Step | Hyp | Ref
| Expression |
1 | | ssrab2 4038 |
. . . . 5
β’ {π€ β Word π· β£ π€:dom π€β1-1βπ·} β Word π· |
2 | | cycpmco2.w |
. . . . . 6
β’ (π β π β dom π) |
3 | | cycpmco2.d |
. . . . . . . 8
β’ (π β π· β π) |
4 | | cycpmco2.c |
. . . . . . . . 9
β’ π = (toCycβπ·) |
5 | | cycpmco2.s |
. . . . . . . . 9
β’ π = (SymGrpβπ·) |
6 | | eqid 2733 |
. . . . . . . . 9
β’
(Baseβπ) =
(Baseβπ) |
7 | 4, 5, 6 | tocycf 32015 |
. . . . . . . 8
β’ (π· β π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
8 | 3, 7 | syl 17 |
. . . . . . 7
β’ (π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
9 | 8 | fdmd 6680 |
. . . . . 6
β’ (π β dom π = {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
10 | 2, 9 | eleqtrd 2836 |
. . . . 5
β’ (π β π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
11 | 1, 10 | sselid 3943 |
. . . 4
β’ (π β π β Word π·) |
12 | | lencl 14427 |
. . . 4
β’ (π β Word π· β (β―βπ) β
β0) |
13 | 11, 12 | syl 17 |
. . 3
β’ (π β (β―βπ) β
β0) |
14 | 13 | nn0cnd 12480 |
. 2
β’ (π β (β―βπ) β
β) |
15 | | 1cnd 11155 |
. 2
β’ (π β 1 β
β) |
16 | | cycpmco2.1 |
. . . . . . 7
β’ π = (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) |
17 | | cycpmco2.e |
. . . . . . . . 9
β’ πΈ = ((β‘πβπ½) + 1) |
18 | | ovexd 7393 |
. . . . . . . . 9
β’ (π β ((β‘πβπ½) + 1) β V) |
19 | 17, 18 | eqeltrid 2838 |
. . . . . . . 8
β’ (π β πΈ β V) |
20 | | cycpmco2.i |
. . . . . . . . . 10
β’ (π β πΌ β (π· β ran π)) |
21 | 20 | eldifad 3923 |
. . . . . . . . 9
β’ (π β πΌ β π·) |
22 | 21 | s1cld 14497 |
. . . . . . . 8
β’ (π β β¨βπΌββ© β Word π·) |
23 | | splval 14645 |
. . . . . . . 8
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©))) |
24 | 2, 19, 19, 22, 23 | syl13anc 1373 |
. . . . . . 7
β’ (π β (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) = (((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) |
25 | 16, 24 | eqtrid 2785 |
. . . . . 6
β’ (π β π = (((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) |
26 | 25 | fveq2d 6847 |
. . . . 5
β’ (π β (β―βπ) = (β―β(((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©)))) |
27 | | pfxcl 14571 |
. . . . . . . 8
β’ (π β Word π· β (π prefix πΈ) β Word π·) |
28 | 11, 27 | syl 17 |
. . . . . . 7
β’ (π β (π prefix πΈ) β Word π·) |
29 | | ccatcl 14468 |
. . . . . . 7
β’ (((π prefix πΈ) β Word π· β§ β¨βπΌββ© β Word π·) β ((π prefix πΈ) ++ β¨βπΌββ©) β Word π·) |
30 | 28, 22, 29 | syl2anc 585 |
