Theorem finsumvtxdg2sstep 30157

Description: Induction step of finsumvtxdg2size 30158: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.)

Hypotheses

Ref	Expression
finsumvtxdg2sstep.v	⊢ 𝑉 = (Vtx‘𝐺)
finsumvtxdg2sstep.e	⊢ 𝐸 = (iEdg‘𝐺)
finsumvtxdg2sstep.k	⊢ 𝐾 = (𝑉 ∖ {𝑁})
finsumvtxdg2sstep.i	⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
finsumvtxdg2sstep.p	⊢ 𝑃 = (𝐸 ↾ 𝐼)
finsumvtxdg2sstep.s	⊢ 𝑆 = ⟨𝐾, 𝑃⟩

Assertion

Ref	Expression
finsumvtxdg2sstep	⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))

Distinct variable groups:   𝑖,𝐸   𝑖,𝐺   𝑖,𝑁   𝑣,𝐸   𝑣,𝐺   𝑣,𝐾   𝑣,𝑁   𝑖,𝑉,𝑣

Allowed substitution hints:   𝑃(𝑣, 𝑖)   𝑆(𝑣, 𝑖)   𝐼(𝑣, 𝑖)   𝐾(𝑖)

Proof of Theorem finsumvtxdg2sstep

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		finsumvtxdg2sstep.p	. . 3 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
2		finresfin 9288	. . . 4 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin)
3	2	ad2antll 742	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸 ↾ 𝐼) ∈ Fin)
4	1, 3	eqeltrid 2868	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5		difsnid 4771	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
6	5	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
7	6	eqcomd 2769	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
8	7	sumeq1d 15893	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9		diffi 9215	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
10	9	adantr 486	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
11	10	adantl 487	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12		simpr 490	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
13	12	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∈ 𝑉)
14		neldifsn 4755	. . . . . . . . 9 ⊢ ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
15	14	nelir 3068	. . . . . . . 8 ⊢ 𝑁 ∉ (𝑉 ∖ {𝑁})
16	15	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17		dmfi 9349	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
18	17	ad2antll 742	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
19	5	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣 ∈ 𝑉))
20	19	biimpd 232	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣 ∈ 𝑉))
21	20	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣 ∈ 𝑉))
22	21	imp 412	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → 𝑣 ∈ 𝑉)
23		finsumvtxdg2sstep.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
24		finsumvtxdg2sstep.e	. . . . . . . . . . 11 ⊢ 𝐸 = (iEdg‘𝐺)
25		eqid 2763	. . . . . . . . . . 11 ⊢ dom 𝐸 = dom 𝐸
26	23, 24, 25	vtxdgfisnn0 30083	. . . . . . . . . 10 ⊢ ((dom 𝐸 ∈ Fin ∧ 𝑣 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
27	18, 22, 26	syl2an2r 698	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
28	27	nn0zd 12742	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
29	28	ralrimiva 3158	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30		fsumsplitsnun 15947	. . . . . . 7 ⊢ (((𝑉 ∖ {𝑁}) ∈ Fin ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ∉ (𝑉 ∖ {𝑁})) ∧ ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣)))
31	11, 13, 16, 29, 30	syl121anc 1403	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣)))
32		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
33	32	adantl 487	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
34	12, 33	csbied 3887	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
35	34	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
36	35	oveq2d 7456	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
37	8, 31, 36	3eqtrd 2802	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
38	37	adantr 486	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
39		finsumvtxdg2sstep.k	. . . . . . . 8 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
40		finsumvtxdg2sstep.i	. . . . . . . 8 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
41		finsumvtxdg2sstep.s	. . . . . . . 8 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
42		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐸‘𝑗) = (𝐸‘𝑖))
43	42	eleq2d 2850	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑁 ∈ (𝐸‘𝑗) ↔ 𝑁 ∈ (𝐸‘𝑖)))
44	43	cbvrabv 3426	. . . . . . . 8 ⊢ {𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
45	23, 24, 39, 40, 1, 41, 44	finsumvtxdg2ssteplem2 30154	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
46	45	oveq2d 7456	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
47	46	adantr 486	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
48	23, 24, 39, 40, 1, 41, 44	finsumvtxdg2ssteplem4 30156	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))))
49	44	fveq2i 6908	. . . . . . . 8 ⊢ (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})
50	49	oveq2i 7451	. . . . . . 7 ⊢ ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))
51	50	oveq2i 7451	. . . . . 6 ⊢ (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})))
52	51	a1i 11	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
53	47, 48, 52	3eqtrd 2802	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
54		eqid 2763	. . . . . . . 8 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
55	23, 24, 39, 40, 1, 41, 54	finsumvtxdg2ssteplem1 30153	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})))
56	55	oveq2d 7456	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
57	56	eqcomd 2769	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))) = (2 · (♯‘𝐸)))
58	57	adantr 486	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))) = (2 · (♯‘𝐸)))
59	38, 53, 58	3eqtrd 2802	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
60	59	ex 418	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
61	4, 60	embantd 60	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  ∀wral 3080  {crab 3416  ⦋csb 3851   ∖ cdif 4048   ∪ cun 4152  {csn 4588  ⟨cop 4838  dom cdm 5655   ↾ cres 5657  ‘cfv 6555  (class class class)co 7440  Fincfn 8998   + caddc 11227   · cmul 11229  2c2 12421  ℕ₀cn0 12630  ℤcz 12717  ♯chash 14498  Σcsu 15879  Vtxcvtx 29601  iEdgciedg 29602  UPGraphcupgr 29685  VtxDegcvtxdg 30073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-oi 9528  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-rp 13145  df-xadd 13266  df-fz 13664  df-fzo 13813  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-sum 15880  df-vtx 29603  df-iedg 29604  df-edg 29653  df-uhgr 29663  df-upgr 29687  df-vtxdg 30074

This theorem is used by:  finsumvtxdg2size  30158