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Theorem finsumvtxdg2size 27384
 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 27385) 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) → Σ𝑣𝑉 (𝐷𝑣) = (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.
StepHypRef Expression
1 upgrop 26931 . . . 4 (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
2 fvex 6668 . . . . . 6 (iEdg‘𝐺) ∈ V
3 fvex 6668 . . . . . . 7 (iEdg‘⟨𝑘, 𝑒⟩) ∈ V
43resex 5870 . . . . . 6 ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ V
5 eleq1 2877 . . . . . . . 8 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
65adantl 485 . . . . . . 7 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
7 simpl 486 . . . . . . . . 9 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → 𝑘 = (Vtx‘𝐺))
8 oveq12 7154 . . . . . . . . . . 11 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑘VtxDeg𝑒) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
98fveq1d 6657 . . . . . . . . . 10 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
109adantr 484 . . . . . . . . 9 (((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) ∧ 𝑣𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
117, 10sumeq12dv 15075 . . . . . . . 8 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
12 fveq2 6655 . . . . . . . . . 10 (𝑒 = (iEdg‘𝐺) → (♯‘𝑒) = (♯‘(iEdg‘𝐺)))
1312oveq2d 7161 . . . . . . . . 9 (𝑒 = (iEdg‘𝐺) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
1413adantl 485 . . . . . . . 8 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
1511, 14eqeq12d 2814 . . . . . . 7 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
166, 15imbi12d 348 . . . . . 6 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑒 ∈ Fin → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))) ↔ ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
17 eleq1 2877 . . . . . . . 8 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1817adantl 485 . . . . . . 7 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
19 simpl 486 . . . . . . . . 9 ((𝑘 = 𝑤𝑒 = 𝑓) → 𝑘 = 𝑤)
20 oveq12 7154 . . . . . . . . . . . 12 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (𝑤VtxDeg𝑓))
21 df-ov 7148 . . . . . . . . . . . 12 (𝑤VtxDeg𝑓) = (VtxDeg‘⟨𝑤, 𝑓⟩)
2220, 21eqtrdi 2849 . . . . . . . . . . 11 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑤, 𝑓⟩))
2322fveq1d 6657 . . . . . . . . . 10 ((𝑘 = 𝑤𝑒 = 𝑓) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
2423adantr 484 . . . . . . . . 9 (((𝑘 = 𝑤𝑒 = 𝑓) ∧ 𝑣𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
2519, 24sumeq12dv 15075 . . . . . . . 8 ((𝑘 = 𝑤𝑒 = 𝑓) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
26 fveq2 6655 . . . . . . . . . 10 (𝑒 = 𝑓 → (♯‘𝑒) = (♯‘𝑓))
2726oveq2d 7161 . . . . . . . . 9 (𝑒 = 𝑓 → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
2827adantl 485 . . . . . . . 8 ((𝑘 = 𝑤𝑒 = 𝑓) → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
2925, 28eqeq12d 2814 . . . . . . 7 ((𝑘 = 𝑤𝑒 = 𝑓) → (Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓))))
3018, 29imbi12d 348 . . . . . 6 ((𝑘 = 𝑤𝑒 = 𝑓) → ((𝑒 ∈ Fin → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))) ↔ (𝑓 ∈ Fin → Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓)))))
31 vex 3445 . . . . . . . . 9 𝑘 ∈ V
32 vex 3445 . . . . . . . . 9 𝑒 ∈ V
3331, 32opvtxfvi 26846 . . . . . . . 8 (Vtx‘⟨𝑘, 𝑒⟩) = 𝑘
3433eqcomi 2807 . . . . . . 7 𝑘 = (Vtx‘⟨𝑘, 𝑒⟩)
35 eqid 2798 . . . . . . 7 (iEdg‘⟨𝑘, 𝑒⟩) = (iEdg‘⟨𝑘, 𝑒⟩)
36 eqid 2798 . . . . . . 7 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
37 eqid 2798 . . . . . . 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)})⟩
3834, 35, 36, 37upgrres 27140 . . . . . 6 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) → ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ ∈ UPGraph)
39 eleq1 2877 . . . . . . . 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))
4039adantl 485 . . . . . . 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 486 . . . . . . . . 9 ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → 𝑤 = (𝑘 ∖ {𝑛}))
42 opeq12 4771 . . . . . . . . . . . 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)})⟩)
4342fveq2d 6659 . . . . . . . . . . 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)})⟩))
4443fveq1d 6657 . . . . . . . . . 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))
4544adantr 484 . . . . . . . . 9 (((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) ∧ 𝑣𝑤) → ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = ((VtxDeg‘⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩)‘𝑣))
4641, 45sumeq12dv 15075 . . . . . . . 8 ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → Σ𝑣𝑤 ((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 6655 . . . . . . . . . 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)})))
4847oveq2d 7161 . . . . . . . . 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)}))))
