Step | Hyp | Ref
| Expression |
1 | | efgval.w |
. . . . . . . . 9
β’ π = ( I βWord (πΌ Γ
2o)) |
2 | | efgval.r |
. . . . . . . . 9
β’ βΌ = (
~FG βπΌ) |
3 | | efgval2.m |
. . . . . . . . 9
β’ π = (π¦ β πΌ, π§ β 2o β¦ β¨π¦, (1o β π§)β©) |
4 | | efgval2.t |
. . . . . . . . 9
β’ π = (π£ β π β¦ (π β (0...(β―βπ£)), π€ β (πΌ Γ 2o) β¦ (π£ splice β¨π, π, β¨βπ€(πβπ€)ββ©β©))) |
5 | | efgred.d |
. . . . . . . . 9
β’ π· = (π β βͺ
π₯ β π ran (πβπ₯)) |
6 | | efgred.s |
. . . . . . . . 9
β’ π = (π β {π‘ β (Word π β {β
}) β£ ((π‘β0) β π· β§ βπ β
(1..^(β―βπ‘))(π‘βπ) β ran (πβ(π‘β(π β 1))))} β¦ (πβ((β―βπ) β 1))) |
7 | 1, 2, 3, 4, 5, 6 | efgsdm 19598 |
. . . . . . . 8
β’ (πΉ β dom π β (πΉ β (Word π β {β
}) β§ (πΉβ0) β π· β§ βπ β (1..^(β―βπΉ))(πΉβπ) β ran (πβ(πΉβ(π β 1))))) |
8 | 7 | simp1bi 1146 |
. . . . . . 7
β’ (πΉ β dom π β πΉ β (Word π β {β
})) |
9 | 8 | adantr 482 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β πΉ β (Word π β {β
})) |
10 | 9 | eldifad 3961 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β πΉ β Word π) |
11 | | fz1ssfz0 13597 |
. . . . . 6
β’
(1...(β―βπΉ)) β (0...(β―βπΉ)) |
12 | | simpr 486 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β π β (1...(β―βπΉ))) |
13 | 11, 12 | sselid 3981 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β π β (0...(β―βπΉ))) |
14 | | pfxres 14629 |
. . . . 5
β’ ((πΉ β Word π β§ π β (0...(β―βπΉ))) β (πΉ prefix π) = (πΉ βΎ (0..^π))) |
15 | 10, 13, 14 | syl2anc 585 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ prefix π) = (πΉ βΎ (0..^π))) |
16 | | pfxcl 14627 |
. . . . 5
β’ (πΉ β Word π β (πΉ prefix π) β Word π) |
17 | 10, 16 | syl 17 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ prefix π) β Word π) |
18 | 15, 17 | eqeltrrd 2835 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ βΎ (0..^π)) β Word π) |
19 | | pfxlen 14633 |
. . . . . . 7
β’ ((πΉ β Word π β§ π β (0...(β―βπΉ))) β (β―β(πΉ prefix π)) = π) |
20 | 10, 13, 19 | syl2anc 585 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (β―β(πΉ prefix π)) = π) |
21 | | elfznn 13530 |
. . . . . . 7
β’ (π β
(1...(β―βπΉ))
β π β
β) |
22 | 21 | adantl 483 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β π β β) |
23 | 20, 22 | eqeltrd 2834 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (β―β(πΉ prefix π)) β β) |
24 | | wrdfin 14482 |
. . . . . 6
β’ ((πΉ prefix π) β Word π β (πΉ prefix π) β Fin) |
25 | | hashnncl 14326 |
. . . . . 6
β’ ((πΉ prefix π) β Fin β ((β―β(πΉ prefix π)) β β β (πΉ prefix π) β β
)) |
26 | 17, 24, 25 | 3syl 18 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β
((β―β(πΉ prefix
π)) β β β
(πΉ prefix π) β β
)) |
27 | 23, 26 | mpbid 231 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ prefix π) β β
) |
