Step | Hyp | Ref
| Expression |
1 | | nnnn0 12427 |
. . . . . . . . . 10
β’ (π β β β π β
β0) |
2 | | 2z 12542 |
. . . . . . . . . . 11
β’ 2 β
β€ |
3 | 2 | a1i 11 |
. . . . . . . . . 10
β’ (π β β β 2 β
β€) |
4 | | nn0pzuz 12837 |
. . . . . . . . . 10
β’ ((π β β0
β§ 2 β β€) β (π + 2) β
(β€β₯β2)) |
5 | 1, 3, 4 | syl2anc 585 |
. . . . . . . . 9
β’ (π β β β (π + 2) β
(β€β₯β2)) |
6 | 5 | anim2i 618 |
. . . . . . . 8
β’ ((π β π β§ π β β) β (π β π β§ (π + 2) β
(β€β₯β2))) |
7 | 6 | 3adant1 1131 |
. . . . . . 7
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π β π β§ (π + 2) β
(β€β₯β2))) |
8 | | numclwwlk.h |
. . . . . . . . 9
β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) |
9 | 8 | numclwwlkovh 29359 |
. . . . . . . 8
β’ ((π β π β§ (π + 2) β
(β€β₯β2)) β (ππ»(π + 2)) = {π€ β ((π + 2) ClWWalksN πΊ) β£ ((π€β0) = π β§ (π€β((π + 2) β 2)) β (π€β0))}) |
10 | 9 | eleq2d 2824 |
. . . . . . 7
β’ ((π β π β§ (π + 2) β
(β€β₯β2)) β (π₯ β (ππ»(π + 2)) β π₯ β {π€ β ((π + 2) ClWWalksN πΊ) β£ ((π€β0) = π β§ (π€β((π + 2) β 2)) β (π€β0))})) |
11 | 7, 10 | syl 17 |
. . . . . 6
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π₯ β (ππ»(π + 2)) β π₯ β {π€ β ((π + 2) ClWWalksN πΊ) β£ ((π€β0) = π β§ (π€β((π + 2) β 2)) β (π€β0))})) |
12 | | fveq1 6846 |
. . . . . . . . 9
β’ (π€ = π₯ β (π€β0) = (π₯β0)) |
13 | 12 | eqeq1d 2739 |
. . . . . . . 8
β’ (π€ = π₯ β ((π€β0) = π β (π₯β0) = π)) |
14 | | fveq1 6846 |
. . . . . . . . 9
β’ (π€ = π₯ β (π€β((π + 2) β 2)) = (π₯β((π + 2) β 2))) |
15 | 14, 12 | neeq12d 3006 |
. . . . . . . 8
β’ (π€ = π₯ β ((π€β((π + 2) β 2)) β (π€β0) β (π₯β((π + 2) β 2)) β (π₯β0))) |
16 | 13, 15 | anbi12d 632 |
. . . . . . 7
β’ (π€ = π₯ β (((π€β0) = π β§ (π€β((π + 2) β 2)) β (π€β0)) β ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) |
17 | 16 | elrab 3650 |
. . . . . 6
β’ (π₯ β {π€ β ((π + 2) ClWWalksN πΊ) β£ ((π€β0) = π β§ (π€β((π + 2) β 2)) β (π€β0))} β (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) |
18 | 11, 17 | bitrdi 287 |
. . . . 5
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π₯ β (ππ»(π + 2)) β (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))))) |
19 | | peano2nn 12172 |
. . . . . . . . . . . 12
β’ (π β β β (π + 1) β
β) |
20 | | nnz 12527 |
. . . . . . . . . . . . . . 15
β’ (π β β β π β
β€) |
21 | 20, 3 | zaddcld 12618 |
. . . . . . . . . . . . . 14
β’ (π β β β (π + 2) β
β€) |
22 | | uzid 12785 |
. . . . . . . . . . . . . 14
β’ ((π + 2) β β€ β
(π + 2) β
(β€β₯β(π + 2))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . 13
β’ (π β β β (π + 2) β
(β€β₯β(π + 2))) |
24 | | nncn 12168 |
. . . . . . . . . . . . . . . 16
β’ (π β β β π β
β) |
25 | | 1cnd 11157 |
. . . . . . . . . . . . . . . 16
β’ (π β β β 1 β
β) |
26 | 24, 25, 25 | addassd 11184 |
. . . . . . . . . . . . . . 15
β’ (π β β β ((π + 1) + 1) = (π + (1 + 1))) |
27 | | 1p1e2 12285 |
. . . . . . . . . . . . . . . . 17
β’ (1 + 1) =
2 |
28 | 27 | a1i 11 |
. . . . . . . . . . . . . . . 16
β’ (π β β β (1 + 1) =
2) |
29 | 28 | oveq2d 7378 |
. . . . . . . . . . . . . . 15
β’ (π β β β (π + (1 + 1)) = (π + 2)) |
30 | 26, 29 | eqtrd 2777 |
. . . . . . . . . . . . . 14
β’ (π β β β ((π + 1) + 1) = (π + 2)) |
31 | 30 | fveq2d 6851 |
. . . . . . . . . . . . 13
β’ (π β β β
(β€β₯β((π + 1) + 1)) =
(β€β₯β(π + 2))) |
32 | 23, 31 | eleqtrrd 2841 |
. . . . . . . . . . . 12
β’ (π β β β (π + 2) β