. . . . . 6
β’ (π β ((π prefix πΈ) ++ β¨βπΌββ©) β Word π·) |
31 | | swrdcl 14539 |
. . . . . . 7
β’ (π β Word π· β (π substr β¨πΈ, (β―βπ)β©) β Word π·) |
32 | 11, 31 | syl 17 |
. . . . . 6
β’ (π β (π substr β¨πΈ, (β―βπ)β©) β Word π·) |
33 | | ccatlen 14469 |
. . . . . 6
β’ ((((π prefix πΈ) ++ β¨βπΌββ©) β Word π· β§ (π substr β¨πΈ, (β―βπ)β©) β Word π·) β (β―β(((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) = ((β―β((π prefix πΈ) ++ β¨βπΌββ©)) + (β―β(π substr β¨πΈ, (β―βπ)β©)))) |
34 | 30, 32, 33 | syl2anc 585 |
. . . . 5
β’ (π β (β―β(((π prefix πΈ) ++ β¨βπΌββ©) ++ (π substr β¨πΈ, (β―βπ)β©))) = ((β―β((π prefix πΈ) ++ β¨βπΌββ©)) + (β―β(π substr β¨πΈ, (β―βπ)β©)))) |
35 | | ccatws1len 14514 |
. . . . . . . 8
β’ ((π prefix πΈ) β Word π· β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = ((β―β(π prefix πΈ)) + 1)) |
36 | 28, 35 | syl 17 |
. . . . . . 7
β’ (π β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = ((β―β(π prefix πΈ)) + 1)) |
37 | | id 22 |
. . . . . . . . . . . . . . . . . 18
β’ (π€ = π β π€ = π) |
38 | | dmeq 5860 |
. . . . . . . . . . . . . . . . . 18
β’ (π€ = π β dom π€ = dom π) |
39 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . 18
β’ (π€ = π β π· = π·) |
40 | 37, 38, 39 | f1eq123d 6777 |
. . . . . . . . . . . . . . . . 17
β’ (π€ = π β (π€:dom π€β1-1βπ· β π:dom πβ1-1βπ·)) |
41 | 40 | elrab 3646 |
. . . . . . . . . . . . . . . 16
β’ (π β {π€ β Word π· β£ π€:dom π€β1-1βπ·} β (π β Word π· β§ π:dom πβ1-1βπ·)) |
42 | 10, 41 | sylib 217 |
. . . . . . . . . . . . . . 15
β’ (π β (π β Word π· β§ π:dom πβ1-1βπ·)) |
43 | | f1cnv 6809 |
. . . . . . . . . . . . . . 15
β’ (π:dom πβ1-1βπ· β β‘π:ran πβ1-1-ontoβdom
π) |
44 | 42, 43 | simpl2im 505 |
. . . . . . . . . . . . . 14
β’ (π β β‘π:ran πβ1-1-ontoβdom
π) |
45 | | f1of 6785 |
. . . . . . . . . . . . . 14
β’ (β‘π:ran πβ1-1-ontoβdom
π β β‘π:ran πβΆdom π) |
46 | 44, 45 | syl 17 |
. . . . . . . . . . . . 13
β’ (π β β‘π:ran πβΆdom π) |
47 | | cycpmco2.j |
. . . . . . . . . . . . 13
β’ (π β π½ β ran π) |
48 | 46, 47 | ffvelcdmd 7037 |
. . . . . . . . . . . 12
β’ (π β (β‘πβπ½) β dom π) |
49 | | wrddm 14415 |
. . . . . . . . . . . . 13
β’ (π β Word π· β dom π = (0..^(β―βπ))) |
50 | 11, 49 | syl 17 |
. . . . . . . . . . . 12
β’ (π β dom π = (0..^(β―βπ))) |
51 | 48, 50 | eleqtrd 2836 |
. . . . . . . . . . 11
β’ (π β (β‘πβπ½) β (0..^(β―βπ))) |
52 | | fzofzp1 13675 |
