Theorem finsumvtxdg2size 29568

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 29569) 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 29111	. . . 4 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
2		fvex 6919	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
3		fvex 6919	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) ∈ V
4	3	resex 6047	. . . . . 6 ⊢ ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ V
5		eleq1 2829	. . . . . . . 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 7440	. . . . . . . . . . 11 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑘VtxDeg𝑒) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
9	8	fveq1d 6908	. . . . . . . . . 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 15742	. . . . . . . 8 ⊢ ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
12		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘𝑒) = (♯‘(iEdg‘𝐺)))
13	12	oveq2d 7447	. . . . . . . . 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 2753	. . . . . . 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 2829	. . . . . . . 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 7440	. . . . . . . . . . . 12 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (𝑤VtxDeg𝑓))
21		df-ov 7434	. . . . . . . . . . . 12 ⊢ (𝑤VtxDeg𝑓) = (VtxDeg‘⟨𝑤, 𝑓⟩)
22	20, 21	eqtrdi 2793	. . . . . . . . . . 11 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑤, 𝑓⟩))
23	22	fveq1d 6908	. . . . . . . . . 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 15742	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
26		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑒 = 𝑓 → (♯‘𝑒) = (♯‘𝑓))
27	26	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑒 = 𝑓 → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
28	27	adantl 481	. . . . . . . 8 ⊢ ((𝑘 = 𝑤 ∧ 𝑒 = 𝑓) → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
29	25, 28	eqeq12d 2753	. . . . . . 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 3484	. . . . . . . . 9 ⊢ 𝑘 ∈ V
32		vex 3484	. . . . . . . . 9 ⊢ 𝑒 ∈ V
33	31, 32	opvtxfvi 29026	. . . . . . . 8 ⊢ (Vtx‘⟨𝑘, 𝑒⟩) = 𝑘
34	33	eqcomi 2746	. . . . . . 7 ⊢ 𝑘 = (Vtx‘⟨𝑘, 𝑒⟩)
35		eqid 2737	. . . . . . 7 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = (iEdg‘⟨𝑘, 𝑒⟩)
36		eqid 2737	. . . . . . 7 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
37		eqid 2737	. . . . . . 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 29323	. . . . . 6 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛 ∈ 𝑘) → ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ ∈ UPGraph)
39		eleq1 2829	. . . . . . . 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 4875	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 6908	. . . . . . . . . 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 15742	. . . . . . . 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 6906	. . . . . . . . . 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 7447	. . . . . . . . 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 2753	. . . . . . 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 14402	. . . . . . . . 9 ⊢ (𝑘 ∈ V → ((♯‘𝑘) = 0 ↔ 𝑘 = ∅))
53	52	elv 3485	. . . . . . . 8 ⊢ ((♯‘𝑘) = 0 ↔ 𝑘 = ∅)
54		2t0e0 12435	. . . . . . . . . 10 ⊢ (2 · 0) = 0
55	54	a1i 11	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · 0) = 0)
56	31, 32	opiedgfvi 29027	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒
57	56	eqcomi 2746	. . . . . . . . . . . 12 ⊢ 𝑒 = (iEdg‘⟨𝑘, 𝑒⟩)
58		upgruhgr 29119	. . . . . . . . . . . . . 14 ⊢ (⟨𝑘, 𝑒⟩ ∈ UPGraph → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
59	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
60	34	eqeq1i 2742	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ∅ ↔ (Vtx‘⟨𝑘, 𝑒⟩) = ∅)
61		uhgr0vb 29089	. . . . . . . . . . . . . 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 2789	. . . . . . . . . . 11 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → 𝑒 = ∅)
65		hasheq0 14402	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ V → ((♯‘𝑒) = 0 ↔ 𝑒 = ∅))
66	65	elv 3485	. . . . . . . . . . 11 ⊢ ((♯‘𝑒) = 0 ↔ 𝑒 = ∅)
67	64, 66	sylibr 234	. . . . . . . . . 10 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (♯‘𝑒) = 0)
68	67	oveq2d 7447	. . . . . . . . 9 ⊢ ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · (♯‘𝑒)) = (2 · 0))
69		sumeq1 15725	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣))
70		sum0 15757	. . . . . . . . . . 11 ⊢ Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣) = 0
71	69, 70	eqtrdi 2793	. . . . . . . . . 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 2788	. . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ ((𝑦 + 1) = (♯‘𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
77	76	eqcoms 2745	. . . . . . . . . 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 14397	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ V → (𝑘 ∈ Fin ↔ (♯‘𝑘) ∈ ℕ0))
80	79	biimprd 248	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin))
81	80	elv 3485	. . . . . . . . . 10 ⊢ ((♯‘𝑘) ∈ ℕ0 → 𝑘 ∈ Fin)
82		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∖ {𝑛}) = (𝑘 ∖ {𝑛})
83		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
84	56	dmeqi 5915	. . . . . . . . . . . . . . . . . 18 ⊢ dom (iEdg‘⟨𝑘, 𝑒⟩) = dom 𝑒
85	84	rabeqi 3450	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
86		eqidd 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → 𝑛 = 𝑛)
87	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ dom 𝑒 → (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒)
88	87	fveq1d 6908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝑒 → ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) = (𝑒‘𝑖))
89	86, 88	neleq12d 3051	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝑒 → (𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) ↔ 𝑛 ∉ (𝑒‘𝑖)))
90	89	rabbiia 3440	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
91	85, 90	eqtri 2765	. . . . . . . . . . . . . . . 16 ⊢ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒 ∣ 𝑛 ∉ (𝑒‘𝑖)}
92	56, 91	reseq12i 5995	. . . . . . . . . . . . . . 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 29567	. . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑘, 𝑒⟩)
95	94	fveq1i 6907	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
96	95	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑘 → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣))
97	96	sumeq2i 15734	. . . . . . . . . . . . . . 15 ⊢ Σ𝑣 ∈ 𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ 𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
98	97	eqeq1i 2742	. . . . . . . . . . . . . 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 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 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 14551	. . . . 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 2832	. . . 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 2832	. . . . 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 29488	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
119	117, 118	eqtrid 2789	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → 𝐷 = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
120	119	fveq1d 6908	. . . . . . 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 15742	. . . . 5 ⊢ (𝐺 ∈ UPGraph → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
123	113	fveq2i 6909	. . . . . . 7 ⊢ (♯‘𝐼) = (♯‘(iEdg‘𝐺))
124	123	oveq2i 7442	. . . . . 6 ⊢ (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺)))
125	124	a1i 11	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺))))
126	122, 125	eqeq12d 2753	. . . 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 1111	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∉ wnel 3046  {crab 3436  Vcvv 3480   ∖ cdif 3948  ∅c0 4333  {csn 4626  ⟨cop 4632  dom cdm 5685   ↾ cres 5687  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  2c2 12321  ℕ₀cn0 12526  ♯chash 14369  Σcsu 15722  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073  UPGraphcupgr 29097  VtxDegcvtxdg 29483

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-vtxdg 29484

This theorem is referenced by:  fusgr1th  29569  finsumvtxdgeven  29570