4948adantl 485 . . . . . . . 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)}))))
5046, 49eqeq12d 2814 . . . . . . 7 ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → (Σ𝑣𝑤 ((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)})))))
5140, 50imbi12d 348 . . . . . 6 ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → ((𝑓 ∈ Fin → Σ𝑣𝑤 ((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 13740 . . . . . . . . 9 (𝑘 ∈ V → ((♯‘𝑘) = 0 ↔ 𝑘 = ∅))
5352elv 3447 . . . . . . . 8 ((♯‘𝑘) = 0 ↔ 𝑘 = ∅)
54 2t0e0 11812 . . . . . . . . . 10 (2 · 0) = 0
5554a1i 11 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · 0) = 0)
5631, 32opiedgfvi 26847 . . . . . . . . . . . . 13 (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒
5756eqcomi 2807 . . . . . . . . . . . 12 𝑒 = (iEdg‘⟨𝑘, 𝑒⟩)
58 upgruhgr 26939 . . . . . . . . . . . . . 14 (⟨𝑘, 𝑒⟩ ∈ UPGraph → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
5958adantr 484 . . . . . . . . . . . . 13 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
6034eqeq1i 2803 . . . . . . . . . . . . . 14 (𝑘 = ∅ ↔ (Vtx‘⟨𝑘, 𝑒⟩) = ∅)
61 uhgr0vb 26909 . . . . . . . . . . . . . 14 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (Vtx‘⟨𝑘, 𝑒⟩) = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
6260, 61sylan2b 596 . . . . . . . . . . . . 13 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
6359, 62mpbid 235 . . . . . . . . . . . 12 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (iEdg‘⟨𝑘, 𝑒⟩) = ∅)
6457, 63syl5eq 2845 . . . . . . . . . . 11 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → 𝑒 = ∅)
65 hasheq0 13740 . . . . . . . . . . . 12 (𝑒 ∈ V → ((♯‘𝑒) = 0 ↔ 𝑒 = ∅))
6665elv 3447 . . . . . . . . . . 11 ((♯‘𝑒) = 0 ↔ 𝑒 = ∅)
6764, 66sylibr 237 . . . . . . . . . 10 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (♯‘𝑒) = 0)
6867oveq2d 7161 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · (♯‘𝑒)) = (2 · 0))
69 sumeq1 15057 . . . . . . . . . . 11 (𝑘 = ∅ → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣))
70 sum0 15090 . . . . . . . . . . 11 Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣) = 0
7169, 70eqtrdi 2849 . . . . . . . . . 10 (𝑘 = ∅ → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
7271adantl 485 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
7355, 68, 723eqtr4rd 2844 . . . . . . . 8 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
7453, 73sylan2b 596 . . . . . . 7 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = 0) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
7574a1d 25 . . . . . 6 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = 0) → (𝑒 ∈ Fin → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
76 eleq1 2877 . . . . . . . . . . 11 ((𝑦 + 1) = (♯‘𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
7776eqcoms 2806 . . . . . . . . . 10 ((♯‘𝑘) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
78773ad2ant2 1131 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
79 hashclb 13735 . . . . . . . . . . . 12 (𝑘 ∈ V → (𝑘 ∈ Fin ↔ (♯‘𝑘) ∈ ℕ0))
8079biimprd 251 . . . . . . . . . . 11 (𝑘 ∈ V → ((♯‘𝑘) ∈ ℕ0𝑘 ∈ Fin))
8180elv 3447 . . . . . . . . . 10 ((♯‘𝑘) ∈ ℕ0𝑘 ∈ Fin)
82 eqid 2798 . . . . . . . . . . . . . . 15 (𝑘 ∖ {𝑛}) = (𝑘 ∖ {𝑛})
83 eqid 2798 . . . . . . . . . . . . . . 15 {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
8456dmeqi 5743 . . . . . . . . . . . . . . . . . 18 dom (iEdg‘⟨𝑘, 𝑒⟩) = dom 𝑒
8584rabeqi 3430 . . . . . . . . . . . . . . . . 17 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
86 eqidd 2799 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝑒𝑛 = 𝑛)
8756a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ dom 𝑒 → (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒)
8887fveq1d 6657 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝑒 → ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) = (𝑒𝑖))
8986, 88neleq12d 3095 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝑒 → (𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) ↔ 𝑛 ∉ (𝑒𝑖)))
9089rabbiia 3420 . . . . . . . . . . . . . . . . 17 {𝑖 ∈ dom 𝑒𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
9185, 90eqtri 2821 . . . . . . . . . . . . . . . 16 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
9256, 91reseq12i 5820 . . . . . . . . . . . . . . 15 ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})
9334, 57, 82, 83, 92, 37finsumvtxdg2sstep 27383 . . . . . . . . . . . . . 14 (((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) ∧ (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)})))) → Σ𝑣𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒))))
94 df-ov 7148 . . . . . . . . . . . . . . . . . 18 (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑘, 𝑒⟩)
9594fveq1i 6656 . . . . . . . . . . . . . . . . 17 ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
9695a1i 11 . . . . . . . . . . . . . . . 16 (𝑣𝑘 → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣))
9796sumeq2i 15068 . . . . . . . . . . . . . . 15 Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
