MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  finsumvtxdg2size Structured version   Visualization version   GIF version

Theorem finsumvtxdg2size 26737
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 26738) 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 26266 . . . 4 (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)
2 fvex 6388 . . . . . 6 (iEdg‘𝐺) ∈ V
3 fvex 6388 . . . . . . 7 (iEdg‘⟨𝑘, 𝑒⟩) ∈ V
43resex 5620 . . . . . 6 ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}) ∈ V
5 eleq1 2832 . . . . . . . 8 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
65adantl 473 . . . . . . 7 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
7 simpl 474 . . . . . . . . 9 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → 𝑘 = (Vtx‘𝐺))
8 oveq12 6851 . . . . . . . . . . 11 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑘VtxDeg𝑒) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
98fveq1d 6377 . . . . . . . . . 10 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
109adantr 472 . . . . . . . . 9 (((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) ∧ 𝑣𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
117, 10sumeq12dv 14724 . . . . . . . 8 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
12 fveq2 6375 . . . . . . . . . 10 (𝑒 = (iEdg‘𝐺) → (♯‘𝑒) = (♯‘(iEdg‘𝐺)))
1312oveq2d 6858 . . . . . . . . 9 (𝑒 = (iEdg‘𝐺) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
1413adantl 473 . . . . . . . 8 ((𝑘 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (2 · (♯‘𝑒)) = (2 · (♯‘(iEdg‘𝐺))))
1511, 14eqeq12d 2780 . . . . . . 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 335 . . . . . 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 2832 . . . . . . . 8 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1817adantl 473 . . . . . . 7 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
19 simpl 474 . . . . . . . . 9 ((𝑘 = 𝑤𝑒 = 𝑓) → 𝑘 = 𝑤)
20 oveq12 6851 . . . . . . . . . . . 12 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (𝑤VtxDeg𝑓))
21 df-ov 6845 . . . . . . . . . . . 12 (𝑤VtxDeg𝑓) = (VtxDeg‘⟨𝑤, 𝑓⟩)
2220, 21syl6eq 2815 . . . . . . . . . . 11 ((𝑘 = 𝑤𝑒 = 𝑓) → (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑤, 𝑓⟩))
2322fveq1d 6377 . . . . . . . . . 10 ((𝑘 = 𝑤𝑒 = 𝑓) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
2423adantr 472 . . . . . . . . 9 (((𝑘 = 𝑤𝑒 = 𝑓) ∧ 𝑣𝑘) → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
2519, 24sumeq12dv 14724 . . . . . . . 8 ((𝑘 = 𝑤𝑒 = 𝑓) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣))
26 fveq2 6375 . . . . . . . . . 10 (𝑒 = 𝑓 → (♯‘𝑒) = (♯‘𝑓))
2726oveq2d 6858 . . . . . . . . 9 (𝑒 = 𝑓 → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
2827adantl 473 . . . . . . . 8 ((𝑘 = 𝑤𝑒 = 𝑓) → (2 · (♯‘𝑒)) = (2 · (♯‘𝑓)))
2925, 28eqeq12d 2780 . . . . . . 7 ((𝑘 = 𝑤𝑒 = 𝑓) → (Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓))))
3018, 29imbi12d 335 . . . . . 6 ((𝑘 = 𝑤𝑒 = 𝑓) → ((𝑒 ∈ Fin → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒))) ↔ (𝑓 ∈ Fin → Σ𝑣𝑤 ((VtxDeg‘⟨𝑤, 𝑓⟩)‘𝑣) = (2 · (♯‘𝑓)))))
31 vex 3353 . . . . . . . . 9 𝑘 ∈ V
32 vex 3353 . . . . . . . . 9 𝑒 ∈ V
3331, 32opvtxfvi 26178 . . . . . . . 8 (Vtx‘⟨𝑘, 𝑒⟩) = 𝑘
3433eqcomi 2774 . . . . . . 7 𝑘 = (Vtx‘⟨𝑘, 𝑒⟩)
35 eqid 2765 . . . . . . 7 (iEdg‘⟨𝑘, 𝑒⟩) = (iEdg‘⟨𝑘, 𝑒⟩)
36 eqid 2765 . . . . . . 7 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
37 eqid 2765 . . . . . . 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 26477 . . . . . 6 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑛𝑘) → ⟨(𝑘 ∖ {𝑛}), ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})⟩ ∈ UPGraph)
39 eleq1 2832 . . . . . . . 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 473 . . . . . . 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 474 . . . . . . . . 9 ((𝑤 = (𝑘 ∖ {𝑛}) ∧ 𝑓 = ((iEdg‘⟨𝑘, 𝑒⟩) ↾ {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)})) → 𝑤 = (𝑘 ∖ {𝑛}))
