Theorem finsumvtxdg2sstep 29526

Description: Induction step of finsumvtxdg2size 29527: 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 9156	. . . 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 2835	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑃 ∈ Fin)
5		difsnid 4762	. . . . . . . . 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 2737	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑉 = ((𝑉 ∖ {𝑁}) ∪ {𝑁}))
8	7	sumeq1d 15604	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣))
9		diffi 9084	. . . . . . . . 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 4744	. . . . . . . . 9 ⊢ ¬ 𝑁 ∈ (𝑉 ∖ {𝑁})
15	14	nelir 3035	. . . . . . . 8 ⊢ 𝑁 ∉ (𝑉 ∖ {𝑁})
16	15	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 𝑁 ∉ (𝑉 ∖ {𝑁}))
17		dmfi 9219	. . . . . . . . . . 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 2817	. . . . . . . . . . . . 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 2731	. . . . . . . . . . 11 ⊢ dom 𝐸 = dom 𝐸
26	23, 24, 25	vtxdgfisnn0 29452	. . . . . . . . . 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 12491	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})) → ((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
29	28	ralrimiva 3124	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ ((𝑉 ∖ {𝑁}) ∪ {𝑁})((VtxDeg‘𝐺)‘𝑣) ∈ ℤ)
30		fsumsplitsnun 15659	. . . . . . 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 1377	. . . . . 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 6822	. . . . . . . . . 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 3886	. . . . . . . 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 7362	. . . . . 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 2770	. . . . 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 6822	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝐸‘𝑗) = (𝐸‘𝑖))
43	42	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑁 ∈ (𝐸‘𝑗) ↔ 𝑁 ∈ (𝐸‘𝑖)))
44	43	cbvrabv 3405	. . . . . . . 8 ⊢ {𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
45	23, 24, 39, 40, 1, 41, 44	finsumvtxdg2ssteplem2 29523	. . . . . . 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 7362	. . . . . 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 29525	. . . . 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 6825	. . . . . . . 8 ⊢ (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)}) = (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})
50	49	oveq2i 7357	. . . . . . 7 ⊢ ((♯‘𝑃) + (♯‘{𝑗 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑗)})) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))
51	50	oveq2i 7357	. . . . . 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 2770	. . . 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 2731	. . . . . . . 8 ⊢ {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
55	23, 24, 39, 40, 1, 41, 54	finsumvtxdg2ssteplem1 29522	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})))
56	55	oveq2d 7362	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝐸)) = (2 · ((♯‘𝑃) + (♯‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}))))
57	56	eqcomd 2737	. . . . 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 2770	. . 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 1541   ∈ wcel 2111   ∉ wnel 3032  ∀wral 3047  {crab 3395  ⦋csb 3850   ∖ cdif 3899   ∪ cun 3900  {csn 4576  ⟨cop 4582  dom cdm 5616   ↾ cres 5618  ‘cfv 6481  (class class class)co 7346  Fincfn 8869   + caddc 11006   · cmul 11008  2c2 12177  ℕ₀cn0 12378  ℤcz 12465  ♯chash 14234  Σcsu 15590  Vtxcvtx 28972  iEdgciedg 28973  UPGraphcupgr 29056  VtxDegcvtxdg 29442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-disj 5059  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-oi 9396  df-dju 9791  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-n0 12379  df-xnn0 12452  df-z 12466  df-uz 12730  df-rp 12888  df-xadd 13009  df-fz 13405  df-fzo 13552  df-seq 13906  df-exp 13966  df-hash 14235  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-clim 15392  df-sum 15591  df-vtx 28974  df-iedg 28975  df-edg 29024  df-uhgr 29034  df-upgr 29058  df-vtxdg 29443

This theorem is referenced by:  finsumvtxdg2size  29527