Step | Hyp | Ref
| Expression |
1 | | wwlksnextbij0.v |
. . 3
β’ π = (VtxβπΊ) |
2 | | wwlksnextbij0.e |
. . 3
β’ πΈ = (EdgβπΊ) |
3 | | wwlksnextbij0.d |
. . 3
β’ π· = {π€ β Word π β£ ((β―βπ€) = (π + 2) β§ (π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ)} |
4 | | wwlksnextbij0.r |
. . 3
β’ π
= {π β π β£ {(lastSβπ), π} β πΈ} |
5 | | wwlksnextbij0.f |
. . 3
β’ πΉ = (π‘ β π· β¦ (lastSβπ‘)) |
6 | 1, 2, 3, 4, 5 | wwlksnextfun 28940 |
. 2
β’ (π β β0
β πΉ:π·βΆπ
) |
7 | | fveq2 6862 |
. . . . . . 7
β’ (π‘ = π β (lastSβπ‘) = (lastSβπ)) |
8 | | fvex 6875 |
. . . . . . 7
β’
(lastSβπ)
β V |
9 | 7, 5, 8 | fvmpt 6968 |
. . . . . 6
β’ (π β π· β (πΉβπ) = (lastSβπ)) |
10 | | fveq2 6862 |
. . . . . . 7
β’ (π‘ = π₯ β (lastSβπ‘) = (lastSβπ₯)) |
11 | | fvex 6875 |
. . . . . . 7
β’
(lastSβπ₯)
β V |
12 | 10, 5, 11 | fvmpt 6968 |
. . . . . 6
β’ (π₯ β π· β (πΉβπ₯) = (lastSβπ₯)) |
13 | 9, 12 | eqeqan12d 2745 |
. . . . 5
β’ ((π β π· β§ π₯ β π·) β ((πΉβπ) = (πΉβπ₯) β (lastSβπ) = (lastSβπ₯))) |
14 | 13 | adantl 482 |
. . . 4
β’ ((π β β0
β§ (π β π· β§ π₯ β π·)) β ((πΉβπ) = (πΉβπ₯) β (lastSβπ) = (lastSβπ₯))) |
15 | | fveqeq2 6871 |
. . . . . . . 8
β’ (π€ = π β ((β―βπ€) = (π + 2) β (β―βπ) = (π + 2))) |
16 | | oveq1 7384 |
. . . . . . . . 9
β’ (π€ = π β (π€ prefix (π + 1)) = (π prefix (π + 1))) |
17 | 16 | eqeq1d 2733 |
. . . . . . . 8
β’ (π€ = π β ((π€ prefix (π + 1)) = π β (π prefix (π + 1)) = π)) |
18 | | fveq2 6862 |
. . . . . . . . . 10
β’ (π€ = π β (lastSβπ€) = (lastSβπ)) |
19 | 18 | preq2d 4721 |
. . . . . . . . 9
β’ (π€ = π β {(lastSβπ), (lastSβπ€)} = {(lastSβπ), (lastSβπ)}) |
20 | 19 | eleq1d 2817 |
. . . . . . . 8
β’ (π€ = π β ({(lastSβπ), (lastSβπ€)} β πΈ β {(lastSβπ), (lastSβπ)} β πΈ)) |
21 | 15, 17, 20 | 3anbi123d 1436 |
. . . . . . 7
β’ (π€ = π β (((β―βπ€) = (π + 2) β§ (π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ) β ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ))) |
22 | 21, 3 | elrab2 3666 |
. . . . . 6
β’ (π β π· β (π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ))) |
23 | | fveqeq2 6871 |
. . . . . . . 8
β’ (π€ = π₯ β ((β―βπ€) = (π + 2) β (β―βπ₯) = (π + 2))) |
24 | | oveq1 7384 |
. . . . . . . . 9
β’ (π€ = π₯ β (π€ prefix (π + 1)) = (π₯ prefix (π + 1))) |
25 | 24 | eqeq1d 2733 |
. . . . . . . 8
β’ (π€ = π₯ β ((π€ prefix (π + 1)) = π β (π₯ prefix (π + 1)) = π)) |
26 | | fveq2 6862 |
. . . . . . . . . 10
β’ (π€ = π₯ β (lastSβπ€) = (lastSβπ₯)) |
27 | 26 | preq2d 4721 |
. . . . . . . . 9
β’ (π€ = π₯ β {(lastSβπ), (lastSβπ€)} = {(lastSβπ), (lastSβπ₯)}) |
28 | 27 | eleq1d 2817 |
. . . . . . . 8
β’ (π€ = π₯ β ({(lastSβπ), (lastSβπ€)} β πΈ β {(lastSβπ), (lastSβπ₯)} β πΈ)) |
29 | 23, 25, 28 | 3anbi123d 1436 |
. . . . . . 7
β’ (π€ = π₯ β (((β―βπ€) = (π + 2) β§ (π€ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ€)} β πΈ) β ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) |
30 | 29, 3 | elrab2 3666 |
. . . . . 6
β’ (π₯ β π· β (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) |
31 | | eqtr3 2757 |
. . . . . . . . . . . . . . . . 17
β’
(((β―βπ)
= (π + 2) β§
(β―βπ₯) = (π + 2)) β
(β―βπ) =
(β―βπ₯)) |
32 | 31 | expcom 414 |
. . . . . . . . . . . . . . . 16
β’
((β―βπ₯) =
(π + 2) β
((β―βπ) = (π + 2) β
(β―βπ) =
(β―βπ₯))) |
33 | 32 | 3ad2ant1 1133 |
. . . . . . . . . . . . . . 15
β’
(((β―βπ₯)
= (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ) β ((β―βπ) = (π + 2) β (β―βπ) = (β―βπ₯))) |
34 | 33 | adantl 482 |
. . . . . . . . . . . . . 14
β’ ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β ((β―βπ) = (π + 2) β (β―βπ) = (β―βπ₯))) |
35 | 34 | com12 32 |
