Step | Hyp | Ref
| Expression |
1 | | cycpmco2.1 |
. . . . 5
β’ π = (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) |
2 | | cycpmco2.w |
. . . . . 6
β’ (π β π β dom π) |
3 | | cycpmco2.e |
. . . . . . 7
β’ πΈ = ((β‘πβπ½) + 1) |
4 | | ovexd 7393 |
. . . . . . 7
β’ (π β ((β‘πβπ½) + 1) β V) |
5 | 3, 4 | eqeltrid 2838 |
. . . . . 6
β’ (π β πΈ β V) |
6 | | cycpmco2.i |
. . . . . . . 8
β’ (π β πΌ β (π· β ran π)) |
7 | 6 | eldifad 3923 |
. . . . . . 7
β’ (π β πΌ β π·) |
8 | 7 | s1cld 14497 |
. . . . . 6
β’ (π β β¨βπΌββ© β Word π·) |
9 | | splval 14645 |
. . . . . 6
β’ ((π β dom π β§ (πΈ β V β§ πΈ β V β§ β¨βπΌββ© β Word π·)) β (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) = (((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) |
10 | 2, 5, 5, 8, 9 | syl13anc 1373 |
. . . . 5
β’ (π β (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) = (((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) |
11 | 1, 10 | eqtrid 2785 |
. . . 4
β’ (π β π = (((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) |
12 | 11 | fveq1d 6845 |
. . 3
β’ (π β (πβπΈ) = ((((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))βπΈ)) |
13 | | ssrab2 4038 |
. . . . . . 7
β’ {π€ β Word π· β£ π€:dom π€β1-1βπ·} β Word π· |
14 | | cycpmco2.d |
. . . . . . . . . 10
β’ (π β π· β π) |
15 | | cycpmco2.c |
. . . . . . . . . . 11
β’ π = (toCycβπ·) |
16 | | cycpmco2.s |
. . . . . . . . . . 11
β’ π = (SymGrpβπ·) |
17 | | eqid 2733 |
. . . . . . . . . . 11
β’
(Baseβπ) =
(Baseβπ) |
18 | 15, 16, 17 | tocycf 32015 |
. . . . . . . . . 10
β’ (π· β π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
19 | 14, 18 | syl 17 |
. . . . . . . . 9
β’ (π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
20 | 19 | fdmd 6680 |
. . . . . . . 8
β’ (π β dom π = {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
21 | 2, 20 | eleqtrd 2836 |
. . . . . . 7
β’ (π β π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
22 | 13, 21 | sselid 3943 |
. . . . . 6
β’ (π β π β Word π·) |
23 | | pfxcl 14571 |
. . . . . 6
β’ (π β Word π· β (π prefix πΈ) β Word π·) |
24 | 22, 23 | syl 17 |
. . . . 5
β’ (π β (π prefix πΈ) β Word π·) |
25 | | ccatcl 14468 |
. . . . 5
β’ (((π prefix πΈ) β Word π· β§ β¨βπΌββ© β Word π·) β ((π prefix πΈ) ++ β¨βπΌββ©) β Word π·) |
26 | 24, 8, 25 | syl2anc 585 |
. . . 4
β’ (π β ((π prefix πΈ) ++ β¨βπΌββ©) β Word π·) |
27 | | swrdcl 14539 |
. . . . 5
β’ (π β Word π· β (π substr β¨πΈ, (β―βπ)β©) β Word π·) |
