Step | Hyp | Ref
| Expression |
1 | | simp1 1137 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β π β
(β€β₯β3)) |
2 | | isclwwlknon 29077 |
. . . . 5
β’ (π β (π(ClWWalksNOnβπΊ)π) β (π β (π ClWWalksN πΊ) β§ (πβ0) = π)) |
3 | 2 | simplbi 499 |
. . . 4
β’ (π β (π(ClWWalksNOnβπΊ)π) β π β (π ClWWalksN πΊ)) |
4 | 3 | 3ad2ant2 1135 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β π β (π ClWWalksN πΊ)) |
5 | | simpr 486 |
. . . . . . . 8
β’ ((π β (π ClWWalksN πΊ) β§ (πβ0) = π) β (πβ0) = π) |
6 | 5 | eqcomd 2743 |
. . . . . . 7
β’ ((π β (π ClWWalksN πΊ) β§ (πβ0) = π) β π = (πβ0)) |
7 | 2, 6 | sylbi 216 |
. . . . . 6
β’ (π β (π(ClWWalksNOnβπΊ)π) β π = (πβ0)) |
8 | 7 | eqeq2d 2748 |
. . . . 5
β’ (π β (π(ClWWalksNOnβπΊ)π) β ((πβ(π β 2)) = π β (πβ(π β 2)) = (πβ0))) |
9 | 8 | biimpa 478 |
. . . 4
β’ ((π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β (πβ(π β 2)) = (πβ0)) |
10 | 9 | 3adant1 1131 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β (πβ(π β 2)) = (πβ0)) |
11 | | clwwnrepclwwn 29330 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π ClWWalksN πΊ) β§ (πβ(π β 2)) = (πβ0)) β (π prefix (π β 2)) β ((π β 2) ClWWalksN πΊ)) |
12 | 1, 4, 10, 11 | syl3anc 1372 |
. 2
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β (π prefix (π β 2)) β ((π β 2) ClWWalksN πΊ)) |
13 | | 2clwwlklem 29329 |
. . . . . 6
β’ ((π β (π ClWWalksN πΊ) β§ π β (β€β₯β3))
β ((π prefix (π β 2))β0) = (πβ0)) |
14 | 3, 13 | sylan 581 |
. . . . 5
β’ ((π β (π(ClWWalksNOnβπΊ)π) β§ π β (β€β₯β3))
β ((π prefix (π β 2))β0) = (πβ0)) |
15 | 14 | ancoms 460 |
. . . 4
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π)) β ((π prefix (π β 2))β0) = (πβ0)) |
16 | 15 | 3adant3 1133 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β ((π prefix (π β 2))β0) = (πβ0)) |
17 | 2 | simprbi 498 |
. . . 4
β’ (π β (π(ClWWalksNOnβπΊ)π) β (πβ0) = π) |
18 | 17 | 3ad2ant2 1135 |
. . 3
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β (πβ0) = π) |
19 | 16, 18 | eqtrd 2777 |
. 2
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β ((π prefix (π β 2))β0) = π) |
20 | | isclwwlknon 29077 |
. 2
β’ ((π prefix (π β 2)) β (π(ClWWalksNOnβπΊ)(π β 2)) β ((π prefix (π β 2)) β ((π β 2) ClWWalksN πΊ) β§ ((π prefix (π β 2))β0) = π)) |
21 | 12, 19, 20 | sylanbrc 584 |
1
β’ ((π β
(β€β₯β3) β§ π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π) β (π prefix (π β 2)) β (π(ClWWalksNOnβπΊ)(π β 2))) |