9897eqeq1i 2803 . . . . . . . . . . . . . 14 𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒)))
9993, 98syl6ibr 255 . . . . . . . . . . . . 13 (((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) ∧ (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)})))) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
10099exp32 424 . . . . . . . . . . . 12 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) → (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)})))) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
101100com34 91 . . . . . . . . . . 11 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) → (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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
1021013adant2 1128 . . . . . . . . . 10 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘) → (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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
10381, 102syl5 34 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘) → ((♯‘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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
10478, 103sylbid 243 . . . . . . . 8 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘) → ((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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))))
105104impcom 411 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘)) → ((((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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))))
106105imp 410 . . . . . 6 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘)) ∧ (((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 → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))))
1072, 4, 16, 30, 38, 51, 75, 106opfi1ind 13876 . . . . 5 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
108107ex 416 . . . 4 (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph → ((Vtx‘𝐺) ∈ Fin → ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
1091, 108syl 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)
111110eleq1i 2880 . . . 4 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
112111a1i 11 . . 3 (𝐺 ∈ UPGraph → (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin))
113 sumvtxdg2size.i . . . . . 6 𝐼 = (iEdg‘𝐺)
114113eleq1i 2880 . . . . 5 (𝐼 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin)
115114a1i 11 . . . 4 (𝐺 ∈ UPGraph → (𝐼 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
116110a1i 11 . . . . . 6 (𝐺 ∈ UPGraph → 𝑉 = (Vtx‘𝐺))
117 sumvtxdg2size.d . . . . . . . . 9 𝐷 = (VtxDeg‘𝐺)
118 vtxdgop 27304 . . . . . . . . 9 (𝐺 ∈ UPGraph → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
119117, 118syl5eq 2845 . . . . . . . 8 (𝐺 ∈ UPGraph → 𝐷 = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
120119fveq1d 6657 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐷𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
121120adantr 484 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑣𝑉) → (𝐷𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
122116, 121sumeq12dv 15075 . . . . 5 (𝐺 ∈ UPGraph → Σ𝑣𝑉 (𝐷𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
123113fveq2i 6658 . . . . . . 7 (♯‘𝐼) = (♯‘(iEdg‘𝐺))
124123oveq2i 7156 . . . . . 6 (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺)))
125124a1i 11 . . . . 5 (𝐺 ∈ UPGraph → (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺))))
126122, 125eqeq12d 2814 . . . 4 (𝐺 ∈ UPGraph → (Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)) ↔ Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
127115, 126imbi12d 348 . . 3 (𝐺 ∈ UPGraph → ((𝐼 ∈ Fin → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼))) ↔ ((iEdg‘𝐺) ∈ Fin → Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺))))))
128109, 112, 1273imtr4d 297 . 2 (𝐺 ∈ UPGraph → (𝑉 ∈ Fin → (𝐼 ∈ Fin → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))))
1291283imp 1108 1 ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091  {crab 3110  Vcvv 3442   ∖ cdif 3880  ∅c0 4246  {csn 4528  ⟨cop 4534  dom cdm 5523   ↾ cres 5525  ‘cfv 6332  (class class class)co 7145  Fincfn 8510  0cc0 10544  1c1 10545   + caddc 10547   · cmul 10549  2c2 11698  ℕ0cn0 11903  ♯chash 13706  Σcsu 15054  Vtxcvtx 26833  iEdgciedg 26834  UHGraphcuhgr 26893  UPGraphcupgr 26917  VtxDegcvtxdg 27299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-disj 5000  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-n0 11904  df-xnn0 11976  df-z 11990  df-uz 12252  df-rp 12398  df-xadd 12516  df-fz 12906  df-fzo 13049  df-seq 13385  df-exp 13446  df-hash 13707  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-clim 14857  df-sum 15055  df-vtx 26835  df-iedg 26836  df-edg 26885  df-uhgr 26895  df-upgr 26919  df-vtxdg 27300 This theorem is referenced by:  fusgr1th  27385  finsumvtxdgeven  27386
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