Step | Hyp | Ref
| Expression |
1 | | vtxdlfgrval.d |
. . . 4
β’ π· = (VtxDegβπΊ) |
2 | 1 | fveq1i 6847 |
. . 3
β’ (π·βπ) = ((VtxDegβπΊ)βπ) |
3 | | vtxdlfgrval.v |
. . . . 5
β’ π = (VtxβπΊ) |
4 | | vtxdlfgrval.i |
. . . . 5
β’ πΌ = (iEdgβπΊ) |
5 | | vtxdlfgrval.a |
. . . . 5
β’ π΄ = dom πΌ |
6 | 3, 4, 5 | vtxdgval 28465 |
. . . 4
β’ (π β π β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π
(β―β{π₯ β
π΄ β£ (πΌβπ₯) = {π}}))) |
7 | 6 | adantl 483 |
. . 3
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β ((VtxDegβπΊ)βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π
(β―β{π₯ β
π΄ β£ (πΌβπ₯) = {π}}))) |
8 | 2, 7 | eqtrid 2785 |
. 2
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β (π·βπ) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π
(β―β{π₯ β
π΄ β£ (πΌβπ₯) = {π}}))) |
9 | | eqid 2733 |
. . . . . . 7
β’ {π₯ β π« π β£ 2 β€
(β―βπ₯)} = {π₯ β π« π β£ 2 β€
(β―βπ₯)} |
10 | 4, 5, 9 | lfgrnloop 28125 |
. . . . . 6
β’ (πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β {π₯ β π΄ β£ (πΌβπ₯) = {π}} = β
) |
11 | 10 | adantr 482 |
. . . . 5
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β {π₯ β π΄ β£ (πΌβπ₯) = {π}} = β
) |
12 | 11 | fveq2d 6850 |
. . . 4
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) =
(β―ββ
)) |
13 | | hash0 14276 |
. . . 4
β’
(β―ββ
) = 0 |
14 | 12, 13 | eqtrdi 2789 |
. . 3
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β (β―β{π₯ β π΄ β£ (πΌβπ₯) = {π}}) = 0) |
15 | 14 | oveq2d 7377 |
. 2
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π
(β―β{π₯ β
π΄ β£ (πΌβπ₯) = {π}})) = ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π
0)) |
16 | 4 | dmeqi 5864 |
. . . . . . 7
β’ dom πΌ = dom (iEdgβπΊ) |
17 | 5, 16 | eqtri 2761 |
. . . . . 6
β’ π΄ = dom (iEdgβπΊ) |
18 | | fvex 6859 |
. . . . . . 7
β’
(iEdgβπΊ)
β V |
19 | 18 | dmex 7852 |
. . . . . 6
β’ dom
(iEdgβπΊ) β
V |
20 | 17, 19 | eqeltri 2830 |
. . . . 5
β’ π΄ β V |
21 | 20 | rabex 5293 |
. . . 4
β’ {π₯ β π΄ β£ π β (πΌβπ₯)} β V |
22 | | hashxnn0 14248 |
. . . 4
β’ ({π₯ β π΄ β£ π β (πΌβπ₯)} β V β (β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) β
β0*) |
23 | | xnn0xr 12498 |
. . . 4
β’
((β―β{π₯
β π΄ β£ π β (πΌβπ₯)}) β β0*
β (β―β{π₯
β π΄ β£ π β (πΌβπ₯)}) β
β*) |
24 | 21, 22, 23 | mp2b 10 |
. . 3
β’
(β―β{π₯
β π΄ β£ π β (πΌβπ₯)}) β
β* |
25 | | xaddid1 13169 |
. . 3
β’
((β―β{π₯
β π΄ β£ π β (πΌβπ₯)}) β β* β
((β―β{π₯ β
π΄ β£ π β (πΌβπ₯)}) +π 0) =
(β―β{π₯ β
π΄ β£ π β (πΌβπ₯)})) |
26 | 24, 25 | mp1i 13 |
. 2
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β ((β―β{π₯ β π΄ β£ π β (πΌβπ₯)}) +π 0) =
(β―β{π₯ β
π΄ β£ π β (πΌβπ₯)})) |
27 | 8, 15, 26 | 3eqtrd 2777 |
1
β’ ((πΌ:π΄βΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β§ π β π) β (π·βπ) = (β―β{π₯ β π΄ β£ π β (πΌβπ₯)})) |