Theorem vtxdginducedm1 27008

Description: The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 17-Dec-2021.)

Hypotheses

Ref	Expression
vtxdginducedm1.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdginducedm1.e	⊢ 𝐸 = (iEdg‘𝐺)
vtxdginducedm1.k	⊢ 𝐾 = (𝑉 ∖ {𝑁})
vtxdginducedm1.i	⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
vtxdginducedm1.p	⊢ 𝑃 = (𝐸 ↾ 𝐼)
vtxdginducedm1.s	⊢ 𝑆 = ⟨𝐾, 𝑃⟩
vtxdginducedm1.j	⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}

Assertion

Ref	Expression
vtxdginducedm1	⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))

Distinct variable groups:   𝑖,𝐸   𝑖,𝑁   𝐸,𝑙   𝐽,𝑙   𝑣,𝑙

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

Proof of Theorem vtxdginducedm1

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdginducedm1.j	. . . . . . . . . . . 12 ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}
2		vtxdginducedm1.i	. . . . . . . . . . . 12 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
3	1, 2	elnelun 4263	. . . . . . . . . . 11 ⊢ (𝐽 ∪ 𝐼) = dom 𝐸
4	3	eqcomi 2804	. . . . . . . . . 10 ⊢ dom 𝐸 = (𝐽 ∪ 𝐼)
5	4	rabeqi 3427	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)}
6		rabun2 4202	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
7	5, 6	eqtri 2819	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
8	7	fveq2i 6541	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = (♯‘({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
9		vtxdginducedm1.e	. . . . . . . . . . 11 ⊢ 𝐸 = (iEdg‘𝐺)
10	9	fvexi 6552	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11	10	dmex 7472	. . . . . . . . 9 ⊢ dom 𝐸 ∈ V
12	1, 11	rab2ex 5129	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
13	2, 11	rab2ex 5129	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
14		ssrab2 3977	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐽
15		ssrab2 3977	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐼
16		ss2in 4133	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐽 ∧ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐼) → ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼))
17	14, 15, 16	mp2an 688	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼)
18	1, 2	elneldisj 4262	. . . . . . . . . . 11 ⊢ (𝐽 ∩ 𝐼) = ∅
19	18	sseq2i 3917	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼) ↔ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ ∅)
20		ss0 4272	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ ∅ → ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ∅)
21	19, 20	sylbi 218	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼) → ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ∅)
22	17, 21	ax-mp 5	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ∅
23		hashunx 13595	. . . . . . . 8 ⊢ (({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V ∧ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V ∧ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ∅) → (♯‘({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})))
24	12, 13, 22, 23	mp3an 1453	. . . . . . 7 ⊢ (♯‘({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
25	8, 24	eqtri 2819	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
26	4	rabeqi 3427	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}}
27		rabun2 4202	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
28	26, 27	eqtri 2819	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
29	28	fveq2i 6541	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
30	1, 11	rab2ex 5129	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
31	2, 11	rab2ex 5129	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
32		ssrab2 3977	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐽
33		ssrab2 3977	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐼
34		ss2in 4133	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐽 ∧ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐼) → ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼))
35	32, 33, 34	mp2an 688	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼)
36	18	sseq2i 3917	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼) ↔ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ ∅)
37		ss0 4272	. . . . . . . . . 10 ⊢ (({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ ∅ → ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) = ∅)
38	36, 37	sylbi 218	. . . . . . . . 9 ⊢ (({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼) → ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) = ∅)
39	35, 38	ax-mp 5	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) = ∅
40		hashunx 13595	. . . . . . . 8 ⊢ (({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V ∧ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V ∧ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) = ∅) → (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})))
41	30, 31, 39, 40	mp3an 1453	. . . . . . 7 ⊢ (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
42	29, 41	eqtri 2819	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
43	25, 42	oveq12i 7028	. . . . 5 ⊢ ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})) = (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})))
44		hashxnn0 13549	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V → (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*)
45	12, 44	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*
46	45	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*)
47		hashxnn0 13549	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V → (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*)
48	13, 47	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*
49	48	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0*)
50		hashxnn0 13549	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*)
51	30, 50	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*
52	51	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*)
53		hashxnn0 13549	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V → (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*)
54	31, 53	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*
55	54	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*)
56	46, 49, 52, 55	xnn0add4d 12547	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))))
57		xnn0xaddcl 12478	. . . . . . . . . 10 ⊢ (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0* ∧ (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*)
58	45, 51, 57	mp2an 688	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
59		xnn0xr 11820	. . . . . . . . 9 ⊢ (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0* → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ*)
60	58, 59	ax-mp 5	. . . . . . . 8 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ*
61		xnn0xaddcl 12478	. . . . . . . . . 10 ⊢ (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0* ∧ (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ∈ ℕ0*) → ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*)
