Theorem finsumvtxdg2size 28807

Description: The sum of the degrees of all vertices of a finite pseudograph of finite size is twice the size of the pseudograph. See equation (1) in section I.1 in [Bollobas] p. 4. Here, the "proof" is simply the statement "Since each edge has two endvertices, the sum of the degrees is exactly twice the number of edges". The formal proof of this theorem (for pseudographs) is much more complicated, taking also the used auxiliary theorems into account. The proof for a (finite) simple graph (see fusgr1th 28808) would be shorter, but nevertheless still laborious. Although this theorem would hold also for infinite pseudographs and pseudographs of infinite size, the proof of this most general version (see theorem "sumvtxdg2size" below) would require many more auxiliary theorems (e.g., the extension of the sum Σ over an arbitrary set).

I dedicate this theorem and its proof to Norman Megill, who deceased too early on December 9, 2021. This proof is an example for the rigor which was the main motivation for Norman Megill to invent and develop Metamath, see section 1.1.6 "Rigor" on page 19 of the Metamath book: "... it is usually assumed in mathematical literature that the person reading the proof is a mathematician familiar with the specialty being described, and that the missing steps are obvious to such a reader or at least the reader is capable of filling them in." I filled in the missing steps of Bollobas' proof as Norm would have liked it... (Contributed by Alexander van der Vekens, 19-Dec-2021.)

Hypotheses

Ref	Expression
sumvtxdg2size.v	⊢ 𝑉 = (Vtx‘𝐺)
sumvtxdg2size.i	⊢ 𝐼 = (iEdg‘𝐺)
sumvtxdg2size.d	⊢ 𝐷 = (VtxDeg‘𝐺)

Assertion

Ref	Expression
finsumvtxdg2size	⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Allowed substitution hints:   𝐷(𝑣)   𝐼(𝑣)