28 | 15, 27 | eqnetrrd 3010 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ βΎ (0..^π)) β β
) |
29 | | eldifsn 4791 |
. . 3
β’ ((πΉ βΎ (0..^π)) β (Word π β {β
}) β ((πΉ βΎ (0..^π)) β Word π β§ (πΉ βΎ (0..^π)) β β
)) |
30 | 18, 28, 29 | sylanbrc 584 |
. 2
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ βΎ (0..^π)) β (Word π β {β
})) |
31 | | lbfzo0 13672 |
. . . . 5
β’ (0 β
(0..^π) β π β
β) |
32 | 22, 31 | sylibr 233 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β 0 β (0..^π)) |
33 | 32 | fvresd 6912 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β ((πΉ βΎ (0..^π))β0) = (πΉβ0)) |
34 | 7 | simp2bi 1147 |
. . . 4
β’ (πΉ β dom π β (πΉβ0) β π·) |
35 | 34 | adantr 482 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉβ0) β π·) |
36 | 33, 35 | eqeltrd 2834 |
. 2
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β ((πΉ βΎ (0..^π))β0) β π·) |
37 | | elfzuz3 13498 |
. . . . . . 7
β’ (π β
(1...(β―βπΉ))
β (β―βπΉ)
β (β€β₯βπ)) |
38 | 37 | adantl 483 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (β―βπΉ) β
(β€β₯βπ)) |
39 | | fzoss2 13660 |
. . . . . 6
β’
((β―βπΉ)
β (β€β₯βπ) β (1..^π) β (1..^(β―βπΉ))) |
40 | 38, 39 | syl 17 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (1..^π) β
(1..^(β―βπΉ))) |
41 | 7 | simp3bi 1148 |
. . . . . 6
β’ (πΉ β dom π β βπ β (1..^(β―βπΉ))(πΉβπ) β ran (πβ(πΉβ(π β 1)))) |
42 | 41 | adantr 482 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β βπ β
(1..^(β―βπΉ))(πΉβπ) β ran (πβ(πΉβ(π β 1)))) |
43 | | ssralv 4051 |
. . . . 5
β’
((1..^π) β
(1..^(β―βπΉ))
β (βπ β
(1..^(β―βπΉ))(πΉβπ) β ran (πβ(πΉβ(π β 1))) β βπ β (1..^π)(πΉβπ) β ran (πβ(πΉβ(π β 1))))) |
44 | 40, 42, 43 | sylc 65 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β βπ β (1..^π)(πΉβπ) β ran (πβ(πΉβ(π β 1)))) |
45 | | fzo0ss1 13662 |
. . . . . . . 8
β’
(1..^π) β
(0..^π) |
46 | 45 | sseli 3979 |
. . . . . . 7
β’ (π β (1..^π) β π β (0..^π)) |
47 | 46 | fvresd 6912 |
. . . . . 6
β’ (π β (1..^π) β ((πΉ βΎ (0..^π))βπ) = (πΉβπ)) |
48 | | elfzoel2 13631 |
. . . . . . . . . . . . 13
β’ (π β (1..^π) β π β β€) |
49 | | peano2zm 12605 |
. . . . . . . . . . . . 13
β’ (π β β€ β (π β 1) β
β€) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . 12
β’ (π β (1..^π) β (π β 1) β β€) |
51 | | uzid 12837 |
. . . . . . . . . . . . . 14
β’ (π β β€ β π β
(β€β₯βπ)) |
52 | 48, 51 | syl 17 |
. . . . . . . . . . . . 13
β’ (π β (1..^π) β π β (β€β₯βπ)) |
53 | 48 | zcnd 12667 |
. . . . . . . . . . . . . . 15
β’ (π β (1..^π) β π β β) |
54 | | ax-1cn 11168 |
. . . . . . . . . . . . . . 15
β’ 1 β
β |
55 | | npcan 11469 |
. . . . . . . . . . . . . . 15
β’ ((π β β β§ 1 β
β) β ((π β
1) + 1) = π) |
56 | 53, 54, 55 | sylancl 587 |
. . . . . . . . . . . . . 14
β’ (π β (1..^π) β ((π β 1) + 1) = π) |
57 | 56 | fveq2d 6896 |
