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Theorem finsumvtxdg2ssteplem4 29070

Description: Lemma for finsumvtxdg2sstep 29071. (Contributed by AV, 12-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⟩
finsumvtxdg2ssteplem.j	⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}

**Assertion**
Ref	Expression
finsumvtxdg2ssteplem4	⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))

Distinct variable groups: 𝑖,𝐸 𝑖,𝐺 𝑖,𝑁 𝑣,𝐸 𝑣,𝐺 𝑣,𝑁 𝑖,𝑉,𝑣 𝑖,𝐽 𝑣,𝐾 Allowed substitution hints: 𝑃(𝑣,𝑖) 𝑆(𝑣,𝑖) 𝐼(𝑣,𝑖) 𝐽(𝑣) 𝐾(𝑖)

**Proof of Theorem finsumvtxdg2ssteplem4**
Step	Hyp	Ref	Expression
1		finsumvtxdg2sstep.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
2		finsumvtxdg2sstep.e	. . . . . . . 8 ⊢ 𝐸 = (iEdg‘𝐺)
3		finsumvtxdg2sstep.k	. . . . . . . 8 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		finsumvtxdg2sstep.i	. . . . . . . 8 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
5		finsumvtxdg2sstep.p	. . . . . . . 8 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
6		finsumvtxdg2sstep.s	. . . . . . . 8 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
7		finsumvtxdg2ssteplem.j	. . . . . . . 8 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
8	1, 2, 3, 4, 5, 6, 7	vtxdginducedm1fi 29066	. . . . . . 7 ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
9	8	ad2antll 725	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
10	9	sumeq2d 15654	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = Σ𝑣 ∈ (𝑉 ∖ {𝑁})(((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
11		diffi 9183	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
12	11	adantr 479	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
13	12	adantl 480	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (𝑉 ∖ {𝑁}) ∈ Fin)
14	5	dmeqi 5905	. . . . . . . . 9 ⊢ dom 𝑃 = dom (𝐸 ↾ 𝐼)
15		finresfin 9274	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐼) ∈ Fin)
16		dmfi 9334	. . . . . . . . . 10 ⊢ ((𝐸 ↾ 𝐼) ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom (𝐸 ↾ 𝐼) ∈ Fin)
18	14, 17	eqeltrid 2835	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → dom 𝑃 ∈ Fin)
19	18	ad2antll 725	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → dom 𝑃 ∈ Fin)
20	3	eqcomi 2739	. . . . . . . . 9 ⊢ (𝑉 ∖ {𝑁}) = 𝐾
21	20	eleq2i 2823	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ 𝑣 ∈ 𝐾)
22	21	biimpi 215	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝐾)
23	6	fveq2i 6895	. . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘⟨𝐾, 𝑃⟩)
24	1	fvexi 6906	. . . . . . . . . . . . 13 ⊢ 𝑉 ∈ V
25	24	difexi 5329	. . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁}) ∈ V
26	3, 25	eqeltri 2827	. . . . . . . . . . 11 ⊢ 𝐾 ∈ V
27	2	fvexi 6906	. . . . . . . . . . . . 13 ⊢ 𝐸 ∈ V
28	27	resex 6030	. . . . . . . . . . . 12 ⊢ (𝐸 ↾ 𝐼) ∈ V
29	5, 28	eqeltri 2827	. . . . . . . . . . 11 ⊢ 𝑃 ∈ V
30	26, 29	opvtxfvi 28534	. . . . . . . . . 10 ⊢ (Vtx‘⟨𝐾, 𝑃⟩) = 𝐾
31	23, 30	eqtr2i 2759	. . . . . . . . 9 ⊢ 𝐾 = (Vtx‘𝑆)
32	1, 2, 3, 4, 5, 6	vtxdginducedm1lem1 29061	. . . . . . . . . 10 ⊢ (iEdg‘𝑆) = 𝑃
33	32	eqcomi 2739	. . . . . . . . 9 ⊢ 𝑃 = (iEdg‘𝑆)
34		eqid 2730	. . . . . . . . 9 ⊢ dom 𝑃 = dom 𝑃
35	31, 33, 34	vtxdgfisnn0 28997	. . . . . . . 8 ⊢ ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ 𝐾) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℕ₀)
36	35	nn0cnd 12540	. . . . . . 7 ⊢ ((dom 𝑃 ∈ Fin ∧ 𝑣 ∈ 𝐾) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℂ)
37	19, 22, 36	syl2an 594	. . . . . 6 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((VtxDeg‘𝑆)‘𝑣) ∈ ℂ)
38		dmfi 9334	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
39		rabfi 9273	. . . . . . . . . . . 12 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ∈ Fin)
40	38, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ∈ Fin)
41	7, 40	eqeltrid 2835	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → 𝐽 ∈ Fin)
42		rabfi 9273	. . . . . . . . . 10 ⊢ (𝐽 ∈ Fin → {𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)} ∈ Fin)
43		hashcl 14322	. . . . . . . . . 10 ⊢ ({𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)} ∈ Fin → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℕ₀)
44	41, 42, 43	3syl 18	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℕ₀)
45	44	nn0cnd 12540	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
46	45	ad2antll 725	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
47	46	adantr 479	. . . . . 6 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
48	13, 37, 47	fsumadd 15692	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})(((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
49	10, 48	eqtrd 2770	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
50	3	sumeq1i 15650	. . . . . 6 ⊢ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣)
51	50	eqeq1i 2735	. . . . 5 ⊢ (Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) ↔ Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)))
52		oveq1 7420	. . . . 5 ⊢ (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) = ((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