. . . . . . . . . . . . 13
β’
((β―βπ) =
(π + 2) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (β―βπ) = (β―βπ₯))) |
36 | 35 | 3ad2ant1 1133 |
. . . . . . . . . . . 12
β’
(((β―βπ)
= (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (β―βπ) = (β―βπ₯))) |
37 | 36 | adantl 482 |
. . . . . . . . . . 11
β’ ((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (β―βπ) = (β―βπ₯))) |
38 | 37 | imp 407 |
. . . . . . . . . 10
β’ (((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β (β―βπ) = (β―βπ₯)) |
39 | 38 | adantr 481 |
. . . . . . . . 9
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β
(β―βπ) =
(β―βπ₯)) |
40 | 39 | adantr 481 |
. . . . . . . 8
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
(β―βπ) =
(β―βπ₯)) |
41 | | simpr 485 |
. . . . . . . 8
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
(lastSβπ) =
(lastSβπ₯)) |
42 | | eqtr3 2757 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π prefix (π + 1)) = π β§ (π₯ prefix (π + 1)) = π) β (π prefix (π + 1)) = (π₯ prefix (π + 1))) |
43 | | 1e2m1 12304 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ 1 = (2
β 1) |
44 | 43 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β0
β 1 = (2 β 1)) |
45 | 44 | oveq2d 7393 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β0
β (π + 1) = (π + (2 β
1))) |
46 | | nn0cn 12447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β0
β π β
β) |
47 | | 2cnd 12255 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β0
β 2 β β) |
48 | | 1cnd 11174 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β β0
β 1 β β) |
49 | 46, 47, 48 | addsubassd 11556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β β0
β ((π + 2) β 1)
= (π + (2 β
1))) |
50 | 45, 49 | eqtr4d 2774 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β β0
β (π + 1) = ((π + 2) β
1)) |
51 | 50 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β β0
β§ (β―βπ) =
(π + 2)) β (π + 1) = ((π + 2) β 1)) |
52 | | oveq1 7384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
((β―βπ) =
(π + 2) β
((β―βπ) β
1) = ((π + 2) β
1)) |
53 | 52 | eqeq2d 2742 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
((β―βπ) =
(π + 2) β ((π + 1) = ((β―βπ) β 1) β (π + 1) = ((π + 2) β 1))) |
54 | 53 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β β0
β§ (β―βπ) =
(π + 2)) β ((π + 1) = ((β―βπ) β 1) β (π + 1) = ((π + 2) β 1))) |
55 | 51, 54 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β β0
β§ (β―βπ) =
(π + 2)) β (π + 1) = ((β―βπ) β 1)) |
56 | | oveq2 7385 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π + 1) = ((β―βπ) β 1) β (π prefix (π + 1)) = (π prefix ((β―βπ) β 1))) |
57 | | oveq2 7385 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π + 1) = ((β―βπ) β 1) β (π₯ prefix (π + 1)) = (π₯ prefix ((β―βπ) β 1))) |
58 | 56, 57 | eqeq12d 2747 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π + 1) = ((β―βπ) β 1) β ((π prefix (π + 1)) = (π₯ prefix (π + 1)) β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)))) |
59 | 55, 58 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β0
β§ (β―βπ) =
(π + 2)) β ((π prefix (π + 1)) = (π₯ prefix (π + 1)) β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)))) |
60 | 59 | biimpd 228 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π β β0
β§ (β―βπ) =
(π + 2)) β ((π prefix (π + 1)) = (π₯ prefix (π + 1)) β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)))) |
61 | 60 | ex 413 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β0
β ((β―βπ) =
(π + 2) β ((π prefix (π + 1)) = (π₯ prefix (π + 1)) β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1))))) |