Proof of Theorem finsumvtxdg2size

Dummy variables 𝑒 𝑘 𝑛 𝑓 𝑖 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrop 28354	. . . 4 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
2		fvex 6905	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
3		fvex 6905	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) ∈ V
4	3	resex 6030	. . . . . 6 ⊢ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ V
5		eleq1 2822	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
6	5	adantl 483	. . . . . . 7 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
7		simpl 484	. . . . . . . . 9 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → 𝑘 = (Vtx‘𝐺))
8		oveq12 7418	. . . . . . . . . . 11 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑘VtxDeg𝑒) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
9	8	fveq1d 6894	. . . . . . . . . 10 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
10	9	adantr 482	. . . . . . . . 9 ⊢ (((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) ∧ 𝑣 ∈ 𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
11	7, 10	sumeq12dv 15652	. . . . . . . 8 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
12		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘𝑒) = (♯‘(iEdg‘𝐺)))
13	12	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝐺) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
14	13	adantl 483	. . . . . . . 8 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
15	11, 14	eqeq12d 2749	. . . . . . 7 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
16	6, 15	imbi12d 345	. . . . . 6 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))) ↔ ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
17		eleq1 2822	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
18	17	adantl 483	. . . . . . 7 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
19		simpl 484	. . . . . . . . 9 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑘 = 𝑤)
20		oveq12 7418	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (𝑤VtxDeg𝑓))
21		df-ov 7412	. . . . . . . . . . . 12 ⊢ (𝑤VtxDeg𝑓) = (VtxDeg‘⟨𝑤, 𝑓⟩)
22	20, 21	eqtrdi 2789	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑤, 𝑓⟩))
23	22	fveq1d 6894	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
24	23	adantr 482	. . . . . . . . 9 ⊢ (((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) ∧ 𝑣 ∈ 𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
25	19, 24	sumeq12dv 15652	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
26		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → (♯‘𝑒) = (♯‘𝑓))
27	26	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
28	27	adantl 483	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
29	25, 28	eqeq12d 2749	. . . . . . 7 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓))))
30	18, 29	imbi12d 345	. . . . . 6 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))) ↔ (𝑓 ∈ Fin → Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓)))))
31		vex 3479	. . . . . . . . 9 ⊢ 𝑘 ∈ V
32		vex 3479	. . . . . . . . 9 ⊢ 𝑒 ∈ V
33	31, 32	opvtxfvi 28269	. . . . . . . 8 ⊢ (Vtx‘⟨𝑘, 𝑒⟩) = 𝑘
34	33	eqcomi 2742	. . . . . . 7 ⊢ 𝑘 = (Vtx‘⟨𝑘, 𝑒⟩)
35		eqid 2733	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = (iEdg‘⟨𝑘, 𝑒⟩)
36		eqid 2733	. . . . . . 7 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
37		eqid 2733	. . . . . . 7 ⊢ ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ = ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩
38	34, 35, 36, 37	upgrres 28563	. . . . . 6 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) → ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ ∈ UPGraph)
39		eleq1 2822	. . . . . . . 8 ⊢ (𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) → (𝑓 ∈ Fin ↔ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin))
40	39	adantl 483	. . . . . . 7 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → (𝑓 ∈ Fin ↔ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin))
41		simpl 484	. . . . . . . . 9 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → 𝑤 = (𝑘 ∖ {𝑛}))
42		opeq12 4876	. . . . . . . . . . . 12 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → ⟨𝑤, 𝑓⟩ = ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)
43	42	fveq2d 6896	. . . . . . . . . . 11 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → (VtxDeg‘⟨𝑤, 𝑓⟩) = (VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩))
44	43	fveq1d 6894	. . . . . . . . . 10 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = ((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣))
45	44	adantr 482	. . . . . . . . 9 ⊢ (((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) ∧ 𝑣 ∈ 𝑤) → ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = ((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣))
46	41, 45	sumeq12dv 15652	. . . . . . . 8 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣))
47		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) → (♯‘𝑓) = (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))
48	47	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) → (2 · (♯‘𝑓)) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}))))
49	48	adantl 483	. . . . . . . 8 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → (2 · (♯‘𝑓)) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}))))
50	46, 49	eqeq12d 2749	. . . . . . 7 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → (Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓)) ↔ Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))))
51	40, 50	imbi12d 345	. . . . . 6 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → ((𝑓 ∈ Fin → Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓))) ↔ (((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}))))))
52		hasheq0 14323	. . . . . . . . 9 ⊢ (𝑘 ∈ V → ((♯‘𝑘) = 0 ↔ 𝑘 = ∅))
53	52	elv 3481	. . . . . . . 8 ⊢ ((♯‘𝑘) = 0 ↔ 𝑘 = ∅)
54		2t0e0 12381	. . . . . . . . . 10 ⊢ (2 · 0) = 0
55	54	a1i 11	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · 0) = 0)
56	31, 32	opiedgfvi 28270	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒
57	56	eqcomi 2742	. . . . . . . . . . . 12 ⊢ 𝑒 = (iEdg‘⟨𝑘, 𝑒⟩)
58		upgruhgr 28362	. . . . . . . . . . . . . 14 ⊢ (⟨𝑘, 𝑒⟩ ∈ UPGraph → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
59	58	adantr 482	. . . . . . . . . . . . 13 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
60	34	eqeq1i 2738	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ∅ ↔ (Vtx‘⟨𝑘, 𝑒⟩) = ∅)
61		uhgr0vb 28332	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (Vtx‘⟨𝑘, 𝑒⟩) = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
62	60, 61	sylan2b 595	. . . . . . . . . . . . 13 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
63	59, 62	mpbid 231	. . . . . . . . . . . 12 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (iEdg‘⟨𝑘, 𝑒⟩) = ∅)
64	57, 63	eqtrid 2785	. . . . . . . . . . 11 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → 𝑒 = ∅)
65		hasheq0 14323	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ V → ((♯‘𝑒) = 0 ↔ 𝑒 = ∅))
66	65	elv 3481	. . . . . . . . . . 11 ⊢ ((♯‘𝑒) = 0 ↔ 𝑒 = ∅)
67	64, 66	sylibr 233	. . . . . . . . . 10 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (♯‘𝑒) = 0)
68	67	oveq2d 7425	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · (♯‘𝑒)) = (2 · 0))
69		sumeq1 15635	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣))
70		sum0 15667	. . . . . . . . . . 11 ⊢ Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣) = 0
71	69, 70	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
72	71	adantl 483	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
73	55, 68, 72	3eqtr4rd 2784	. . . . . . . 8 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
74	53, 73	sylan2b 595	. . . . . . 7 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = 0) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
75	74	a1d 25	. . . . . 6 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = 0) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
76		eleq1 2822	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) = (♯‘𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
77	76	eqcoms 2741	. . . . . . . . . 10 ⊢ ((♯‘𝑘) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
78	77	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
79		hashclb 14318	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ V → (𝑘 ∈ Fin ↔ (♯‘𝑘) ∈ ℕ0))
80	79	biimprd 247	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin))
81	80	elv 3481	. . . . . . . . . 10 ⊢ ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin)
82		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∖ {𝑛}) = (𝑘 ∖ {𝑛})
83		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
84	56	dmeqi 5905	. . . . . . . . . . . . . . . . . 18 ⊢ dom (iEdg‘⟨𝑘, 𝑒⟩) = dom 𝑒
85	84	rabeqi 3446	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
86		eqidd 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → 𝑛 = 𝑛)
87	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ dom 𝑒 → (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒)
88	87	fveq1d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) = (𝑒‘𝑖))
89	86, 88	neleq12d 3052	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝑒 → (𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) ↔ 𝑛 ∉ (𝑒‘𝑖)))
90	89	rabbiia 3437	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
91	85, 90	eqtri 2761	. . . . . . . . . . . . . . . 16 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
92	56, 91	reseq12i 5980	. . . . . . . . . . . . . . 15 ⊢ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) = (𝑒 ↾ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)})
93	34, 57, 82, 83, 92, 37	finsumvtxdg2sstep 28806	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) ∧ (𝑘 ∈ Fin ∧ 𝑒 ∈ Fin)) → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒))))
94		df-ov 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑘, 𝑒⟩)
95	94	fveq1i 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
96	95	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑘 → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣))
97	96	sumeq2i 15645	. . . . . . . . . . . . . . 15 ⊢ Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
98	97	eqeq1i 2738	. . . . . . . . . . . . . 14 ⊢ (Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒)))
99	93, 98	imbitrrdi 251	. . . . . . . . . . . . 13 ⊢ (((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) ∧ (𝑘 ∈ Fin ∧ 𝑒 ∈ Fin)) → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
100	99	exp32 422	. . . . . . . . . . . 12 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) → (𝑘 ∈ Fin → (𝑒 ∈ Fin → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
101	100	com34 91	. . . . . . . . . . 11 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) → (𝑘 ∈ Fin → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
102	101	3adant2 1132	. . . . . . . . . 10 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘) → (𝑘 ∈ Fin → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
103	81, 102	syl5 34	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘) → ((♯‘𝑘) ∈ ℕ0 → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
104	78, 103	sylbid 239	. . . . . . . 8 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘) → ((𝑦 + 1) ∈ ℕ0 → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
105	104	impcom 409	. . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘)) → ((((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))))
106	105	imp 408	. . . . . 6 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘)) ∧ (((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ Fin → Σ𝑣 ∈ (𝑘 ∖ {𝑛})((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣) = (2 · (♯‘((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}))))) → (𝑒 ∈ Fin → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
107	2, 4, 16, 30, 38, 51, 75, 106	opfi1ind 14463	. . . . 5 ⊢ ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
108	107	ex 414	. . . 4 ⊢ (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph → ((Vtx‘𝐺) ∈ Fin → ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
109	1, 108	syl 17	. . 3 ⊢ (𝐺 ∈ UPGraph → ((Vtx‘𝐺) ∈ Fin → ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
110		sumvtxdg2size.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
111	110	eleq1i 2825	. . . 4 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
112	111	a1i 11	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin))
113		sumvtxdg2size.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
114	113	eleq1i 2825	. . . . 5 ⊢ (𝐼 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin)
115	114	a1i 11	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐼 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
116	110	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝑉 = (Vtx‘𝐺))
117		sumvtxdg2size.d	. . . . . . . . 9 ⊢ 𝐷 = (VtxDeg‘𝐺)
118		vtxdgop 28727	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
119	117, 118	eqtrid 2785	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → 𝐷 = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
120	119	fveq1d 6894	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (𝐷‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
121	120	adantr 482	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑣 ∈ 𝑉) → (𝐷‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
122	116, 121	sumeq12dv 15652	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
123	113	fveq2i 6895	. . . . . . 7 ⊢ (♯‘𝐼) = (♯‘(iEdg‘𝐺))
124	123	oveq2i 7420	. . . . . 6 ⊢ (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺)))
125	124	a1i 11	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺))))
126	122, 125	eqeq12d 2749	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)) ↔ Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
127	115, 126	imbi12d 345	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝐼 ∈ Fin → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼))) ↔ ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
128	109, 112, 127	3imtr4d 294	. 2 ⊢ (𝐺 ∈ UPGraph → (𝑉 ∈ Fin → (𝐼 ∈ Fin → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))))
129	128	3imp 1112	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∉ wnel 3047  {crab 3433  Vcvv 3475   ∖ cdif 3946  ∅c0 4323  {csn 4629  ⟨cop 4635  dom cdm 5677   ↾ cres 5679  ‘cfv 6544  (class class class)co 7409  Fincfn 8939  0cc0 11110  1c1 11111   + caddc 11113   · cmul 11115  2c2 12267  ℕ₀cn0 12472  ♯chash 14290  Σcsu 15632  Vtxcvtx 28256  iEdgciedg 28257  UHGraphcuhgr 28316  UPGraphcupgr 28340  VtxDegcvtxdg 28722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-oi 9505  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-xadd 13093  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-vtx 28258  df-iedg 28259  df-edg 28308  df-uhgr 28318  df-upgr 28342  df-vtxdg 28723

This theorem is referenced by:  fusgr1th  28808  finsumvtxdgeven  28809