Theorem finsumvtxdg2size 29496

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 29497) 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 29039	. . . 4 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
2		fvex 6835	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
3		fvex 6835	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) ∈ V
4	3	resex 5980	. . . . . 6 ⊢ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ V
5		eleq1 2816	. . . . . . . 8 ⊢ (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
6	5	adantl 481	. . . . . . 7 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
7		simpl 482	. . . . . . . . 9 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → 𝑘 = (Vtx‘𝐺))
8		oveq12 7358	. . . . . . . . . . 11 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑘VtxDeg𝑒) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
9	8	fveq1d 6824	. . . . . . . . . 10 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
10	9	adantr 480	. . . . . . . . 9 ⊢ (((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) ∧ 𝑣 ∈ 𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
11	7, 10	sumeq12dv 15613	. . . . . . . 8 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
12		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘𝑒) = (♯‘(iEdg‘𝐺)))
13	12	oveq2d 7365	. . . . . . . . 9 ⊢ (𝑒 = (iEdg‘𝐺) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
15	11, 14	eqeq12d 2745	. . . . . . 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 344	. . . . . 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 2816	. . . . . . . 8 ⊢ (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
18	17	adantl 481	. . . . . . 7 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
19		simpl 482	. . . . . . . . 9 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑘 = 𝑤)
20		oveq12 7358	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (𝑤VtxDeg𝑓))
21		df-ov 7352	. . . . . . . . . . . 12 ⊢ (𝑤VtxDeg𝑓) = (VtxDeg‘⟨𝑤, 𝑓⟩)
22	20, 21	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑤, 𝑓⟩))
23	22	fveq1d 6824	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
24	23	adantr 480	. . . . . . . . 9 ⊢ (((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) ∧ 𝑣 ∈ 𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
25	19, 24	sumeq12dv 15613	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
26		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → (♯‘𝑒) = (♯‘𝑓))
27	26	oveq2d 7365	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
28	27	adantl 481	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
29	25, 28	eqeq12d 2745	. . . . . . 7 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓))))
30	18, 29	imbi12d 344	. . . . . 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 3440	. . . . . . . . 9 ⊢ 𝑘 ∈ V
32		vex 3440	. . . . . . . . 9 ⊢ 𝑒 ∈ V
33	31, 32	opvtxfvi 28954	. . . . . . . 8 ⊢ (Vtx‘⟨𝑘, 𝑒⟩) = 𝑘
34	33	eqcomi 2738	. . . . . . 7 ⊢ 𝑘 = (Vtx‘⟨𝑘, 𝑒⟩)
35		eqid 2729	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = (iEdg‘⟨𝑘, 𝑒⟩)
36		eqid 2729	. . . . . . 7 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
37		eqid 2729	. . . . . . 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 29251	. . . . . 6 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) → ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ ∈ UPGraph)
39		eleq1 2816	. . . . . . . 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 481	. . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → 𝑤 = (𝑘 ∖ {𝑛}))
42		opeq12 4826	. . . . . . . . . . . 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 6826	. . . . . . . . . . 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 6824	. . . . . . . . . 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 480	. . . . . . . . 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 15613	. . . . . . . 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 6822	. . . . . . . . . 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 7365	. . . . . . . . 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 481	. . . . . . . 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 2745	. . . . . . 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 344	. . . . . 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 14270	. . . . . . . . 9 ⊢ (𝑘 ∈ V → ((♯‘𝑘) = 0 ↔ 𝑘 = ∅))
53	52	elv 3441	. . . . . . . 8 ⊢ ((♯‘𝑘) = 0 ↔ 𝑘 = ∅)
54		2t0e0 12292	. . . . . . . . . 10 ⊢ (2 · 0) = 0
55	54	a1i 11	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · 0) = 0)
56	31, 32	opiedgfvi 28955	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒
57	56	eqcomi 2738	. . . . . . . . . . . 12 ⊢ 𝑒 = (iEdg‘⟨𝑘, 𝑒⟩)
58		upgruhgr 29047	. . . . . . . . . . . . . 14 ⊢ (⟨𝑘, 𝑒⟩ ∈ UPGraph → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
59	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
60	34	eqeq1i 2734	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ∅ ↔ (Vtx‘⟨𝑘, 𝑒⟩) = ∅)
61		uhgr0vb 29017	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (Vtx‘⟨𝑘, 𝑒⟩) = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
62	60, 61	sylan2b 594	. . . . . . . . . . . . 13 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
63	59, 62	mpbid 232	. . . . . . . . . . . 12 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (iEdg‘⟨𝑘, 𝑒⟩) = ∅)
64	57, 63	eqtrid 2776	. . . . . . . . . . 11 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → 𝑒 = ∅)
