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Theorem finsumvtxdg2sstep 26799
Description: Induction step of finsumvtxdg2size 26800: 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 8428 . . . 4 (𝐸 ∈ Fin → (𝐸𝐼) ∈ Fin)
32ad2antll 721 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸𝐼) ∈ Fin)
41, 3syl5eqel 2882 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5 difsnid 4529 . . . . . . . . 9 (𝑁𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
65ad2antlr 719 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
76eqcomd 2805 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
87sumeq1d 14772 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9 diffi 8434 . . . . . . . . 9 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
109adantr 473 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
1110adantl 474 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12 simpr 478 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
1312adantr 473 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁𝑉)
14 neldifsn 4511 . . . . . . . . 9 ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
1514nelir 3077 . . . . . . . 8 𝑁 ∉ (𝑉 ∖ {𝑁})
1615a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17 dmfi 8486 . . . . . . . . . . 11 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
1817ad2antll 721 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
195eleq2d 2864 . . . . . . . . . . . . 13 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣𝑉))
2019biimpd 221 . . . . . . . . . . . 12 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2120ad2antlr 719 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2221imp 396 . . . . . . . . . 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 2799 . . . . . . . . . . 11 dom 𝐸 = dom 𝐸
2623, 24, 25vtxdgfisnn0 26725 . . . . . . . . . 10 ((dom 𝐸 ∈ Fin ∧ 𝑣𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2718, 22, 26syl2an2r 676 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2827nn0zd 11770 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
2928ralrimiva 3147 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30 fsumsplitsnun 14825 . . . . . . 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 1495 . . . . . 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 6411 . . . . . . . . . 10 (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3332adantl 474 . . . . . . . . 9 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3412, 33csbied 3755 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3534adantr 473 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3635oveq2d 6894 . . . . . 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 2837 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
3837adantr 473 . . . 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 6411 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
4342eleq2d 2864 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑁 ∈ (𝐸𝑗) ↔ 𝑁 ∈ (𝐸𝑖)))
4443cbvrabv 3383 . . . . . . . 8 {𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
4523, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem2 26796 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
4645oveq2d 6894 . . . . . 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 473 . . . . 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 26798 . . . . 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 6414 . . . . . . . 8 (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
5049oveq2i 6889 . . . . . . 7 ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
5150oveq2i 6889 . . . . . 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 2837 . . . 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 2799 . . . . . . . 8 {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
5523, 24, 39, 40, 1, 41, 54finsumvtxdg2ssteplem1 26795 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5655oveq2d 6894 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5756eqcomd 2805 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5857adantr 473 . . . 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 2837 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
6059ex 402 . 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 385   = wceq 1653  wcel 2157  wnel 3074  wral 3089  {crab 3093  csb 3728  cdif 3766  cun 3767  {csn 4368  cop 4374  dom cdm 5312  cres 5314  cfv 6101  (class class class)co 6878  Fincfn 8195   + caddc 10227   · cmul 10229  2c2 11368  0cn0 11580  cz 11666  chash 13370  Σcsu 14757  Vtxcvtx 26231  iEdgciedg 26232  UPGraphcupgr 26315  VtxDegcvtxdg 26715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-disj 4812  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-sup 8590  df-oi 8657  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-rp 12075  df-xadd 12194  df-fz 12581  df-fzo 12721  df-seq 13056  df-exp 13115  df-hash 13371  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-clim 14560  df-sum 14758  df-vtx 26233  df-iedg 26234  df-edg 26283  df-uhgr 26293  df-upgr 26317  df-vtxdg 26716
This theorem is referenced by:  finsumvtxdg2size  26800
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