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Theorem finsumvtxdg2sstep 26680
 Description: Induction step of finsumvtxdg2size 26681: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.)
Hypotheses
Ref Expression
finsumvtxdg2sstep.v 𝑉 = (Vtx‘𝐺)
finsumvtxdg2sstep.e 𝐸 = (iEdg‘𝐺)
finsumvtxdg2sstep.k 𝐾 = (𝑉 ∖ {𝑁})
finsumvtxdg2sstep.i 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
finsumvtxdg2sstep.p 𝑃 = (𝐸𝐼)
finsumvtxdg2sstep.s 𝑆 = ⟨𝐾, 𝑃
Assertion
Ref Expression
finsumvtxdg2sstep (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
Distinct variable groups:   𝑖,𝐸   𝑖,𝐺   𝑖,𝑁   𝑣,𝐸   𝑣,𝐺   𝑣,𝐾   𝑣,𝑁   𝑖,𝑉,𝑣
Allowed substitution hints:   𝑃(𝑣,𝑖)   𝑆(𝑣,𝑖)   𝐼(𝑣,𝑖)   𝐾(𝑖)

Proof of Theorem finsumvtxdg2sstep
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 finsumvtxdg2sstep.p . . 3 𝑃 = (𝐸𝐼)
2 finresfin 8346 . . . 4 (𝐸 ∈ Fin → (𝐸𝐼) ∈ Fin)
32ad2antll 708 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸𝐼) ∈ Fin)
41, 3syl5eqel 2854 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5 difsnid 4477 . . . . . . . . 9 (𝑁𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
65ad2antlr 706 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
76eqcomd 2777 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
87sumeq1d 14639 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9 diffi 8352 . . . . . . . . 9 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
109adantr 466 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
1110adantl 467 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12 simpr 471 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
1312adantr 466 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁𝑉)
14 neldifsn 4459 . . . . . . . . 9 ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
1514nelir 3049 . . . . . . . 8 𝑁 ∉ (𝑉 ∖ {𝑁})
1615a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17 dmfi 8404 . . . . . . . . . . 11 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
1817ad2antll 708 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
195eleq2d 2836 . . . . . . . . . . . . 13 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣𝑉))
2019biimpd 219 . . . . . . . . . . . 12 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2120ad2antlr 706 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2221imp 393 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → 𝑣𝑉)
23 finsumvtxdg2sstep.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
24 finsumvtxdg2sstep.e . . . . . . . . . . 11 𝐸 = (iEdg‘𝐺)
25 eqid 2771 . . . . . . . . . . 11 dom 𝐸 = dom 𝐸
2623, 24, 25vtxdgfisnn0 26606 . . . . . . . . . 10 ((dom 𝐸 ∈ Fin ∧ 𝑣𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2718, 22, 26syl2an2r 664 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2827nn0zd 11687 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
2928ralrimiva 3115 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30 fsumsplitsnun 14692 . . . . . . 7 (((𝑉 ∖ {𝑁}) ∈ Fin ∧ (𝑁𝑉𝑁 ∉ (𝑉 ∖ {𝑁})) ∧ ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)))
3111, 13, 16, 29, 30syl121anc 1481 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)))
32 fveq2 6333 . . . . . . . . . 10 (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3332adantl 467 . . . . . . . . 9 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3412, 33csbied 3709 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3534adantr 466 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3635oveq2d 6812 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
378, 31, 363eqtrd 2809 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
3837adantr 466 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
39 finsumvtxdg2sstep.k . . . . . . . 8 𝐾 = (𝑉 ∖ {𝑁})
40 finsumvtxdg2sstep.i . . . . . . . 8 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
41 finsumvtxdg2sstep.s . . . . . . . 8 𝑆 = ⟨𝐾, 𝑃
42 fveq2 6333 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
4342eleq2d 2836 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑁 ∈ (𝐸𝑗) ↔ 𝑁 ∈ (𝐸𝑖)))
4443cbvrabv 3349 . . . . . . . 8 {𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
4523, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem2 26677 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
4645oveq2d 6812 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))))
4746adantr 466 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))))
4823, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem4 26679 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))))
4944fveq2i 6336 . . . . . . . 8 (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
5049oveq2i 6807 . . . . . . 7 ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
5150oveq2i 6807 . . . . . 6 (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5251a1i 11 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5347, 48, 523eqtrd 2809 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
54 eqid 2771 . . . . . . . 8 {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
5523, 24, 39, 40, 1, 41, 54finsumvtxdg2ssteplem1 26676 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5655oveq2d 6812 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5756eqcomd 2777 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5857adantr 466 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5938, 53, 583eqtrd 2809 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
6059ex 397 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
614, 60embantd 59 1 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∉ wnel 3046  ∀wral 3061  {crab 3065  ⦋csb 3682   ∖ cdif 3720   ∪ cun 3721  {csn 4317  ⟨cop 4323  dom cdm 5250   ↾ cres 5252  ‘cfv 6030  (class class class)co 6796  Fincfn 8113   + caddc 10145   · cmul 10147  2c2 11276  ℕ0cn0 11499  ℤcz 11584  ♯chash 13321  Σcsu 14624  Vtxcvtx 26095  iEdgciedg 26096  UPGraphcupgr 26196  VtxDegcvtxdg 26596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-oi 8575  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-rp 12036  df-xadd 12152  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-vtx 26097  df-iedg 26098  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-vtxdg 26597 This theorem is referenced by:  finsumvtxdg2size  26681
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