42 opeq12 4561 . . . . . . . . . . . 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 6379 . . . . . . . . . . 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 6377 . . . . . . . . . 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 472 . . . . . . . . 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 14724 . . . . . . . 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 6375 . . . . . . . . . 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 6858 . . . . . . . . 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 473 . . . . . . . 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 2780 . . . . . . 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 335 . . . . . 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 13356 . . . . . . . . 9 (𝑘 ∈ V → ((♯‘𝑘) = 0 ↔ 𝑘 = ∅))
5331, 52ax-mp 5 . . . . . . . 8 ((♯‘𝑘) = 0 ↔ 𝑘 = ∅)
54 2t0e0 11447 . . . . . . . . . 10 (2 · 0) = 0
5554a1i 11 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · 0) = 0)
5631, 32opiedgfvi 26179 . . . . . . . . . . . . 13 (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒
5756eqcomi 2774 . . . . . . . . . . . 12 𝑒 = (iEdg‘⟨𝑘, 𝑒⟩)
58 upgruhgr 26274 . . . . . . . . . . . . . 14 (⟨𝑘, 𝑒⟩ ∈ UPGraph → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
5958adantr 472 . . . . . . . . . . . . 13 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → ⟨𝑘, 𝑒⟩ ∈ UHGraph)
6034eqeq1i 2770 . . . . . . . . . . . . . 14 (𝑘 = ∅ ↔ (Vtx‘⟨𝑘, 𝑒⟩) = ∅)
61 uhgr0vb 26244 . . . . . . . . . . . . . 14 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (Vtx‘⟨𝑘, 𝑒⟩) = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
6260, 61sylan2b 587 . . . . . . . . . . . . 13 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (⟨𝑘, 𝑒⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑘, 𝑒⟩) = ∅))
6359, 62mpbid 223 . . . . . . . . . . . 12 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (iEdg‘⟨𝑘, 𝑒⟩) = ∅)
6457, 63syl5eq 2811 . . . . . . . . . . 11 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → 𝑒 = ∅)
65 hasheq0 13356 . . . . . . . . . . . 12 (𝑒 ∈ V → ((♯‘𝑒) = 0 ↔ 𝑒 = ∅))
6632, 65ax-mp 5 . . . . . . . . . . 11 ((♯‘𝑒) = 0 ↔ 𝑒 = ∅)
6764, 66sylibr 225 . . . . . . . . . 10 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (♯‘𝑒) = 0)
6867oveq2d 6858 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → (2 · (♯‘𝑒)) = (2 · 0))
69 sumeq1 14706 . . . . . . . . . . 11 (𝑘 = ∅ → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣))
70 sum0 14739 . . . . . . . . . . 11 Σ𝑣 ∈ ∅ ((𝑘VtxDeg𝑒)‘𝑣) = 0
7169, 70syl6eq 2815 . . . . . . . . . 10 (𝑘 = ∅ → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
7271adantl 473 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = 0)
7355, 68, 723eqtr4rd 2810 . . . . . . . 8 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ 𝑘 = ∅) → Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)))
7453, 73sylan2b 587 . . . . . . 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 2832 . . . . . . . . . . 11 ((𝑦 + 1) = (♯‘𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
7776eqcoms 2773 . . . . . . . . . 10 ((♯‘𝑘) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
78773ad2ant2 1164 . . . . . . . . 9 ((⟨𝑘, 𝑒⟩ ∈ UPGraph ∧ (♯‘𝑘) = (𝑦 + 1) ∧ 𝑛𝑘) → ((𝑦 + 1) ∈ ℕ0 ↔ (♯‘𝑘) ∈ ℕ0))
79 hashclb 13351 . . . . . . . . . . . 12 (𝑘 ∈ V → (𝑘 ∈ Fin ↔ (♯‘𝑘) ∈ ℕ0))
8079biimprd 239 . . . . . . . . . . 11 (𝑘 ∈ V → ((♯‘𝑘) ∈ ℕ0𝑘 ∈ Fin))
8131, 80ax-mp 5 . . . . . . . . . 10 ((♯‘𝑘) ∈ ℕ0𝑘 ∈ Fin)
82 eqid 2765 . . . . . . . . . . . . . . 15 (𝑘 ∖ {𝑛}) = (𝑘 ∖ {𝑛})
83 eqid 2765 . . . . . . . . . . . . . . 15 {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
8456dmeqi 5493 . . . . . . . . . . . . . . . . . 18 dom (iEdg‘⟨𝑘, 𝑒⟩) = dom 𝑒
8584rabeqi 3342 . . . . . . . . . . . . . . . . 17 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)}
86 eqidd 2766 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝑒𝑛 = 𝑛)
8756a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ dom 𝑒 → (iEdg‘⟨𝑘, 𝑒⟩) = 𝑒)
8887fveq1d 6377 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ dom 𝑒 → ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) = (𝑒𝑖))