65		hasheq0 14270	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ V → ((♯‘𝑒) = 0 ↔ 𝑒 = ∅))
66	65	elv 3441	. . . . . . . . . . 11 ⊢ ((♯‘𝑒) = 0 ↔ 𝑒 = ∅)
67	64, 66	sylibr 234	. . . . . . . . . 10 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (♯‘𝑒) = 0)
68	67	oveq2d 7365	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · (♯‘𝑒)) = (2 · 0))
69		sumeq1 15596	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣))
70		sum0 15628	. . . . . . . . . . 11 ⊢ Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣) = 0
71	69, 70	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
72	71	adantl 481	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
73	55, 68, 72	3eqtr4rd 2775	. . . . . . . 8 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
74	53, 73	sylan2b 594	. . . . . . 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 2816	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) = (♯‘𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
77	76	eqcoms 2737	. . . . . . . . . 10 ⊢ ((♯‘𝑘) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
78	77	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
79		hashclb 14265	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ V → (𝑘 ∈ Fin ↔ (♯‘𝑘) ∈ ℕ0))
80	79	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin))
81	80	elv 3441	. . . . . . . . . 10 ⊢ ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin)
82		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∖ {𝑛}) = (𝑘 ∖ {𝑛})
83		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
84	56	dmeqi 5847	. . . . . . . . . . . . . . . . . 18 ⊢ dom (iEdg‘⟨𝑘, 𝑒⟩) = dom 𝑒
85	84	rabeqi 3408	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
86		eqidd 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → 𝑛 = 𝑛)
87	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ dom 𝑒 → (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒)
88	87	fveq1d 6824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) = (𝑒‘𝑖))
89	86, 88	neleq12d 3034	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝑒 → (𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) ↔ 𝑛 ∉ (𝑒‘𝑖)))
90	89	rabbiia 3398	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
91	85, 90	eqtri 2752	. . . . . . . . . . . . . . . 16 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
92	56, 91	reseq12i 5928	. . . . . . . . . . . . . . 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 29495	. . . . . . . . . . . . . 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 7352	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑘, 𝑒⟩)
95	94	fveq1i 6823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
96	95	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑘 → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣))
97	96	sumeq2i 15605	. . . . . . . . . . . . . . 15 ⊢ Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
98	97	eqeq1i 2734	. . . . . . . . . . . . . 14 ⊢ (Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒)))
99	93, 98	imbitrrdi 252	. . . . . . . . . . . . 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 420	. . . . . . . . . . . 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 1131	. . . . . . . . . 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 240	. . . . . . . 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 407	. . . . . . 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 406	. . . . . 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 14419	. . . . 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 412	. . . 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 2819	. . . 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 2819	. . . . 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 29416	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
119	117, 118	eqtrid 2776	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → 𝐷 = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
120	119	fveq1d 6824	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (𝐷‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
121	120	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑣 ∈ 𝑉) → (𝐷‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
122	116, 121	sumeq12dv 15613	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
123	113	fveq2i 6825	. . . . . . 7 ⊢ (♯‘𝐼) = (♯‘(iEdg‘𝐺))
124	123	oveq2i 7360	. . . . . 6 ⊢ (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺)))
125	124	a1i 11	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺))))
126	122, 125	eqeq12d 2745	. . . 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 344	. . 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 1110	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  {crab 3394  Vcvv 3436   ∖ cdif 3900  ∅c0 4284  {csn 4577  ⟨cop 4583  dom cdm 5619   ↾ cres 5621  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  2c2 12183  ℕ₀cn0 12384  ♯chash 14237  Σcsu 15593  Vtxcvtx 28941  iEdgciedg 28942  UHGraphcuhgr 29001  UPGraphcupgr 29025  VtxDegcvtxdg 29411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-rp 12894  df-xadd 13015  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-vtx 28943  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-upgr 29027  df-vtxdg 29412

This theorem is referenced by:  fusgr1th  29497  finsumvtxdgeven  29498