. . . . . . . . . . . . 13
β’ (π β (1..^π) β
(β€β₯β((π β 1) + 1)) =
(β€β₯βπ)) |
58 | 52, 57 | eleqtrrd 2837 |
. . . . . . . . . . . 12
β’ (π β (1..^π) β π β
(β€β₯β((π β 1) + 1))) |
59 | | peano2uzr 12887 |
. . . . . . . . . . . 12
β’ (((π β 1) β β€ β§
π β
(β€β₯β((π β 1) + 1))) β π β (β€β₯β(π β 1))) |
60 | 50, 58, 59 | syl2anc 585 |
. . . . . . . . . . 11
β’ (π β (1..^π) β π β (β€β₯β(π β 1))) |
61 | | fzoss2 13660 |
. . . . . . . . . . 11
β’ (π β
(β€β₯β(π β 1)) β (0..^(π β 1)) β (0..^π)) |
62 | 60, 61 | syl 17 |
. . . . . . . . . 10
β’ (π β (1..^π) β (0..^(π β 1)) β (0..^π)) |
63 | | elfzo1elm1fzo0 13733 |
. . . . . . . . . 10
β’ (π β (1..^π) β (π β 1) β (0..^(π β 1))) |
64 | 62, 63 | sseldd 3984 |
. . . . . . . . 9
β’ (π β (1..^π) β (π β 1) β (0..^π)) |
65 | 64 | fvresd 6912 |
. . . . . . . 8
β’ (π β (1..^π) β ((πΉ βΎ (0..^π))β(π β 1)) = (πΉβ(π β 1))) |
66 | 65 | fveq2d 6896 |
. . . . . . 7
β’ (π β (1..^π) β (πβ((πΉ βΎ (0..^π))β(π β 1))) = (πβ(πΉβ(π β 1)))) |
67 | 66 | rneqd 5938 |
. . . . . 6
β’ (π β (1..^π) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))) = ran (πβ(πΉβ(π β 1)))) |
68 | 47, 67 | eleq12d 2828 |
. . . . 5
β’ (π β (1..^π) β (((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))) β (πΉβπ) β ran (πβ(πΉβ(π β 1))))) |
69 | 68 | ralbiia 3092 |
. . . 4
β’
(βπ β
(1..^π)((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))) β βπ β (1..^π)(πΉβπ) β ran (πβ(πΉβ(π β 1)))) |
70 | 44, 69 | sylibr 233 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β βπ β (1..^π)((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1)))) |
71 | 15 | fveq2d 6896 |
. . . . . 6
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (β―β(πΉ prefix π)) = (β―β(πΉ βΎ (0..^π)))) |
72 | 71, 20 | eqtr3d 2775 |
. . . . 5
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (β―β(πΉ βΎ (0..^π))) = π) |
73 | 72 | oveq2d 7425 |
. . . 4
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β
(1..^(β―β(πΉ
βΎ (0..^π)))) =
(1..^π)) |
74 | 73 | raleqdv 3326 |
. . 3
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (βπ β
(1..^(β―β(πΉ
βΎ (0..^π))))((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))) β βπ β (1..^π)((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))))) |
75 | 70, 74 | mpbird 257 |
. 2
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β βπ β
(1..^(β―β(πΉ
βΎ (0..^π))))((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1)))) |
76 | 1, 2, 3, 4, 5, 6 | efgsdm 19598 |
. 2
β’ ((πΉ βΎ (0..^π)) β dom π β ((πΉ βΎ (0..^π)) β (Word π β {β
}) β§ ((πΉ βΎ (0..^π))β0) β π· β§ βπ β (1..^(β―β(πΉ βΎ (0..^π))))((πΉ βΎ (0..^π))βπ) β ran (πβ((πΉ βΎ (0..^π))β(π β 1))))) |
77 | 30, 36, 75, 76 | syl3anbrc 1344 |
1
β’ ((πΉ β dom π β§ π β (1...(β―βπΉ))) β (πΉ βΎ (0..^π)) β dom π) |