53	51, 52	sylbi 216	. . . 4 ⊢ (Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝑆)‘𝑣) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) = ((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
54	49, 53	sylan9eq 2790	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = ((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})))
55	54	oveq1d 7428	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
56	45	adantl 480	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
57	56	adantr 479	. . . . . . . . 9 ⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
58	12, 57	fsumcl 15685	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
59		hashcl 14322	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Fin → (♯‘𝐽) ∈ ℕ₀)
60	41, 59	syl 17	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (♯‘𝐽) ∈ ℕ₀)
61	60	nn0cnd 12540	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘𝐽) ∈ ℂ)
62	61	adantl 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (♯‘𝐽) ∈ ℂ)
63		rabfi 9273	. . . . . . . . . . 11 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
64		hashcl 14322	. . . . . . . . . . 11 ⊢ ({𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin → (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) ∈ ℕ₀)
65	38, 63, 64	3syl 18	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) ∈ ℕ₀)
66	65	nn0cnd 12540	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) ∈ ℂ)
67	66	adantl 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) ∈ ℂ)
68	58, 62, 67	add12d 11446	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = ((♯‘𝐽) + (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
69	68	adantl 480	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = ((♯‘𝐽) + (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))))
70	1, 2, 3, 4, 5, 6, 7	finsumvtxdg2ssteplem3 29069	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = (♯‘𝐽))
71	70	oveq2d 7429	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((♯‘𝐽) + (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = ((♯‘𝐽) + (♯‘𝐽)))
72	61	2timesd 12461	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → (2 · (♯‘𝐽)) = ((♯‘𝐽) + (♯‘𝐽)))
73	72	eqcomd 2736	. . . . . . 7 ⊢ (𝐸 ∈ Fin → ((♯‘𝐽) + (♯‘𝐽)) = (2 · (♯‘𝐽)))
74	73	ad2antll 725	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((♯‘𝐽) + (♯‘𝐽)) = (2 · (♯‘𝐽)))
75	69, 71, 74	3eqtrd 2774	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · (♯‘𝐽)))
76	75	oveq2d 7429	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((2 · (♯‘𝑃)) + (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))) = ((2 · (♯‘𝑃)) + (2 · (♯‘𝐽))))
77		2cnd 12296	. . . . . . 7 ⊢ (𝐸 ∈ Fin → 2 ∈ ℂ)
78	5, 15	eqeltrid 2835	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → 𝑃 ∈ Fin)
79		hashcl 14322	. . . . . . . . 9 ⊢ (𝑃 ∈ Fin → (♯‘𝑃) ∈ ℕ₀)
80	78, 79	syl 17	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → (♯‘𝑃) ∈ ℕ₀)
81	80	nn0cnd 12540	. . . . . . 7 ⊢ (𝐸 ∈ Fin → (♯‘𝑃) ∈ ℂ)
82	77, 81	mulcld 11240	. . . . . 6 ⊢ (𝐸 ∈ Fin → (2 · (♯‘𝑃)) ∈ ℂ)
83	82	ad2antll 725	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · (♯‘𝑃)) ∈ ℂ)
84	58	adantl 480	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) ∈ ℂ)
85	61, 66	addcld 11239	. . . . . 6 ⊢ (𝐸 ∈ Fin → ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) ∈ ℂ)
86	85	ad2antll 725	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) ∈ ℂ)
87	83, 84, 86	addassd 11242	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = ((2 · (♯‘𝑃)) + (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))))
88		2cnd 12296	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → 2 ∈ ℂ)
89	81	ad2antll 725	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝑃) ∈ ℂ)
90	61	ad2antll 725	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐽) ∈ ℂ)
91	88, 89, 90	adddid 11244	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (2 · ((♯‘𝑃) + (♯‘𝐽))) = ((2 · (♯‘𝑃)) + (2 · (♯‘𝐽))))
92	76, 87, 91	3eqtr4d 2780	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))
93	92	adantr 479	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (((2 · (♯‘𝑃)) + Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)})) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))
94	55, 93	eqtrd 2770	1 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∉ wnel 3044 ∀wral 3059 {crab 3430 Vcvv 3472 ∖ cdif 3946 {csn 4629 ⟨cop 4635 dom cdm 5677 ↾ cres 5679 ‘cfv 6544 (class class class)co 7413 Fincfn 8943 ℂcc 11112 + caddc 11117 · cmul 11119 2c2 12273 ℕ₀cn0 12478 ♯chash 14296 Σcsu 15638 Vtxcvtx 28521 iEdgciedg 28522 UPGraphcupgr 28605 VtxDegcvtxdg 28987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-rp 12981 df-xadd 13099 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14034 df-hash 14297 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-vtx 28523 df-iedg 28524 df-edg 28573 df-uhgr 28583 df-upgr 28607 df-vtxdg 28988

This theorem is referenced by: finsumvtxdg2sstep 29071

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