62 | 61 | com13 88 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π prefix (π + 1)) = (π₯ prefix (π + 1)) β ((β―βπ) = (π + 2) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
63 | 42, 62 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π prefix (π + 1)) = π β§ (π₯ prefix (π + 1)) = π) β ((β―βπ) = (π + 2) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
64 | 63 | ex 413 |
. . . . . . . . . . . . . . . . . 18
β’ ((π prefix (π + 1)) = π β ((π₯ prefix (π + 1)) = π β ((β―βπ) = (π + 2) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1)))))) |
65 | 64 | com23 86 |
. . . . . . . . . . . . . . . . 17
β’ ((π prefix (π + 1)) = π β ((β―βπ) = (π + 2) β ((π₯ prefix (π + 1)) = π β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1)))))) |
66 | 65 | impcom 408 |
. . . . . . . . . . . . . . . 16
β’
(((β―βπ)
= (π + 2) β§ (π prefix (π + 1)) = π) β ((π₯ prefix (π + 1)) = π β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
67 | 66 | com12 32 |
. . . . . . . . . . . . . . 15
β’ ((π₯ prefix (π + 1)) = π β (((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
68 | 67 | 3ad2ant2 1134 |
. . . . . . . . . . . . . 14
β’
(((β―βπ₯)
= (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ) β (((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
69 | 68 | adantl 482 |
. . . . . . . . . . . . 13
β’ ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
70 | 69 | com12 32 |
. . . . . . . . . . . 12
β’
(((β―βπ)
= (π + 2) β§ (π prefix (π + 1)) = π) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
71 | 70 | 3adant3 1132 |
. . . . . . . . . . 11
β’
(((β―βπ)
= (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
72 | 71 | adantl 482 |
. . . . . . . . . 10
β’ ((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (π β β0 β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β
1))))) |
73 | 72 | imp31 418 |
. . . . . . . . 9
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1))) |
74 | 73 | adantr 481 |
. . . . . . . 8
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
(π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1))) |
75 | | simpl 483 |
. . . . . . . . . . . . 13
β’ ((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β π β Word π) |
76 | | simpl 483 |
. . . . . . . . . . . . 13
β’ ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β π₯ β Word π) |
77 | 75, 76 | anim12i 613 |
. . . . . . . . . . . 12
β’ (((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β (π β Word π β§ π₯ β Word π)) |
78 | 77 | adantr 481 |
. . . . . . . . . . 11
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β (π β Word π β§ π₯ β Word π)) |
79 | | nn0re 12446 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β0
β π β
β) |
80 | | 2re 12251 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ 2 β
β |
81 | 80 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β0
β 2 β β) |
82 | | nn0ge0 12462 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β0
β 0 β€ π) |
83 | | 2pos 12280 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ 0 <
2 |
84 | 83 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β0
β 0 < 2) |
85 | 79, 81, 82, 84 | addgegt0d 11752 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β β0
β 0 < (π +
2)) |
86 | 85 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
β’
(((β―βπ)
= (π + 2) β§ π β β0)
β 0 < (π +
2)) |
87 | | breq2 5129 |
. . . . . . . . . . . . . . . . . . . 20
β’
((β―βπ) =
(π + 2) β (0 <
(β―βπ) β 0
< (π +
2))) |
88 | 87 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
β’
(((β―βπ)
= (π + 2) β§ π β β0)
β (0 < (β―βπ) β 0 < (π + 2))) |
