106, 107 | syl 17 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
(lastSβ(π ++
β¨βπββ©)) = π) |
109 | 63 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β 0 < (π β
1)) |
110 | 109 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β 0
< (π β
1)) |
111 | 65 | adantl 483 |
. . . . . . . . . . . . . . . 16
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β (0
< (β―βπ)
β 0 < (π β
1))) |
112 | 110, 111 | mpbird 257 |
. . . . . . . . . . . . . . 15
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β 0
< (β―βπ)) |
113 | | ccatfv0 14533 |
. . . . . . . . . . . . . . 15
β’ ((π β Word π β§ β¨βπββ© β Word π β§ 0 < (β―βπ)) β ((π ++ β¨βπββ©)β0) = (πβ0)) |
114 | 40, 43, 112, 113 | syl3anc 1372 |
. . . . . . . . . . . . . 14
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
((π ++ β¨βπββ©)β0) = (πβ0)) |
115 | 108, 114 | preq12d 4746 |
. . . . . . . . . . . . 13
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (β―βπ) =
(π β 1)) β
{(lastSβ(π ++
β¨βπββ©)), ((π ++ β¨βπββ©)β0)} = {π, (πβ0)}) |
116 | 115 | ex 414 |
. . . . . . . . . . . 12
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―βπ) =
(π β 1) β
{(lastSβ(π ++
β¨βπββ©)), ((π ++ β¨βπββ©)β0)} = {π, (πβ0)})) |
117 | 24, 116 | syld 47 |
. . . . . . . . . . 11
β’ ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ((β―β(π
++ β¨βπββ©)) = π β {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} = {π, (πβ0)})) |
118 | 117 | impcom 409 |
. . . . . . . . . 10
β’
(((β―β(π
++ β¨βπββ©)) = π β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β {(lastSβ(π ++
β¨βπββ©)), ((π ++ β¨βπββ©)β0)} = {π, (πβ0)}) |
119 | 118 | eleq1d 2819 |
. . . . . . . . 9
β’
(((β―β(π
++ β¨βπββ©)) = π β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β ({(lastSβ(π ++
β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ β {π, (πβ0)} β πΈ)) |
120 | 119 | biimpcd 248 |
. . . . . . . 8
β’
({(lastSβ(π ++
β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ β (((β―β(π ++ β¨βπββ©)) = π β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β {π, (πβ0)} β πΈ)) |
121 | 120 | 3ad2ant3 1136 |
. . . . . . 7
β’ (((π ++ β¨βπββ©) β Word π β§ βπ β
(0..^((β―β(π ++
β¨βπββ©)) β 1)){((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β (((β―β(π ++ β¨βπββ©)) = π β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β {π, (πβ0)} β πΈ)) |
122 | 121 | impl 457 |
. . . . . 6
β’
(((((π ++
β¨βπββ©) β Word π β§ βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β§ (β―β(π ++ β¨βπββ©)) = π) β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β {π, (πβ0)} β πΈ) |
123 | 105, 122 | jca 513 |
. . . . 5
β’
(((((π ++
β¨βπββ©) β Word π β§ βπ β (0..^((β―β(π ++ β¨βπββ©)) β
1)){((π ++
β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β§ (β―β(π ++ β¨βπββ©)) = π) β§ (π β Word π β§ π β π β§ π β (β€β₯β2)))
β ({(lastSβπ),
π} β πΈ β§ {π, (πβ0)} β πΈ)) |
124 | 123 | ex 414 |
. . . 4
β’ ((((π ++ β¨βπββ©) β Word π β§ βπ β
(0..^((β―β(π ++
β¨βπββ©)) β 1)){((π ++ β¨βπββ©)βπ), ((π ++ β¨βπββ©)β(π + 1))} β πΈ β§ {(lastSβ(π ++ β¨βπββ©)), ((π ++ β¨βπββ©)β0)} β πΈ) β§ (β―β(π ++ β¨βπββ©)) = π) β ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ({(lastSβπ),
π} β πΈ β§ {π, (πβ0)} β πΈ))) |
125 | 4, 124 | syl6bi 253 |
. . 3
β’ (π β β β ((π ++ β¨βπββ©) β (π ClWWalksN πΊ) β ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ({(lastSβπ),
π} β πΈ β§ {π, (πβ0)} β πΈ)))) |
126 | 1, 125 | mpcom 38 |
. 2
β’ ((π ++ β¨βπββ©) β (π ClWWalksN πΊ) β ((π β Word π β§ π β π β§ π β (β€β₯β2))
β ({(lastSβπ),
π} β πΈ β§ {π, (πβ0)} β πΈ))) |
127 | 126 | impcom 409 |
1
β’ (((π β Word π β§ π β π β§ π β (β€β₯β2))
β§ (π ++
β¨βπββ©) β (π ClWWalksN πΊ)) β ({(lastSβπ), π} β πΈ β§ {π, (πβ0)} β πΈ)) |