(β€β₯β((π + 1) + 1))) |
33 | 19, 32 | jca 513 |
. . . . . . . . . . 11
β’ (π β β β ((π + 1) β β β§
(π + 2) β
(β€β₯β((π + 1) + 1)))) |
34 | 33 | 3ad2ant3 1136 |
. . . . . . . . . 10
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β ((π + 1) β β β§ (π + 2) β
(β€β₯β((π + 1) + 1)))) |
35 | 34 | adantr 482 |
. . . . . . . . 9
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((π + 1) β β β§ (π + 2) β
(β€β₯β((π + 1) + 1)))) |
36 | | simprl 770 |
. . . . . . . . 9
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β π₯ β ((π + 2) ClWWalksN πΊ)) |
37 | | wwlksubclwwlk 29044 |
. . . . . . . . 9
β’ (((π + 1) β β β§
(π + 2) β
(β€β₯β((π + 1) + 1))) β (π₯ β ((π + 2) ClWWalksN πΊ) β (π₯ prefix (π + 1)) β (((π + 1) β 1) WWalksN πΊ))) |
38 | 35, 36, 37 | sylc 65 |
. . . . . . . 8
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (π₯ prefix (π + 1)) β (((π + 1) β 1) WWalksN πΊ)) |
39 | | pncan1 11586 |
. . . . . . . . . . . . . 14
β’ (π β β β ((π + 1) β 1) = π) |
40 | 39 | eqcomd 2743 |
. . . . . . . . . . . . 13
β’ (π β β β π = ((π + 1) β 1)) |
41 | 24, 40 | syl 17 |
. . . . . . . . . . . 12
β’ (π β β β π = ((π + 1) β 1)) |
42 | 41 | oveq1d 7377 |
. . . . . . . . . . 11
β’ (π β β β (π WWalksN πΊ) = (((π + 1) β 1) WWalksN πΊ)) |
43 | 42 | eleq2d 2824 |
. . . . . . . . . 10
β’ (π β β β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β (π₯ prefix (π + 1)) β (((π + 1) β 1) WWalksN πΊ))) |
44 | 43 | 3ad2ant3 1136 |
. . . . . . . . 9
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β (π₯ prefix (π + 1)) β (((π + 1) β 1) WWalksN πΊ))) |
45 | 44 | adantr 482 |
. . . . . . . 8
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β (π₯ prefix (π + 1)) β (((π + 1) β 1) WWalksN πΊ))) |
46 | 38, 45 | mpbird 257 |
. . . . . . 7
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (π₯ prefix (π + 1)) β (π WWalksN πΊ)) |
47 | | numclwwlk.v |
. . . . . . . . . . . . 13
β’ π = (VtxβπΊ) |
48 | 47 | clwwlknbp 29021 |
. . . . . . . . . . . 12
β’ (π₯ β ((π + 2) ClWWalksN πΊ) β (π₯ β Word π β§ (β―βπ₯) = (π + 2))) |
49 | | simprl 770 |
. . . . . . . . . . . . . . . 16
β’
(((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β (π₯β0) = π) |
50 | | simprr 772 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β π₯ β Word π) |
51 | | peano2nn0 12460 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β β0
β (π + 1) β
β0) |
52 | 1, 51 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β β (π + 1) β
β0) |
53 | | nnre 12167 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β β β π β
β) |
54 | 53 | lep1d 12093 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β β π β€ (π + 1)) |
55 | | elfz2nn0 13539 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β (0...(π + 1)) β (π β β0 β§ (π + 1) β β0
β§ π β€ (π + 1))) |
56 | 1, 52, 54, 55 | syl3anbrc 1344 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β β π β (0...(π + 1))) |
57 | | 2cnd 12238 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β β β 2 β
β) |
58 | | addsubass 11418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((π β β β§ 2 β
β β§ 1 β β) β ((π + 2) β 1) = (π + (2 β 1))) |
59 | | 2m1e1 12286 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (2
β 1) = 1 |
60 | 59 | oveq2i 7373 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (π + (2 β 1)) = (π + 1) |
61 | 58, 60 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((π β β β§ 2 β