89 | 86, 88 | mpbird 256 |
. . . . . . . . . . . . . . . . . 18
β’
(((β―βπ)
= (π + 2) β§ π β β0)
β 0 < (β―βπ)) |
90 | | hashgt0n0 14290 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β Word π β§ 0 < (β―βπ)) β π β β
) |
91 | 89, 90 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
β’ ((π β Word π β§ ((β―βπ) = (π + 2) β§ π β β0)) β π β β
) |
92 | 91 | exp32 421 |
. . . . . . . . . . . . . . . 16
β’ (π β Word π β ((β―βπ) = (π + 2) β (π β β0 β π β
β
))) |
93 | 92 | com12 32 |
. . . . . . . . . . . . . . 15
β’
((β―βπ) =
(π + 2) β (π β Word π β (π β β0 β π β
β
))) |
94 | 93 | 3ad2ant1 1133 |
. . . . . . . . . . . . . 14
β’
(((β―βπ)
= (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ) β (π β Word π β (π β β0 β π β
β
))) |
95 | 94 | impcom 408 |
. . . . . . . . . . . . 13
β’ ((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β (π β β0 β π β β
)) |
96 | 95 | adantr 481 |
. . . . . . . . . . . 12
β’ (((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β (π β β0 β π β β
)) |
97 | 96 | imp 407 |
. . . . . . . . . . 11
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β π β β
) |
98 | 85 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
β’
(((β―βπ₯)
= (π + 2) β§ π β β0)
β 0 < (π +
2)) |
99 | | breq2 5129 |
. . . . . . . . . . . . . . . . . . . 20
β’
((β―βπ₯) =
(π + 2) β (0 <
(β―βπ₯) β 0
< (π +
2))) |
100 | 99 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
β’
(((β―βπ₯)
= (π + 2) β§ π β β0)
β (0 < (β―βπ₯) β 0 < (π + 2))) |
101 | 98, 100 | mpbird 256 |
. . . . . . . . . . . . . . . . . 18
β’
(((β―βπ₯)
= (π + 2) β§ π β β0)
β 0 < (β―βπ₯)) |
102 | | hashgt0n0 14290 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β Word π β§ 0 < (β―βπ₯)) β π₯ β β
) |
103 | 101, 102 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
β’ ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ π β β0)) β π₯ β β
) |
104 | 103 | exp32 421 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β Word π β ((β―βπ₯) = (π + 2) β (π β β0 β π₯ β
β
))) |
105 | 104 | com12 32 |
. . . . . . . . . . . . . . 15
β’
((β―βπ₯) =
(π + 2) β (π₯ β Word π β (π β β0 β π₯ β
β
))) |
106 | 105 | 3ad2ant1 1133 |
. . . . . . . . . . . . . 14
β’
(((β―βπ₯)
= (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ) β (π₯ β Word π β (π β β0 β π₯ β
β
))) |
107 | 106 | impcom 408 |
. . . . . . . . . . . . 13
β’ ((π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ)) β (π β β0 β π₯ β β
)) |
108 | 107 | adantl 482 |
. . . . . . . . . . . 12
β’ (((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β (π β β0 β π₯ β β
)) |
109 | 108 | imp 407 |
. . . . . . . . . . 11
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β π₯ β β
) |
110 | 78, 97, 109 | jca32 516 |
. . . . . . . . . 10
β’ ((((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β ((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
))) |
111 | 110 | adantr 481 |
. . . . . . . . 9
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
))) |
112 | | simpl 483 |
. . . . . . . . . . . 12
β’ ((π β Word π β§ π₯ β Word π) β π β Word π) |
113 | 112 | adantr 481 |
. . . . . . . . . . 11
β’ (((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
)) β π β Word π) |
114 | | simpr 485 |
. . . . . . . . . . . 12
β’ ((π β Word π β§ π₯ β Word π) β π₯ β Word π) |
115 | 114 | adantr 481 |
. . . . . . . . . . 11
β’ (((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
)) β π₯ β Word π) |
116 | | hashneq0 14289 |
. . . . . . . . . . . . . . . 16
β’ (π β Word π β (0 < (β―βπ) β π β β
)) |
117 | 116 | biimprd 247 |
. . . . . . . . . . . . . . 15
