Step | Hyp | Ref
| Expression |
1 | | sseqval.1 |
. . . 4
β’ (π β π β V) |
2 | | sseqval.2 |
. . . 4
β’ (π β π β Word π) |
3 | | sseqval.3 |
. . . 4
β’ π = (Word π β© (β‘β― β
(β€β₯β(β―βπ)))) |
4 | | sseqval.4 |
. . . 4
β’ (π β πΉ:πβΆπ) |
5 | 1, 2, 3, 4 | sseqval 32991 |
. . 3
β’ (π β (πseqstrπΉ) = (π βͺ (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))))) |
6 | 5 | fveq1d 6845 |
. 2
β’ (π β ((πseqstrπΉ)βπ) = ((π βͺ (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))))βπ)) |
7 | | wrdfn 14417 |
. . . 4
β’ (π β Word π β π Fn (0..^(β―βπ))) |
8 | 2, 7 | syl 17 |
. . 3
β’ (π β π Fn (0..^(β―βπ))) |
9 | | fvex 6856 |
. . . . . 6
β’ (π₯β((β―βπ₯) β 1)) β
V |
10 | | df-lsw 14452 |
. . . . . 6
β’ lastS =
(π₯ β V β¦ (π₯β((β―βπ₯) β 1))) |
11 | 9, 10 | fnmpti 6645 |
. . . . 5
β’ lastS Fn
V |
12 | 11 | a1i 11 |
. . . 4
β’ (π β lastS Fn
V) |
13 | | lencl 14422 |
. . . . . . 7
β’ (π β Word π β (β―βπ) β
β0) |
14 | 2, 13 | syl 17 |
. . . . . 6
β’ (π β (β―βπ) β
β0) |
15 | 14 | nn0zd 12526 |
. . . . 5
β’ (π β (β―βπ) β
β€) |
16 | | seqfn 13919 |
. . . . 5
β’
((β―βπ)
β β€ β seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) Fn
(β€β₯β(β―βπ))) |
17 | 15, 16 | syl 17 |
. . . 4
β’ (π β seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) Fn
(β€β₯β(β―βπ))) |
18 | | ssv 3969 |
. . . . 5
β’ ran
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) β
V |
19 | 18 | a1i 11 |
. . . 4
β’ (π β ran
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) β
V) |
20 | | fnco 6619 |
. . . 4
β’ ((lastS
Fn V β§ seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) Fn
(β€β₯β(β―βπ)) β§ ran seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) β V)
β (lastS β seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) Fn
(β€β₯β(β―βπ))) |
21 | 12, 17, 19, 20 | syl3anc 1372 |
. . 3
β’ (π β (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) Fn
(β€β₯β(β―βπ))) |
22 | | fzouzdisj 13609 |
. . . 4
β’
((0..^(β―βπ)) β©
(β€β₯β(β―βπ))) = β
|
23 | 22 | a1i 11 |
. . 3
β’ (π β ((0..^(β―βπ)) β©
(β€β₯β(β―βπ))) = β
) |
24 | | sseqfv2.4 |
. . 3
β’ (π β π β
(β€β₯β(β―βπ))) |
25 | | fvun2 6934 |
. . 3
β’ ((π Fn (0..^(β―βπ)) β§ (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) Fn
(β€β₯β(β―βπ)) β§ (((0..^(β―βπ)) β©
(β€β₯β(β―βπ))) = β
β§ π β
(β€β₯β(β―βπ)))) β ((π βͺ (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))))βπ) = ((lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})))βπ)) |
26 | 8, 21, 23, 24, 25 | syl112anc 1375 |
. 2
β’ (π β ((π βͺ (lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))))βπ) = ((lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})))βπ)) |
27 | | fnfun 6603 |
. . . 4
β’
(seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) Fn
(β€β₯β(β―βπ)) β Fun seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) |
28 | 17, 27 | syl 17 |
. . 3
β’ (π β Fun
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) |
29 | | fvexd 6858 |
. . . . . 6
β’ (π β ((β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})β(β―βπ)) β V) |
30 | | ovexd 7393 |
. . . . . 6
β’ ((π β§ (π β V β§ π β V)) β (π(π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©))π) β V) |
31 | | eqid 2737 |
. . . . . 6
β’
(β€β₯β(β―βπ)) =
(β€β₯β(β―βπ)) |
32 | | fvexd 6858 |
. . . . . 6
β’ ((π β§ π β
(β€β₯β((β―βπ) + 1))) β ((β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})βπ) β V) |
33 | 29, 30, 31, 15, 32 | seqf2 13928 |
. . . . 5
β’ (π β seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})):(β€β₯β(β―βπ))βΆV) |
34 | 33 | fdmd 6680 |
. . . 4
β’ (π β dom
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) =
(β€β₯β(β―βπ))) |
35 | 24, 34 | eleqtrrd 2841 |
. . 3
β’ (π β π β dom seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) |
36 | | fvco 6940 |
. . 3
β’ ((Fun
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})) β§ π β dom
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))) β
((lastS β seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})))βπ) =
(lastSβ(seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))βπ))) |
37 | 28, 35, 36 | syl2anc 585 |
. 2
β’ (π β ((lastS β
seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)})))βπ) =
(lastSβ(seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))βπ))) |
38 | 6, 26, 37 | 3eqtrd 2781 |
1
β’ (π β ((πseqstrπΉ)βπ) = (lastSβ(seq(β―βπ)((π₯ β V, π¦ β V β¦ (π₯ ++ β¨β(πΉβπ₯)ββ©)), (β0
Γ {(π ++
β¨β(πΉβπ)ββ©)}))βπ))) |