. . . . . . . . . . 11
β’ ((β‘πβπ½) β (0..^(β―βπ)) β ((β‘πβπ½) + 1) β (0...(β―βπ))) |
53 | 51, 52 | syl 17 |
. . . . . . . . . 10
β’ (π β ((β‘πβπ½) + 1) β (0...(β―βπ))) |
54 | 17, 53 | eqeltrid 2838 |
. . . . . . . . 9
β’ (π β πΈ β (0...(β―βπ))) |
55 | | pfxlen 14577 |
. . . . . . . . 9
β’ ((π β Word π· β§ πΈ β (0...(β―βπ))) β (β―β(π prefix πΈ)) = πΈ) |
56 | 11, 54, 55 | syl2anc 585 |
. . . . . . . 8
β’ (π β (β―β(π prefix πΈ)) = πΈ) |
57 | 56 | oveq1d 7373 |
. . . . . . 7
β’ (π β ((β―β(π prefix πΈ)) + 1) = (πΈ + 1)) |
58 | 36, 57 | eqtrd 2773 |
. . . . . 6
β’ (π β (β―β((π prefix πΈ) ++ β¨βπΌββ©)) = (πΈ + 1)) |
59 | | nn0fz0 13545 |
. . . . . . . 8
β’
((β―βπ)
β β0 β (β―βπ) β (0...(β―βπ))) |
60 | 13, 59 | sylib 217 |
. . . . . . 7
β’ (π β (β―βπ) β
(0...(β―βπ))) |
61 | | swrdlen 14541 |
. . . . . . 7
β’ ((π β Word π· β§ πΈ β (0...(β―βπ)) β§ (β―βπ) β
(0...(β―βπ)))
β (β―β(π
substr β¨πΈ,
(β―βπ)β©)) =
((β―βπ) β
πΈ)) |
62 | 11, 54, 60, 61 | syl3anc 1372 |
. . . . . 6
β’ (π β (β―β(π substr β¨πΈ, (β―βπ)β©)) = ((β―βπ) β πΈ)) |
63 | 58, 62 | oveq12d 7376 |
. . . . 5
β’ (π β ((β―β((π prefix πΈ) ++ β¨βπΌββ©)) + (β―β(π substr β¨πΈ, (β―βπ)β©))) = ((πΈ + 1) + ((β―βπ) β πΈ))) |
64 | 26, 34, 63 | 3eqtrd 2777 |
. . . 4
β’ (π β (β―βπ) = ((πΈ + 1) + ((β―βπ) β πΈ))) |
65 | | fz0ssnn0 13542 |
. . . . . . . . 9
β’
(0...(β―βπ)) β
β0 |
66 | 65, 54 | sselid 3943 |
. . . . . . . 8
β’ (π β πΈ β
β0) |
67 | 66 | nn0zd 12530 |
. . . . . . 7
β’ (π β πΈ β β€) |
68 | 67 | peano2zd 12615 |
. . . . . 6
β’ (π β (πΈ + 1) β β€) |
69 | 68 | zcnd 12613 |
. . . . 5
β’ (π β (πΈ + 1) β β) |
70 | 66 | nn0cnd 12480 |
. . . . 5
β’ (π β πΈ β β) |
71 | 69, 14, 70 | addsubassd 11537 |
. . . 4
β’ (π β (((πΈ + 1) + (β―βπ)) β πΈ) = ((πΈ + 1) + ((β―βπ) β πΈ))) |
72 | 70, 15, 14 | addassd 11182 |
. . . . 5
β’ (π β ((πΈ + 1) + (β―βπ)) = (πΈ + (1 + (β―βπ)))) |
73 | 72 | oveq1d 7373 |
. . . 4
β’ (π β (((πΈ + 1) + (β―βπ)) β πΈ) = ((πΈ + (1 + (β―βπ))) β πΈ)) |
74 | 64, 71, 73 | 3eqtr2d 2779 |
. . 3
β’ (π β (β―βπ) = ((πΈ + (1 + (β―βπ))) β πΈ)) |
75 | 15, 14 | addcld 11179 |
. . . 4
β’ (π β (1 + (β―βπ)) β
β) |
76 | 70, 75 | pncan2d 11519 |
. . 3
β’ (π β ((πΈ + (1 + (β―βπ))) β πΈ) = (1 + (β―βπ))) |
77 | 15, 14 | addcomd 11362 |
. . 3
β’ (π β (1 + (β―βπ)) = ((β―βπ) + 1)) |
78 | 74, 76, 77 | 3eqtrd 2777 |
. 2
β’ (π β (β―βπ) = ((β―βπ) + 1)) |
79 | 14, 15, 78 | mvrraddd 11572 |
1
β’ (π β ((β―βπ) β 1) =
(β―βπ)) |