Step |
Hyp |
Ref |
Expression |
1 |
|
finsumvtxdg2sstep.v |
|- V = ( Vtx ` G ) |
2 |
|
finsumvtxdg2sstep.e |
|- E = ( iEdg ` G ) |
3 |
|
finsumvtxdg2sstep.k |
|- K = ( V \ { N } ) |
4 |
|
finsumvtxdg2sstep.i |
|- I = { i e. dom E | N e/ ( E ` i ) } |
5 |
|
finsumvtxdg2sstep.p |
|- P = ( E |` I ) |
6 |
|
finsumvtxdg2sstep.s |
|- S = <. K , P >. |
7 |
|
finsumvtxdg2ssteplem.j |
|- J = { i e. dom E | N e. ( E ` i ) } |
8 |
|
upgruhgr |
|- ( G e. UPGraph -> G e. UHGraph ) |
9 |
2
|
uhgrfun |
|- ( G e. UHGraph -> Fun E ) |
10 |
8 9
|
syl |
|- ( G e. UPGraph -> Fun E ) |
11 |
10
|
ad2antrr |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> Fun E ) |
12 |
|
simprr |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> E e. Fin ) |
13 |
4
|
ssrab3 |
|- I C_ dom E |
14 |
13
|
a1i |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> I C_ dom E ) |
15 |
|
hashreshashfun |
|- ( ( Fun E /\ E e. Fin /\ I C_ dom E ) -> ( # ` E ) = ( ( # ` ( E |` I ) ) + ( # ` ( dom E \ I ) ) ) ) |
16 |
11 12 14 15
|
syl3anc |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( # ` E ) = ( ( # ` ( E |` I ) ) + ( # ` ( dom E \ I ) ) ) ) |
17 |
5
|
eqcomi |
|- ( E |` I ) = P |
18 |
17
|
fveq2i |
|- ( # ` ( E |` I ) ) = ( # ` P ) |
19 |
18
|
a1i |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( # ` ( E |` I ) ) = ( # ` P ) ) |
20 |
|
notrab |
|- ( dom E \ { i e. dom E | N e/ ( E ` i ) } ) = { i e. dom E | -. N e/ ( E ` i ) } |
21 |
4
|
difeq2i |
|- ( dom E \ I ) = ( dom E \ { i e. dom E | N e/ ( E ` i ) } ) |
22 |
|
nnel |
|- ( -. N e/ ( E ` i ) <-> N e. ( E ` i ) ) |
23 |
22
|
bicomi |
|- ( N e. ( E ` i ) <-> -. N e/ ( E ` i ) ) |
24 |
23
|
rabbii |
|- { i e. dom E | N e. ( E ` i ) } = { i e. dom E | -. N e/ ( E ` i ) } |
25 |
7 24
|
eqtri |
|- J = { i e. dom E | -. N e/ ( E ` i ) } |
26 |
20 21 25
|
3eqtr4i |
|- ( dom E \ I ) = J |
27 |
26
|
a1i |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( dom E \ I ) = J ) |
28 |
27
|
fveq2d |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( # ` ( dom E \ I ) ) = ( # ` J ) ) |
29 |
19 28
|
oveq12d |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( ( # ` ( E |` I ) ) + ( # ` ( dom E \ I ) ) ) = ( ( # ` P ) + ( # ` J ) ) ) |
30 |
16 29
|
eqtrd |
|- ( ( ( G e. UPGraph /\ N e. V ) /\ ( V e. Fin /\ E e. Fin ) ) -> ( # ` E ) = ( ( # ` P ) + ( # ` J ) ) ) |