β β§ 1 β β) β ((π + 2) β 1) = (π + 1)) |
62 | 24, 57, 25, 61 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β β ((π + 2) β 1) = (π + 1)) |
63 | 62 | oveq2d 7378 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β β
(0...((π + 2) β 1)) =
(0...(π +
1))) |
64 | 56, 63 | eleqtrrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β β β π β (0...((π + 2) β 1))) |
65 | | elfzp1b 13525 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((π β β€ β§ (π + 2) β β€) β
(π β (0...((π + 2) β 1)) β (π + 1) β (1...(π + 2)))) |
66 | 20, 21, 65 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β β β (π β (0...((π + 2) β 1)) β (π + 1) β (1...(π + 2)))) |
67 | 64, 66 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π β β β (π + 1) β (1...(π + 2))) |
68 | 67 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β (π + 1) β (1...(π + 2))) |
69 | | oveq2 7370 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
((β―βπ₯) =
(π + 2) β
(1...(β―βπ₯)) =
(1...(π +
2))) |
70 | 69 | eleq2d 2824 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’
((β―βπ₯) =
(π + 2) β ((π + 1) β
(1...(β―βπ₯))
β (π + 1) β
(1...(π +
2)))) |
71 | 70 | ad2antrl 727 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β ((π + 1) β (1...(β―βπ₯)) β (π + 1) β (1...(π + 2)))) |
72 | 68, 71 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β (π + 1) β (1...(β―βπ₯))) |
73 | | pfxfv0 14587 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π₯ β Word π β§ (π + 1) β (1...(β―βπ₯))) β ((π₯ prefix (π + 1))β0) = (π₯β0)) |
74 | 50, 72, 73 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β ((π₯ prefix (π + 1))β0) = (π₯β0)) |
75 | 74 | ex 414 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β β
(((β―βπ₯) =
(π + 2) β§ π₯ β Word π) β ((π₯ prefix (π + 1))β0) = (π₯β0))) |
76 | 75 | adantl 483 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β π β§ π β β) β
(((β―βπ₯) =
(π + 2) β§ π₯ β Word π) β ((π₯ prefix (π + 1))β0) = (π₯β0))) |
77 | 76 | impcom 409 |
. . . . . . . . . . . . . . . . . . 19
β’
((((β―βπ₯)
= (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β ((π₯ prefix (π + 1))β0) = (π₯β0)) |
78 | 77 | ad2antrl 727 |
. . . . . . . . . . . . . . . . . 18
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((π₯ prefix (π + 1))β0) = (π₯β0)) |
79 | | simpl 484 |
. . . . . . . . . . . . . . . . . 18
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (π₯β0) = π) |
80 | 78, 79 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((π₯ prefix (π + 1))β0) = π) |
81 | | pfxfvlsw 14590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ ((π₯ β Word π β§ (π + 1) β (1...(β―βπ₯))) β (lastSβ(π₯ prefix (π + 1))) = (π₯β((π + 1) β 1))) |
82 | 50, 72, 81 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β (lastSβ(π₯ prefix (π + 1))) = (π₯β((π + 1) β 1))) |
83 | 24, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (π β β β ((π + 1) β 1) = π) |
84 | 24, 57 | pncand 11520 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (π β β β ((π + 2) β 2) = π) |
85 | 83, 84 | eqtr4d 2780 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β β β ((π + 1) β 1) = ((π + 2) β
2)) |
86 | 85 | fveq2d 6851 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β β (π₯β((π + 1) β 1)) = (π₯β((π + 2) β 2))) |
87 | 86 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β (π₯β((π + 1) β 1)) = (π₯β((π + 2) β 2))) |
