Step | Hyp | Ref
| Expression |
1 | | swrdcl 14540 |
. . . . . 6
β’ (π β Word π΄ β (π substr β¨π, πβ©) β Word π΄) |
2 | 1 | adantr 482 |
. . . . 5
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π substr β¨π, πβ©) β Word π΄) |
3 | | swrdcl 14540 |
. . . . . 6
β’ (π β Word π΄ β (π substr β¨π, πβ©) β Word π΄) |
4 | 3 | adantr 482 |
. . . . 5
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π substr β¨π, πβ©) β Word π΄) |
5 | | ccatcl 14469 |
. . . . 5
β’ (((π substr β¨π, πβ©) β Word π΄ β§ (π substr β¨π, πβ©) β Word π΄) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) β Word π΄) |
6 | 2, 4, 5 | syl2anc 585 |
. . . 4
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) β Word π΄) |
7 | | wrdfn 14423 |
. . . 4
β’ (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) β Word π΄ β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) Fn (0..^(β―β((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))))) |
8 | 6, 7 | syl 17 |
. . 3
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) Fn (0..^(β―β((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))))) |
9 | | ccatlen 14470 |
. . . . . . 7
β’ (((π substr β¨π, πβ©) β Word π΄ β§ (π substr β¨π, πβ©) β Word π΄) β (β―β((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))) = ((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©)))) |
10 | 2, 4, 9 | syl2anc 585 |
. . . . . 6
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(β―β((π substr
β¨π, πβ©) ++ (π substr β¨π, πβ©))) = ((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©)))) |
11 | | simpl 484 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β Word π΄) |
12 | | simpr1 1195 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β (0...π)) |
13 | | simpr2 1196 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β (0...π)) |
14 | | simpr3 1197 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β (0...(β―βπ))) |
15 | | fzass4 13486 |
. . . . . . . . . . 11
β’ ((π β
(0...(β―βπ))
β§ π β (π...(β―βπ))) β (π β (0...π) β§ π β (0...(β―βπ)))) |
16 | 15 | biimpri 227 |
. . . . . . . . . 10
β’ ((π β (0...π) β§ π β (0...(β―βπ))) β (π β (0...(β―βπ)) β§ π β (π...(β―βπ)))) |
17 | 16 | simpld 496 |
. . . . . . . . 9
β’ ((π β (0...π) β§ π β (0...(β―βπ))) β π β (0...(β―βπ))) |
18 | 13, 14, 17 | syl2anc 585 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β (0...(β―βπ))) |
19 | | swrdlen 14542 |
. . . . . . . 8
β’ ((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β (β―β(π substr β¨π, πβ©)) = (π β π)) |
20 | 11, 12, 18, 19 | syl3anc 1372 |
. . . . . . 7
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(β―β(π substr
β¨π, πβ©)) = (π β π)) |
21 | | swrdlen 14542 |
. . . . . . . 8
β’ ((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β (β―β(π substr β¨π, πβ©)) = (π β π)) |
22 | 21 | 3adant3r1 1183 |
. . . . . . 7
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(β―β(π substr
β¨π, πβ©)) = (π β π)) |
23 | 20, 22 | oveq12d 7380 |
. . . . . 6
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
((β―β(π substr
β¨π, πβ©)) + (β―β(π substr β¨π, πβ©))) = ((π β π) + (π β π))) |
24 | 13 | elfzelzd 13449 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β€) |
25 | 24 | zcnd 12615 |
. . . . . . 7
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β) |
26 | 12 | elfzelzd 13449 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β€) |
27 | 26 | zcnd 12615 |
. . . . . . 7
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β) |
28 | 14 | elfzelzd 13449 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β€) |
29 | 28 | zcnd 12615 |
. . . . . . 7
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β β) |
30 | 25, 27, 29 | npncan3d 11555 |
. . . . . 6
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π β π) + (π β π)) = (π β π)) |
31 | 10, 23, 30 | 3eqtrd 2781 |
