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 ∈ dom 𝐸 ∣ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } |
5 |
|
finsumvtxdg2sstep.p |
⊢ 𝑃 = ( 𝐸 ↾ 𝐼 ) |
6 |
|
finsumvtxdg2sstep.s |
⊢ 𝑆 = 〈 𝐾 , 𝑃 〉 |
7 |
|
finsumvtxdg2ssteplem.j |
⊢ 𝐽 = { 𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ ( 𝐸 ‘ 𝑖 ) } |
8 |
|
upgruhgr |
⊢ ( 𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) |
9 |
2
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun 𝐸 ) |
10 |
8 9
|
syl |
⊢ ( 𝐺 ∈ UPGraph → Fun 𝐸 ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → Fun 𝐸 ) |
12 |
|
simprr |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → 𝐸 ∈ Fin ) |
13 |
4
|
ssrab3 |
⊢ 𝐼 ⊆ dom 𝐸 |
14 |
13
|
a1i |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → 𝐼 ⊆ dom 𝐸 ) |
15 |
|
hashreshashfun |
⊢ ( ( Fun 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝐼 ⊆ dom 𝐸 ) → ( ♯ ‘ 𝐸 ) = ( ( ♯ ‘ ( 𝐸 ↾ 𝐼 ) ) + ( ♯ ‘ ( dom 𝐸 ∖ 𝐼 ) ) ) ) |
16 |
11 12 14 15
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( ♯ ‘ 𝐸 ) = ( ( ♯ ‘ ( 𝐸 ↾ 𝐼 ) ) + ( ♯ ‘ ( dom 𝐸 ∖ 𝐼 ) ) ) ) |
17 |
5
|
eqcomi |
⊢ ( 𝐸 ↾ 𝐼 ) = 𝑃 |
18 |
17
|
fveq2i |
⊢ ( ♯ ‘ ( 𝐸 ↾ 𝐼 ) ) = ( ♯ ‘ 𝑃 ) |
19 |
18
|
a1i |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( ♯ ‘ ( 𝐸 ↾ 𝐼 ) ) = ( ♯ ‘ 𝑃 ) ) |
20 |
|
notrab |
⊢ ( dom 𝐸 ∖ { 𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } ) = { 𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } |
21 |
4
|
difeq2i |
⊢ ( dom 𝐸 ∖ 𝐼 ) = ( dom 𝐸 ∖ { 𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } ) |
22 |
|
nnel |
⊢ ( ¬ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) ↔ 𝑁 ∈ ( 𝐸 ‘ 𝑖 ) ) |
23 |
22
|
bicomi |
⊢ ( 𝑁 ∈ ( 𝐸 ‘ 𝑖 ) ↔ ¬ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) ) |
24 |
23
|
rabbii |
⊢ { 𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ ( 𝐸 ‘ 𝑖 ) } = { 𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } |
25 |
7 24
|
eqtri |
⊢ 𝐽 = { 𝑖 ∈ dom 𝐸 ∣ ¬ 𝑁 ∉ ( 𝐸 ‘ 𝑖 ) } |
26 |
20 21 25
|
3eqtr4i |
⊢ ( dom 𝐸 ∖ 𝐼 ) = 𝐽 |
27 |
26
|
a1i |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( dom 𝐸 ∖ 𝐼 ) = 𝐽 ) |
28 |
27
|
fveq2d |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( ♯ ‘ ( dom 𝐸 ∖ 𝐼 ) ) = ( ♯ ‘ 𝐽 ) ) |
29 |
19 28
|
oveq12d |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( ( ♯ ‘ ( 𝐸 ↾ 𝐼 ) ) + ( ♯ ‘ ( dom 𝐸 ∖ 𝐼 ) ) ) = ( ( ♯ ‘ 𝑃 ) + ( ♯ ‘ 𝐽 ) ) ) |
30 |
16 29
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑉 ∈ Fin ∧ 𝐸 ∈ Fin ) ) → ( ♯ ‘ 𝐸 ) = ( ( ♯ ‘ 𝑃 ) + ( ♯ ‘ 𝐽 ) ) ) |