Theorem finsumvtxdg2sstep 29582

Description: Induction step of finsumvtxdg2size 29583: 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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
finsumvtxdg2sstep.p	⊢ 𝑃 = (𝐸 ↾ 𝐼)
finsumvtxdg2sstep.s	⊢ 𝑆 = ⟨𝐾, 𝑃⟩

Assertion

Ref	Expression
finsumvtxdg2sstep	⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((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.

Step	Hyp	Ref	Expression
1		finsumvtxdg2sstep.p	. . 3 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
2		finresfin 9302	. . . 4 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin)
3	2	ad2antll 729	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝐸 ↾ 𝐼) ∈ Fin)
4	1, 3	eqeltrid 2843	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5		difsnid 4815	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
6	5	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑉 ∖ {𝑁}) ∪ {𝑁}) = 𝑉)
7	6	eqcomd 2741	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
8	7	sumeq1d 15733	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9		diffi 9214	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
12		simpr 484	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
13	12	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∈ 𝑉)
14		neldifsn 4797	. . . . . . . . 9 ⊢ ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
15	14	nelir 3047	. . . . . . . 8 ⊢ 𝑁 ∉ (𝑉 ∖ {𝑁})
16	15	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17		dmfi 9373	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
18	17	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝐸 ∈ Fin)
19	5	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) ↔ 𝑣 ∈ 𝑉))
20	19	biimpd 229	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣 ∈ 𝑉))
21	20	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁}) → 𝑣 ∈ 𝑉))
22	21	imp 406	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → 𝑣 ∈ 𝑉)
23		finsumvtxdg2sstep.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
24		finsumvtxdg2sstep.e	. . . . . . . . . . 11 ⊢ 𝐸 = (iEdg‘𝐺)
25		eqid 2735	. . . . . . . . . . 11 ⊢ dom 𝐸 = dom 𝐸
26	23, 24, 25	vtxdgfisnn0 29508	. . . . . . . . . 10 ⊢ ((dom 𝐸 ∈ Fin ∧ 𝑣 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
27	18, 22, 26	syl2an2r 685	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
28	27	nn0zd 12637	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
29	28	ralrimiva 3144	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30		fsumsplitsnun 15788	. . . . . . 7 ⊢ (((𝑉 ∖ {𝑁}) ∈ Fin ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ∉ (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)))
31	11, 13, 16, 29, 30	syl121anc 1374	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣)))
32		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑣 = 𝑁 → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
33	32	adantl 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 = 𝑁) → ((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
34	12, 33	csbied 3946	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
35	34	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣) = ((VtxDeg‘𝐺)‘𝑁))
36	35	oveq2d 7447	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ⦋𝑁 / 𝑣⦌((VtxDeg‘𝐺)‘𝑣)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
37	8, 31, 36	3eqtrd 2779	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)))
38	37	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
41		finsumvtxdg2sstep.s	. . . . . . . 8 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
42		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐸‘𝑗) = (𝐸‘𝑖))
43	42	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑁 ∈ (𝐸‘𝑗) ↔ 𝑁 ∈ (𝐸‘𝑖)))
44	43	cbvrabv 3444	. . . . . . . 8 ⊢ {𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
45	23, 24, 39, 40, 1, 41, 44	finsumvtxdg2ssteplem2 29579	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
46	45	oveq2d 7447	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
47	46	adantr 480	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
48	23, 24, 39, 40, 1, 41, 44	finsumvtxdg2ssteplem4 29581	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))))
49	44	fveq2i 6910	. . . . . . . 8 ⊢ (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})
50	49	oveq2i 7442	. . . . . . 7 ⊢ ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))
51	50	oveq2i 7442	. . . . . 6 ⊢ (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})))
52	51	a1i 11	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}))) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
53	47, 48, 52	3eqtrd 2779	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((VtxDeg‘𝐺)‘𝑁)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
54		eqid 2735	. . . . . . . 8 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
55	23, 24, 39, 40, 1, 41, 54	finsumvtxdg2ssteplem1 29578	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})))
56	55	oveq2d 7447	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
57	56	eqcomd 2741	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))) = (2 · (♯‘𝐸)))
58	57	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))) = (2 · (♯‘𝐸)))
59	38, 53, 58	3eqtrd 2779	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸)))
60	59	ex 412	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
61	4, 60	embantd 59	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∉ wnel 3044  ∀wral 3059  {crab 3433  ⦋csb 3908   ∖ cdif 3960   ∪ cun 3961  {csn 4631  ⟨cop 4637  dom cdm 5689   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431  Fincfn 8984   + caddc 11156   · cmul 11158  2c2 12319  ℕ₀cn0 12524  ℤcz 12611  ♯chash 14366  Σcsu 15719  Vtxcvtx 29028  iEdgciedg 29029  UPGraphcupgr 29112  VtxDegcvtxdg 29498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-xadd 13153  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-vtx 29030  df-iedg 29031  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-vtxdg 29499

This theorem is referenced by:  finsumvtxdg2size  29583