. . . . 5
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(β―β((π substr
β¨π, πβ©) ++ (π substr β¨π, πβ©))) = (π β π)) |
32 | 31 | oveq2d 7378 |
. . . 4
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(0..^(β―β((π
substr β¨π, πβ©) ++ (π substr β¨π, πβ©)))) = (0..^(π β π))) |
33 | 32 | fneq2d 6601 |
. . 3
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) Fn (0..^(β―β((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)))) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) Fn (0..^(π β π)))) |
34 | 8, 33 | mpbid 231 |
. 2
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) Fn (0..^(π β π))) |
35 | | swrdcl 14540 |
. . . . 5
β’ (π β Word π΄ β (π substr β¨π, πβ©) β Word π΄) |
36 | 35 | adantr 482 |
. . . 4
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π substr β¨π, πβ©) β Word π΄) |
37 | | wrdfn 14423 |
. . . 4
β’ ((π substr β¨π, πβ©) β Word π΄ β (π substr β¨π, πβ©) Fn (0..^(β―β(π substr β¨π, πβ©)))) |
38 | 36, 37 | syl 17 |
. . 3
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π substr β¨π, πβ©) Fn (0..^(β―β(π substr β¨π, πβ©)))) |
39 | | fzass4 13486 |
. . . . . . . . 9
β’ ((π β (0...π) β§ π β (π...π)) β (π β (0...π) β§ π β (0...π))) |
40 | 39 | biimpri 227 |
. . . . . . . 8
β’ ((π β (0...π) β§ π β (0...π)) β (π β (0...π) β§ π β (π...π))) |
41 | 40 | simpld 496 |
. . . . . . 7
β’ ((π β (0...π) β§ π β (0...π)) β π β (0...π)) |
42 | 12, 13, 41 | syl2anc 585 |
. . . . . 6
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β π β (0...π)) |
43 | | swrdlen 14542 |
. . . . . 6
β’ ((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β (β―β(π substr β¨π, πβ©)) = (π β π)) |
44 | 11, 42, 14, 43 | syl3anc 1372 |
. . . . 5
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(β―β(π substr
β¨π, πβ©)) = (π β π)) |
45 | 44 | oveq2d 7378 |
. . . 4
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(0..^(β―β(π
substr β¨π, πβ©))) = (0..^(π β π))) |
46 | 45 | fneq2d 6601 |
. . 3
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π substr β¨π, πβ©) Fn (0..^(β―β(π substr β¨π, πβ©))) β (π substr β¨π, πβ©) Fn (0..^(π β π)))) |
47 | 38, 46 | mpbid 231 |
. 2
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π substr β¨π, πβ©) Fn (0..^(π β π))) |
48 | 24, 26 | zsubcld 12619 |
. . . . . 6
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π β π) β β€) |
49 | 48 | anim1ci 617 |
. . . . 5
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (π₯ β (0..^(π β π)) β§ (π β π) β β€)) |
50 | | fzospliti 13611 |
. . . . 5
β’ ((π₯ β (0..^(π β π)) β§ (π β π) β β€) β (π₯ β (0..^(π β π)) β¨ π₯ β ((π β π)..^(π β π)))) |
51 | 49, 50 | syl 17 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (π₯ β (0..^(π β π)) β¨ π₯ β ((π β π)..^(π β π)))) |
52 | 1 | ad2antrr 725 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (π substr β¨π, πβ©) β Word π΄) |
53 | 3 | ad2antrr 725 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (π substr β¨π, πβ©) β Word π΄) |
54 | 20 | oveq2d 7378 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
(0..^(β―β(π
substr β¨π, πβ©))) = (0..^(π β π))) |
55 | 54 | eleq2d 2824 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π₯ β (0..^(β―β(π substr β¨π, πβ©))) β π₯ β (0..^(π β π)))) |
56 | 55 | biimpar 479 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π₯ β (0..^(β―β(π substr β¨π, πβ©)))) |
57 | | ccatval1 14472 |
. . . . . . 7
β’ (((π substr β¨π, πβ©) β Word π΄ β§ (π substr β¨π, πβ©) β Word π΄ β§ π₯ β (0..^(β―β(π substr β¨π, πβ©)))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = ((π substr β¨π, πβ©)βπ₯)) |
58 | 52, 53, 56, 57 | syl3anc 1372 |
. . . . . 6
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = ((π substr β¨π, πβ©)βπ₯)) |
59 | | simpll 766 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β Word π΄) |