88 | 82, 87 | eqtr2d 2778 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π β β β§
((β―βπ₯) = (π + 2) β§ π₯ β Word π)) β (π₯β((π + 2) β 2)) = (lastSβ(π₯ prefix (π + 1)))) |
89 | 88 | ex 414 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π β β β
(((β―βπ₯) =
(π + 2) β§ π₯ β Word π) β (π₯β((π + 2) β 2)) = (lastSβ(π₯ prefix (π + 1))))) |
90 | 89 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β π β§ π β β) β
(((β―βπ₯) =
(π + 2) β§ π₯ β Word π) β (π₯β((π + 2) β 2)) = (lastSβ(π₯ prefix (π + 1))))) |
91 | 90 | impcom 409 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’
((((β―βπ₯)
= (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β (π₯β((π + 2) β 2)) = (lastSβ(π₯ prefix (π + 1)))) |
92 | 91 | neeq1d 3004 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
((((β―βπ₯)
= (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β ((π₯β((π + 2) β 2)) β (π₯β0) β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
93 | 92 | biimpcd 249 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯β((π + 2) β 2)) β (π₯β0) β ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
94 | 93 | adantl 483 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)) β ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
95 | 94 | impcom 409 |
. . . . . . . . . . . . . . . . . . 19
β’
(((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β (lastSβ(π₯ prefix (π + 1))) β (π₯β0)) |
96 | 95 | adantl 483 |
. . . . . . . . . . . . . . . . . 18
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (lastSβ(π₯ prefix (π + 1))) β (π₯β0)) |
97 | | neeq2 3008 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π = (π₯β0) β ((lastSβ(π₯ prefix (π + 1))) β π β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
98 | 97 | eqcoms 2745 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯β0) = π β ((lastSβ(π₯ prefix (π + 1))) β π β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
99 | 98 | adantr 482 |
. . . . . . . . . . . . . . . . . 18
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((lastSβ(π₯ prefix (π + 1))) β π β (lastSβ(π₯ prefix (π + 1))) β (π₯β0))) |
100 | 96, 99 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (lastSβ(π₯ prefix (π + 1))) β π) |
101 | 80, 100 | jca 513 |
. . . . . . . . . . . . . . . 16
β’ (((π₯β0) = π β§ ((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)) |
102 | 49, 101 | mpancom 687 |
. . . . . . . . . . . . . . 15
β’
(((((β―βπ₯) = (π + 2) β§ π₯ β Word π) β§ (π β π β§ π β β)) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)) |
103 | 102 | exp31 421 |
. . . . . . . . . . . . . 14
β’
(((β―βπ₯)
= (π + 2) β§ π₯ β Word π) β ((π β π β§ π β β) β (((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
104 | 103 | com23 86 |
. . . . . . . . . . . . 13
β’
(((β―βπ₯)
= (π + 2) β§ π₯ β Word π) β (((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)) β ((π β π β§ π β β) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
105 | 104 | ancoms 460 |
. . . . . . . . . . . 12
β’ ((π₯ β Word π β§ (β―βπ₯) = (π + 2)) β (((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)) β ((π β π β§ π β β) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
106 | 48, 105 | syl 17 |
. . . . . . . . . . 11
β’ (π₯ β ((π + 2) ClWWalksN πΊ) β (((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)) β ((π β π β§ π β β) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