β’ (π β Word π β (π β β
β 0 <
(β―βπ))) |
118 | 117 | adantr 481 |
. . . . . . . . . . . . . 14
β’ ((π β Word π β§ π₯ β Word π) β (π β β
β 0 <
(β―βπ))) |
119 | 118 | com12 32 |
. . . . . . . . . . . . 13
β’ (π β β
β ((π β Word π β§ π₯ β Word π) β 0 < (β―βπ))) |
120 | 119 | adantr 481 |
. . . . . . . . . . . 12
β’ ((π β β
β§ π₯ β β
) β ((π β Word π β§ π₯ β Word π) β 0 < (β―βπ))) |
121 | 120 | impcom 408 |
. . . . . . . . . . 11
β’ (((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
)) β 0 <
(β―βπ)) |
122 | | pfxsuff1eqwrdeq 14614 |
. . . . . . . . . . 11
β’ ((π β Word π β§ π₯ β Word π β§ 0 < (β―βπ)) β (π = π₯ β ((β―βπ) = (β―βπ₯) β§ ((π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)) β§ (lastSβπ) = (lastSβπ₯))))) |
123 | 113, 115,
121, 122 | syl3anc 1371 |
. . . . . . . . . 10
β’ (((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
)) β (π = π₯ β ((β―βπ) = (β―βπ₯) β§ ((π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)) β§ (lastSβπ) = (lastSβπ₯))))) |
124 | | ancom 461 |
. . . . . . . . . . . 12
β’ (((π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1)) β§
(lastSβπ) =
(lastSβπ₯)) β
((lastSβπ) =
(lastSβπ₯) β§
(π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1)))) |
125 | 124 | anbi2i 623 |
. . . . . . . . . . 11
β’
(((β―βπ)
= (β―βπ₯) β§
((π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1)) β§ (lastSβπ) =
(lastSβπ₯))) β
((β―βπ) =
(β―βπ₯) β§
((lastSβπ) =
(lastSβπ₯) β§
(π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1))))) |
126 | | 3anass 1095 |
. . . . . . . . . . 11
β’
(((β―βπ)
= (β―βπ₯) β§
(lastSβπ) =
(lastSβπ₯) β§
(π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1))) β ((β―βπ) = (β―βπ₯) β§ ((lastSβπ) = (lastSβπ₯) β§ (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1))))) |
127 | 125, 126 | bitr4i 277 |
. . . . . . . . . 10
β’
(((β―βπ)
= (β―βπ₯) β§
((π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1)) β§ (lastSβπ) =
(lastSβπ₯))) β
((β―βπ) =
(β―βπ₯) β§
(lastSβπ) =
(lastSβπ₯) β§
(π prefix
((β―βπ) β
1)) = (π₯ prefix
((β―βπ) β
1)))) |
128 | 123, 127 | bitrdi 286 |
. . . . . . . . 9
β’ (((π β Word π β§ π₯ β Word π) β§ (π β β
β§ π₯ β β
)) β (π = π₯ β ((β―βπ) = (β―βπ₯) β§ (lastSβπ) = (lastSβπ₯) β§ (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1))))) |
129 | 111, 128 | syl 17 |
. . . . . . . 8
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
(π = π₯ β ((β―βπ) = (β―βπ₯) β§ (lastSβπ) = (lastSβπ₯) β§ (π prefix ((β―βπ) β 1)) = (π₯ prefix ((β―βπ) β 1))))) |
130 | 40, 41, 74, 129 | mpbir3and 1342 |
. . . . . . 7
β’
(((((π β Word
π β§
((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β§ π β β0) β§
(lastSβπ) =
(lastSβπ₯)) β
π = π₯) |
131 | 130 | exp31 420 |
. . . . . 6
β’ (((π β Word π β§ ((β―βπ) = (π + 2) β§ (π prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ)} β πΈ)) β§ (π₯ β Word π β§ ((β―βπ₯) = (π + 2) β§ (π₯ prefix (π + 1)) = π β§ {(lastSβπ), (lastSβπ₯)} β πΈ))) β (π β β0 β
((lastSβπ) =
(lastSβπ₯) β
π = π₯))) |
132 | 22, 30, 131 | syl2anb 598 |
. . . . 5
β’ ((π β π· β§ π₯ β π·) β (π β β0 β
((lastSβπ) =
(lastSβπ₯) β
π = π₯))) |
133 | 132 | impcom 408 |
. . . 4
β’ ((π β β0
β§ (π β π· β§ π₯ β π·)) β ((lastSβπ) = (lastSβπ₯) β π = π₯)) |
134 | 14, 133 | sylbid 239 |
. . 3
β’ ((π β β0
β§ (π β π· β§ π₯ β π·)) β ((πΉβπ) = (πΉβπ₯) β π = π₯)) |
135 | 134 | ralrimivva 3199 |
. 2
β’ (π β β0
β βπ β
π· βπ₯ β π· ((πΉβπ) = (πΉβπ₯) β π = π₯)) |
136 | | dff13 7222 |
. 2
β’ (πΉ:π·β1-1βπ
β (πΉ:π·βΆπ
β§ βπ β π· βπ₯ β π· ((πΉβπ) = (πΉβπ₯) β π = π₯))) |
137 | 6, 135, 136 | sylanbrc 583 |
1
β’ (π β β0
β πΉ:π·β1-1βπ
) |