60 | | simplr1 1216 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β (0...π)) |
61 | 18 | adantr 482 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β (0...(β―βπ))) |
62 | | simpr 486 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π₯ β (0..^(π β π))) |
63 | | swrdfv 14543 |
. . . . . . 7
β’ (((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β§ π₯ β (0..^(π β π))) β ((π substr β¨π, πβ©)βπ₯) = (πβ(π₯ + π))) |
64 | 59, 60, 61, 62, 63 | syl31anc 1374 |
. . . . . 6
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β ((π substr β¨π, πβ©)βπ₯) = (πβ(π₯ + π))) |
65 | 58, 64 | eqtrd 2777 |
. . . . 5
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = (πβ(π₯ + π))) |
66 | 1 | ad2antrr 725 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π substr β¨π, πβ©) β Word π΄) |
67 | 3 | ad2antrr 725 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π substr β¨π, πβ©) β Word π΄) |
68 | 23, 30 | eqtrd 2777 |
. . . . . . . . . 10
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
((β―β(π substr
β¨π, πβ©)) + (β―β(π substr β¨π, πβ©))) = (π β π)) |
69 | 20, 68 | oveq12d 7380 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β
((β―β(π substr
β¨π, πβ©))..^((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©)))) = ((π β π)..^(π β π))) |
70 | 69 | eleq2d 2824 |
. . . . . . . 8
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π₯ β ((β―β(π substr β¨π, πβ©))..^((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©)))) β π₯ β ((π β π)..^(π β π)))) |
71 | 70 | biimpar 479 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π₯ β ((β―β(π substr β¨π, πβ©))..^((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©))))) |
72 | | ccatval2 14473 |
. . . . . . 7
β’ (((π substr β¨π, πβ©) β Word π΄ β§ (π substr β¨π, πβ©) β Word π΄ β§ π₯ β ((β―β(π substr β¨π, πβ©))..^((β―β(π substr β¨π, πβ©)) + (β―β(π substr β¨π, πβ©))))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = ((π substr β¨π, πβ©)β(π₯ β (β―β(π substr β¨π, πβ©))))) |
73 | 66, 67, 71, 72 | syl3anc 1372 |
. . . . . 6
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = ((π substr β¨π, πβ©)β(π₯ β (β―β(π substr β¨π, πβ©))))) |
74 | | simpll 766 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π β Word π΄) |
75 | | simplr2 1217 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π β (0...π)) |
76 | | simplr3 1218 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π β (0...(β―βπ))) |
77 | 20 | oveq2d 7378 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π₯ β (β―β(π substr β¨π, πβ©))) = (π₯ β (π β π))) |
78 | 77 | adantr 482 |
. . . . . . . 8
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π₯ β (β―β(π substr β¨π, πβ©))) = (π₯ β (π β π))) |
79 | 30 | oveq2d 7378 |
. . . . . . . . . . 11
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π β π)..^((π β π) + (π β π))) = ((π β π)..^(π β π))) |
80 | 79 | eleq2d 2824 |
. . . . . . . . . 10
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π₯ β ((π β π)..^((π β π) + (π β π))) β π₯ β ((π β π)..^(π β π)))) |
81 | 80 | biimpar 479 |
. . . . . . . . 9
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π₯ β ((π β π)..^((π β π) + (π β π)))) |
82 | 28, 24 | zsubcld 12619 |
. . . . . . . . . 10
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π β π) β β€) |
83 | 82 | adantr 482 |
. . . . . . . . 9
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π β π) β β€) |
84 | | fzosubel3 13640 |
. . . . . . . . 9
β’ ((π₯ β ((π β π)..^((π β π) + (π β π))) β§ (π β π) β β€) β (π₯ β (π β π)) β (0..^(π β π))) |
85 | 81, 83, 84 | syl2anc 585 |
. . . . . . . 8
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π₯ β (π β π)) β (0..^(π β π))) |
86 | 78, 85 | eqeltrd 2838 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π₯ β (β―β(π substr β¨π, πβ©))) β (0..^(π β π))) |