8986, 88neleq12d 3044 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ dom 𝑒 → (𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖) ↔ 𝑛 ∉ (𝑒𝑖)))
9089rabbiia 3333 . . . . . . . . . . . . . . . . 17 {𝑖 ∈ dom 𝑒𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
9185, 90eqtri 2787 . . . . . . . . . . . . . . . 16 {𝑖 ∈ dom (iEdg‘⟨𝑘, 𝑒⟩) ∣ 𝑛 ∉ ((iEdg‘⟨𝑘, 𝑒⟩)‘𝑖)} = {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}
9256, 91reseq12i 5563 . . . . . . . . . . . . . . 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 26736 . . . . . . . . . . . . . 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 6845 . . . . . . . . . . . . . . . . . 18 (𝑘VtxDeg𝑒) = (VtxDeg‘⟨𝑘, 𝑒⟩)
9594fveq1i 6376 . . . . . . . . . . . . . . . . 17 ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
9695a1i 11 . . . . . . . . . . . . . . . 16 (𝑣𝑘 → ((𝑘VtxDeg𝑒)‘𝑣) = ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣))
9796sumeq2i 14716 . . . . . . . . . . . . . . 15 Σ𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = Σ𝑣𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣)
9897eqeq1i 2770 . . . . . . . . . . . . . 14 𝑣𝑘 ((𝑘VtxDeg𝑒)‘𝑣) = (2 · (♯‘𝑒)) ↔ Σ𝑣𝑘 ((VtxDeg‘⟨𝑘, 𝑒⟩)‘𝑣) = (2 · (♯‘𝑒)))
9993, 98syl6ibr 243 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 1161 . . . . . . . . . 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 231 . . . . . . . 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 396 . . . . . . 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 395 . . . . . 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 13485 . . . . 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 401 . . . 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 2835 . . . 4 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
112111a1i 11 . . 3 (𝐺 ∈ UPGraph → (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin))
113 sumvtxdg2size.i . . . . . 6 𝐼 = (iEdg‘𝐺)
114113eleq1i 2835 . . . . 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 26657 . . . . . . . . 9 (𝐺 ∈ UPGraph → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
119117, 118syl5eq 2811 . . . . . . . 8 (𝐺 ∈ UPGraph → 𝐷 = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))
120119fveq1d 6377 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐷𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
121120adantr 472 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑣𝑉) → (𝐷𝑣) = (((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
122116, 121sumeq12dv 14724 . . . . 5 (𝐺 ∈ UPGraph → Σ𝑣𝑉 (𝐷𝑣) = Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣))
123113fveq2i 6378 . . . . . . 7 (♯‘𝐼) = (♯‘(iEdg‘𝐺))
124123oveq2i 6853 . . . . . 6 (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺)))
125124a1i 11 . . . . 5 (𝐺 ∈ UPGraph → (2 · (♯‘𝐼)) = (2 · (♯‘(iEdg‘𝐺))))
126122, 125eqeq12d 2780 . . . 4 (𝐺 ∈ UPGraph → (Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)) ↔ Σ𝑣 ∈ (Vtx‘𝐺)(((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))‘𝑣) = (2 · (♯‘(iEdg‘𝐺)))))
127115, 126imbi12d 335 . . 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 285 . 2 (𝐺 ∈ UPGraph → (𝑉 ∈ Fin → (𝐼 ∈ Fin → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))))
1291283imp 1137 1 ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wnel 3040  {crab 3059  Vcvv 3350  cdif 3729  c0 4079  {csn 4334  cop 4340  dom cdm 5277  cres 5279  cfv 6068  (class class class)co 6842  Fincfn 8160  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  2c2 11327  0cn0 11538  chash 13321  Σcsu 14703  Vtxcvtx 26165  iEdgciedg 26166  UHGraphcuhgr 26228  UPGraphcupgr 26252  VtxDegcvtxdg 26652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-rp 12029  df-xadd 12147  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-sum 14704  df-vtx 26167  df-iedg 26168  df-edg 26217  df-uhgr 26230  df-upgr 26254  df-vtxdg 26653
This theorem is referenced by:  fusgr1th  26738  finsumvtxdgeven  26739
  Copyright terms: Public domain W3C validator