28 | 22, 27 | syl 17 |
. . . 4
β’ (π β (π substr β¨πΈ, (β―βπ)β©) β Word π·) |
29 | | fz0ssnn0 13542 |
. . . . . . 7
β’
(0...(β―βπ)) β
β0 |
30 | | id 22 |
. . . . . . . . . . . . . . . 16
β’ (π€ = π β π€ = π) |
31 | | dmeq 5860 |
. . . . . . . . . . . . . . . 16
β’ (π€ = π β dom π€ = dom π) |
32 | | eqidd 2734 |
. . . . . . . . . . . . . . . 16
β’ (π€ = π β π· = π·) |
33 | 30, 31, 32 | f1eq123d 6777 |
. . . . . . . . . . . . . . 15
β’ (π€ = π β (π€:dom π€β1-1βπ· β π:dom πβ1-1βπ·)) |
34 | 33 | elrab 3646 |
. . . . . . . . . . . . . 14
β’ (π β {π€ β Word π· β£ π€:dom π€β1-1βπ·} β (π β Word π· β§ π:dom πβ1-1βπ·)) |
35 | 21, 34 | sylib 217 |
. . . . . . . . . . . . 13
β’ (π β (π β Word π· β§ π:dom πβ1-1βπ·)) |
36 | 35 | simprd 497 |
. . . . . . . . . . . 12
β’ (π β π:dom πβ1-1βπ·) |
37 | | f1cnv 6809 |
. . . . . . . . . . . 12
β’ (π:dom πβ1-1βπ· β β‘π:ran πβ1-1-ontoβdom
π) |
38 | | f1of 6785 |
. . . . . . . . . . . 12
β’ (β‘π:ran πβ1-1-ontoβdom
π β β‘π:ran πβΆdom π) |
39 | 36, 37, 38 | 3syl 18 |
. . . . . . . . . . 11
β’ (π β β‘π:ran πβΆdom π) |
40 | | cycpmco2.j |
. . . . . . . . . . 11
β’ (π β π½ β ran π) |
41 | 39, 40 | ffvelcdmd 7037 |
. . . . . . . . . 10
β’ (π β (β‘πβπ½) β dom π) |
42 | | wrddm 14415 |
. . . . . . . . . . 11
β’ (π β Word π· β dom π = (0..^(β―βπ))) |
43 | 22, 42 | syl 17 |
. . . . . . . . . 10
β’ (π β dom π = (0..^(β―βπ))) |
44 | 41, 43 | eleqtrd 2836 |
. . . . . . . . 9
β’ (π β (β‘πβπ½) β (0..^(β―βπ))) |
45 | | fzofzp1 13675 |
. . . . . . . . 9
β’ ((β‘πβπ½) β (0..^(β―βπ)) β ((β‘πβπ½) + 1) β (0...(β―βπ))) |
46 | 44, 45 | syl 17 |
. . . . . . . 8
β’ (π β ((β‘πβπ½) + 1) β (0...(β―βπ))) |
47 | 3, 46 | eqeltrid 2838 |
. . . . . . 7
β’ (π β πΈ β (0...(β―βπ))) |
48 | 29, 47 | sselid 3943 |
. . . . . 6
β’ (π β πΈ β
β0) |
49 | | fzonn0p1 13655 |
. . . . . 6
β’ (πΈ β β0
β πΈ β (0..^(πΈ + 1))) |
50 | 48, 49 | syl 17 |
. . . . 5
β’ (π β πΈ β (0..^(πΈ + 1))) |
51 | | ccatws1len 14514 |
. . . . . . . 8
β’ ((π prefix πΈ) β Word π· β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = ((β―β(π prefix πΈ)) + 1)) |
52 | 22, 23, 51 | 3syl 18 |
. . . . . . 7
β’ (π β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = ((β―β(π prefix πΈ)) + 1)) |
53 | | pfxlen 14577 |
. . . . . . . . 9
β’ ((π β Word π· β§ πΈ β (0...(β―βπ))) β (β―β(π prefix πΈ)) = πΈ) |
54 | 22, 47, 53 | syl2anc 585 |
. . . . . . . 8
β’ (π β (β―β(π prefix πΈ)) = πΈ) |
55 | 54 | oveq1d 7373 |
. . . . . . 7
