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Theorem finsumvtxdg2sstep 27337
Description: Induction step of finsumvtxdg2size 27338: 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 8737 . . . 4 (𝐸 ∈ Fin → (𝐸𝐼) ∈ Fin)
32ad2antll 728 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸𝐼) ∈ Fin)
41, 3eqeltrid 2920 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5 difsnid 4728 . . . . . . . . 9 (𝑁𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
65ad2antlr 726 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
76eqcomd 2830 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
87sumeq1d 15056 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9 diffi 8743 . . . . . . . . 9 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
109adantr 484 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
1110adantl 485 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12 simpr 488 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
1312adantr 484 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁𝑉)
14 neldifsn 4710 . . . . . . . . 9 ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
1514nelir 3121 . . . . . . . 8 𝑁 ∉ (𝑉 ∖ {𝑁})
1615a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17 dmfi 8795 . . . . . . . . . . 11 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
1817ad2antll 728 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
195eleq2d 2901 . . . . . . . . . . . . 13 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣𝑉))
2019biimpd 232 . . . . . . . . . . . 12 (𝑁𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2120ad2antlr 726 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣𝑉))
2221imp 410 . . . . . . . . . 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 2824 . . . . . . . . . . 11 dom 𝐸 = dom 𝐸
2623, 24, 25vtxdgfisnn0 27263 . . . . . . . . . 10 ((dom 𝐸 ∈ Fin ∧ 𝑣𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2718, 22, 26syl2an2r 684 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
2827nn0zd 12080 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
2928ralrimiva 3177 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30 fsumsplitsnun 15108 . . . . . . 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 1372 . . . . . 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 6659 . . . . . . . . . 10 (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3332adantl 485 . . . . . . . . 9 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3412, 33csbied 3902 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3534adantr 484 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 / 𝑣((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
3635oveq2d 7162 . . . . . 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 2863 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
3837adantr 484 . . . 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 6659 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
4342eleq2d 2901 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑁 ∈ (𝐸𝑗) ↔ 𝑁 ∈ (𝐸𝑖)))
4443cbvrabv 3477 . . . . . . . 8 {𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
4523, 24, 39, 40, 1, 41, 44finsumvtxdg2ssteplem2 27334 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
4645oveq2d 7162 . . . . . 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 484 . . . . 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 27336 . . . . 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 6662 . . . . . . . 8 (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
5049oveq2i 7157 . . . . . . 7 ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸𝑁 ∈ (𝐸𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
5150oveq2i 7157 . . . . . 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 2863 . . . 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 2824 . . . . . . . 8 {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}
5523, 24, 39, 40, 1, 41, 54finsumvtxdg2ssteplem1 27333 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})))
5655oveq2d 7162 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))))
5756eqcomd 2830 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))) = (2 · (♯‘𝐸)))
5857adantr 484 . . . 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 2863 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
6059ex 416 . 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 399   = wceq 1538  wcel 2115  wnel 3118  wral 3133  {crab 3137  csb 3866  cdif 3916  cun 3917  {csn 4550  cop 4556  dom cdm 5543  cres 5545  cfv 6344  (class class class)co 7146  Fincfn 8501   + caddc 10534   · cmul 10536  2c2 11687  0cn0 11892  cz 11976  chash 13693  Σcsu 15040  Vtxcvtx 26787  iEdgciedg 26788  UPGraphcupgr 26871  VtxDegcvtxdg 27253
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-inf2 9097  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8899  df-oi 8967  df-dju 9323  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11695  df-3 11696  df-n0 11893  df-xnn0 11963  df-z 11977  df-uz 12239  df-rp 12385  df-xadd 12503  df-fz 12893  df-fzo 13036  df-seq 13372  df-exp 13433  df-hash 13694  df-cj 14456  df-re 14457  df-im 14458  df-sqrt 14592  df-abs 14593  df-clim 14843  df-sum 15041  df-vtx 26789  df-iedg 26790  df-edg 26839  df-uhgr 26849  df-upgr 26873  df-vtxdg 27254
This theorem is referenced by:  finsumvtxdg2size  27338
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