87 | | swrdfv 14543 |
. . . . . . 7
β’ (((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β§ (π₯ β (β―β(π substr β¨π, πβ©))) β (0..^(π β π))) β ((π substr β¨π, πβ©)β(π₯ β (β―β(π substr β¨π, πβ©)))) = (πβ((π₯ β (β―β(π substr β¨π, πβ©))) + π))) |
88 | 74, 75, 76, 86, 87 | syl31anc 1374 |
. . . . . 6
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β ((π substr β¨π, πβ©)β(π₯ β (β―β(π substr β¨π, πβ©)))) = (πβ((π₯ β (β―β(π substr β¨π, πβ©))) + π))) |
89 | 77 | oveq1d 7377 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π₯ β (β―β(π substr β¨π, πβ©))) + π) = ((π₯ β (π β π)) + π)) |
90 | 89 | adantr 482 |
. . . . . . . 8
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β ((π₯ β (β―β(π substr β¨π, πβ©))) + π) = ((π₯ β (π β π)) + π)) |
91 | | elfzoelz 13579 |
. . . . . . . . . . 11
β’ (π₯ β ((π β π)..^(π β π)) β π₯ β β€) |
92 | 91 | zcnd 12615 |
. . . . . . . . . 10
β’ (π₯ β ((π β π)..^(π β π)) β π₯ β β) |
93 | 92 | adantl 483 |
. . . . . . . . 9
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π₯ β β) |
94 | 25, 27 | subcld 11519 |
. . . . . . . . . 10
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π β π) β β) |
95 | 94 | adantr 482 |
. . . . . . . . 9
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π β π) β β) |
96 | 25 | adantr 482 |
. . . . . . . . 9
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β π β β) |
97 | 93, 95, 96 | subadd23d 11541 |
. . . . . . . 8
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β ((π₯ β (π β π)) + π) = (π₯ + (π β (π β π)))) |
98 | 25, 27 | nncand 11524 |
. . . . . . . . . 10
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π β (π β π)) = π) |
99 | 98 | oveq2d 7378 |
. . . . . . . . 9
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β (π₯ + (π β (π β π))) = (π₯ + π)) |
100 | 99 | adantr 482 |
. . . . . . . 8
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (π₯ + (π β (π β π))) = (π₯ + π)) |
101 | 90, 97, 100 | 3eqtrd 2781 |
. . . . . . 7
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β ((π₯ β (β―β(π substr β¨π, πβ©))) + π) = (π₯ + π)) |
102 | 101 | fveq2d 6851 |
. . . . . 6
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (πβ((π₯ β (β―β(π substr β¨π, πβ©))) + π)) = (πβ(π₯ + π))) |
103 | 73, 88, 102 | 3eqtrd 2781 |
. . . . 5
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β ((π β π)..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = (πβ(π₯ + π))) |
104 | 65, 103 | jaodan 957 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ (π₯ β (0..^(π β π)) β¨ π₯ β ((π β π)..^(π β π)))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = (πβ(π₯ + π))) |
105 | 51, 104 | syldan 592 |
. . 3
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = (πβ(π₯ + π))) |
106 | | simpll 766 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β Word π΄) |
107 | 42 | adantr 482 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β (0...π)) |
108 | | simplr3 1218 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π β (0...(β―βπ))) |
109 | | simpr 486 |
. . . 4
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β π₯ β (0..^(π β π))) |
110 | | swrdfv 14543 |
. . . 4
β’ (((π β Word π΄ β§ π β (0...π) β§ π β (0...(β―βπ))) β§ π₯ β (0..^(π β π))) β ((π substr β¨π, πβ©)βπ₯) = (πβ(π₯ + π))) |
111 | 106, 107,
108, 109, 110 | syl31anc 1374 |
. . 3
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β ((π substr β¨π, πβ©)βπ₯) = (πβ(π₯ + π))) |
112 | 105, 111 | eqtr4d 2780 |
. 2
β’ (((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β§ π₯ β (0..^(π β π))) β (((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©))βπ₯) = ((π substr β¨π, πβ©)βπ₯)) |
113 | 34, 47, 112 | eqfnfvd 6990 |
1
β’ ((π β Word π΄ β§ (π β (0...π) β§ π β (0...π) β§ π β (0...(β―βπ)))) β ((π substr β¨π, πβ©) ++ (π substr β¨π, πβ©)) = (π substr β¨π, πβ©)) |