β’ (π β ((β―β(π prefix πΈ)) + 1) = (πΈ + 1)) |
56 | 52, 55 | eqtrd 2773 |
. . . . . 6
β’ (π β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = (πΈ + 1)) |
57 | 56 | oveq2d 7374 |
. . . . 5
β’ (π β
(0..^(β―β((π
prefix πΈ) ++
β¨βπΌββ©))) = (0..^(πΈ + 1))) |
58 | 50, 57 | eleqtrrd 2837 |
. . . 4
β’ (π β πΈ β (0..^(β―β((π prefix πΈ) ++ β¨βπΌββ©)))) |
59 | | ccatval1 14471 |
. . . 4
β’ ((((π prefix πΈ) ++ β¨βπΌββ©) β Word π· β§ (π substr β¨πΈ, (β―βπ)β©) β Word π· β§ πΈ β (0..^(β―β((π prefix πΈ) ++ β¨βπΌββ©)))) β ((((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))βπΈ) = (((π prefix πΈ) ++ β¨βπΌββ©)βπΈ)) |
60 | 26, 28, 58, 59 | syl3anc 1372 |
. . 3
β’ (π β ((((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))βπΈ) = (((π prefix πΈ) ++ β¨βπΌββ©)βπΈ)) |
61 | 48 | nn0zd 12530 |
. . . . . 6
β’ (π β πΈ β β€) |
62 | | elfzomin 13650 |
. . . . . 6
β’ (πΈ β β€ β πΈ β (πΈ..^(πΈ + 1))) |
63 | 61, 62 | syl 17 |
. . . . 5
β’ (π β πΈ β (πΈ..^(πΈ + 1))) |
64 | | s1len 14500 |
. . . . . . . 8
β’
(β―ββ¨βπΌββ©) = 1 |
65 | 64 | a1i 11 |
. . . . . . 7
β’ (π β
(β―ββ¨βπΌββ©) = 1) |
66 | 54, 65 | oveq12d 7376 |
. . . . . 6
β’ (π β ((β―β(π prefix πΈ)) + (β―ββ¨βπΌββ©)) = (πΈ + 1)) |
67 | 54, 66 | oveq12d 7376 |
. . . . 5
β’ (π β ((β―β(π prefix πΈ))..^((β―β(π prefix πΈ)) + (β―ββ¨βπΌββ©))) = (πΈ..^(πΈ + 1))) |
68 | 63, 67 | eleqtrrd 2837 |
. . . 4
β’ (π β πΈ β ((β―β(π prefix πΈ))..^((β―β(π prefix πΈ)) + (β―ββ¨βπΌββ©)))) |
69 | | ccatval2 14472 |
. . . 4
β’ (((π prefix πΈ) β Word π· β§ β¨βπΌββ© β Word π· β§ πΈ β ((β―β(π prefix πΈ))..^((β―β(π prefix πΈ)) + (β―ββ¨βπΌββ©)))) β
(((π prefix πΈ) ++ β¨βπΌββ©)βπΈ) = (β¨βπΌββ©β(πΈ β (β―β(π prefix πΈ))))) |
70 | 24, 8, 68, 69 | syl3anc 1372 |
. . 3
β’ (π β (((π prefix πΈ) ++ β¨βπΌββ©)βπΈ) = (β¨βπΌββ©β(πΈ β (β―β(π prefix πΈ))))) |
71 | 12, 60, 70 | 3eqtrd 2777 |
. 2
β’ (π β (πβπΈ) = (β¨βπΌββ©β(πΈ β (β―β(π prefix πΈ))))) |
72 | 54 | oveq2d 7374 |
. . . 4
β’ (π β (πΈ β (β―β(π prefix πΈ))) = (πΈ β πΈ)) |
73 | 48 | nn0cnd 12480 |
. . . . 5
β’ (π β πΈ β β) |
74 | 73 | subidd 11505 |
. . . 4
β’ (π β (πΈ β πΈ) = 0) |
75 | 72, 74 | eqtrd 2773 |
. . 3
β’ (π β (πΈ β (β―β(π prefix πΈ))) = 0) |
76 | 75 | fveq2d 6847 |
. 2
β’ (π β (β¨βπΌββ©β(πΈ β (β―β(π prefix πΈ)))) = (β¨βπΌββ©β0)) |
77 | | s1fv 14504 |
. . 3
β’ (πΌ β (π· β ran π) β (β¨βπΌββ©β0) = πΌ) |
78 | 6, 77 | syl 17 |
. 2
β’ (π β (β¨βπΌββ©β0) = πΌ) |
79 | 71, 76, 78 | 3eqtrd 2777 |
1
β’ (π β (πβπΈ) = πΌ) |