107 | 106 | imp 408 |
. . . . . . . . . 10
β’ ((π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β ((π β π β§ π β β) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
108 | 107 | com12 32 |
. . . . . . . . 9
β’ ((π β π β§ π β β) β ((π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
109 | 108 | 3adant1 1131 |
. . . . . . . 8
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β ((π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
110 | 109 | imp 408 |
. . . . . . 7
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)) |
111 | 46, 110 | jca 513 |
. . . . . 6
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ (π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0)))) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
112 | 111 | ex 414 |
. . . . 5
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β ((π₯ β ((π + 2) ClWWalksN πΊ) β§ ((π₯β0) = π β§ (π₯β((π + 2) β 2)) β (π₯β0))) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
113 | 18, 112 | sylbid 239 |
. . . 4
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π₯ β (ππ»(π + 2)) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
114 | 113 | imp 408 |
. . 3
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
115 | | 3simpc 1151 |
. . . . . . 7
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π β π β§ π β β)) |
116 | 115 | adantr 482 |
. . . . . 6
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β (π β π β§ π β β)) |
117 | | numclwwlk.q |
. . . . . . 7
β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) |
118 | 47, 117 | numclwwlkovq 29360 |
. . . . . 6
β’ ((π β π β§ π β β) β (πππ) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
119 | 116, 118 | syl 17 |
. . . . 5
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β (πππ) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
120 | 119 | eleq2d 2824 |
. . . 4
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β ((π₯ prefix (π + 1)) β (πππ) β (π₯ prefix (π + 1)) β {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)})) |
121 | | fveq1 6846 |
. . . . . . 7
β’ (π€ = (π₯ prefix (π + 1)) β (π€β0) = ((π₯ prefix (π + 1))β0)) |
122 | 121 | eqeq1d 2739 |
. . . . . 6
β’ (π€ = (π₯ prefix (π + 1)) β ((π€β0) = π β ((π₯ prefix (π + 1))β0) = π)) |
123 | | fveq2 6847 |
. . . . . . 7
β’ (π€ = (π₯ prefix (π + 1)) β (lastSβπ€) = (lastSβ(π₯ prefix (π + 1)))) |
124 | 123 | neeq1d 3004 |
. . . . . 6
β’ (π€ = (π₯ prefix (π + 1)) β ((lastSβπ€) β π β (lastSβ(π₯ prefix (π + 1))) β π)) |
125 | 122, 124 | anbi12d 632 |
. . . . 5
β’ (π€ = (π₯ prefix (π + 1)) β (((π€β0) = π β§ (lastSβπ€) β π) β (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
126 | 125 | elrab 3650 |
. . . 4
β’ ((π₯ prefix (π + 1)) β {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π))) |
127 | 120, 126 | bitrdi 287 |
. . 3
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β ((π₯ prefix (π + 1)) β (πππ) β ((π₯ prefix (π + 1)) β (π WWalksN πΊ) β§ (((π₯ prefix (π + 1))β0) = π β§ (lastSβ(π₯ prefix (π + 1))) β π)))) |
128 | 114, 127 | mpbird 257 |
. 2
β’ (((πΊ β FriendGraph β§ π β π β§ π β β) β§ π₯ β (ππ»(π + 2))) β (π₯ prefix (π + 1)) β (πππ)) |
129 | | numclwwlk.r |
. 2
β’ π
= (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) |
130 | 128, 129 | fmptd 7067 |
1
β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β π
:(ππ»(π + 2))βΆ(πππ)) |