62	48, 54, 61	mp2an 688	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
63		xnn0xr 11820	. . . . . . . . 9 ⊢ (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0* → ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ*)
64	62, 63	ax-mp 5	. . . . . . . 8 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ*
65		xaddcom 12483	. . . . . . . 8 ⊢ ((((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ* ∧ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℝ*) → (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}))))
66	60, 64, 65	mp2an 688	. . . . . . 7 ⊢ (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})))
67		vtxdginducedm1.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
68		vtxdginducedm1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
69		vtxdginducedm1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
70		vtxdginducedm1.s	. . . . . . . . . . . 12 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
71	67, 9, 68, 2, 69, 70, 1	vtxdginducedm1lem4 27007	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) = 0)
72	71	oveq2d 7032	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 0))
73		xnn0xr 11820	. . . . . . . . . . . 12 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℕ0* → (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℝ*)
74	45, 73	ax-mp 5	. . . . . . . . . . 11 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℝ*
75		xaddid1 12484	. . . . . . . . . . 11 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ∈ ℝ* → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 0) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
76	74, 75	ax-mp 5	. . . . . . . . . 10 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 0) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)})
77	72, 76	syl6eq 2847	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
78		fveq2 6538	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝐸‘𝑘) = (𝐸‘𝑙))
79	78	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑣 ∈ (𝐸‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑙)))
80	79	cbvrabv 3434	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}
81	80	fveq2i 6541	. . . . . . . . 9 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})
82	77, 81	syl6eq 2847	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
83	82	oveq2d 7032	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
84	66, 83	syl5eq 2843	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
85	56, 84	eqtrd 2831	. . . . 5 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})) +𝑒 ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
86	43, 85	syl5eq 2843	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})) = (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
87	67, 9, 68, 2, 69, 70	vtxdginducedm1lem2 27005	. . . . . . . . . 10 ⊢ dom (iEdg‘𝑆) = 𝐼
88	87	rabeqi 3427	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}
89	67, 9, 68, 2, 69, 70	vtxdginducedm1lem3 27006	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐼 → ((iEdg‘𝑆)‘𝑘) = (𝐸‘𝑘))
90	89	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (𝑣 ∈ ((iEdg‘𝑆)‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑘)))
91	90	rabbiia 3418	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
92	88, 91	eqtri 2819	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
93	92	fveq2i 6541	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) = (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
94	87	rabeqi 3427	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}
95	89	eqeq1d 2797	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (((iEdg‘𝑆)‘𝑘) = {𝑣} ↔ (𝐸‘𝑘) = {𝑣}))
96	95	rabbiia 3418	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
97	94, 96	eqtri 2819	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
98	97	fveq2i 6541	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}) = (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
99	93, 98	oveq12i 7028	. . . . . 6 ⊢ ((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
100	99	eqcomi 2804	. . . . 5 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}))
101	100	oveq1i 7026	. . . 4 ⊢ (((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) = (((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
102	86, 101	syl6eq 2847	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})) = (((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
103		eldifi 4024	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
104		eqid 2795	. . . . 5 ⊢ dom 𝐸 = dom 𝐸
105	67, 9, 104	vtxdgval 26933	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) = ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})))
106	103, 105	syl 17	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})))
107	70	fveq2i 6541	. . . . . . . 8 ⊢ (Vtx‘𝑆) = (Vtx‘⟨𝐾, 𝑃⟩)
108	67	fvexi 6552	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
109		difexg 5122	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
110	68, 109	syl5eqel 2887	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → 𝐾 ∈ V)
111	108, 110	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ V
112		resexg 5779	. . . . . . . . . . 11 ⊢ (𝐸 ∈ V → (𝐸 ↾ 𝐼) ∈ V)
113	69, 112	syl5eqel 2887	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → 𝑃 ∈ V)
114	10, 113	ax-mp 5	. . . . . . . . 9 ⊢ 𝑃 ∈ V
115	111, 114	opvtxfvi 26477	. . . . . . . 8 ⊢ (Vtx‘⟨𝐾, 𝑃⟩) = 𝐾
116	107, 115	eqtri 2819	. . . . . . 7 ⊢ (Vtx‘𝑆) = 𝐾
117	116	eleq2i 2874	. . . . . 6 ⊢ (𝑣 ∈ (Vtx‘𝑆) ↔ 𝑣 ∈ 𝐾)
118	68	eleq2i 2874	. . . . . 6 ⊢ (𝑣 ∈ 𝐾 ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))
119	117, 118	sylbbr 237	. . . . 5 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ (Vtx‘𝑆))
120		eqid 2795	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
121		eqid 2795	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
122		eqid 2795	. . . . . 6 ⊢ dom (iEdg‘𝑆) = dom (iEdg‘𝑆)
123	120, 121, 122	vtxdgval 26933	. . . . 5 ⊢ (𝑣 ∈ (Vtx‘𝑆) → ((VtxDeg‘𝑆)‘𝑣) = ((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})))
124	119, 123	syl 17	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝑆)‘𝑣) = ((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})))
125	124	oveq1d 7031	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) = (((♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}})) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
126	102, 106, 125	3eqtr4d 2841	. 2 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
127	126	rgen 3115	1 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))

Syntax hints:   = wceq 1522   ∈ wcel 2081   ∉ wnel 3090  ∀wral 3105  {crab 3109  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  {csn 4472  ⟨cop 4478  dom cdm 5443   ↾ cres 5445  ‘cfv 6225  (class class class)co 7016  0cc0 10383  ℝ^*cxr 10520  ℕ₀^*cxnn0 11815   +_𝑒 cxad 12355  ♯chash 13540  Vtxcvtx 26464  iEdgciedg 26465  VtxDegcvtxdg 26930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-xnn0 11816  df-z 11830  df-uz 12094  df-xadd 12358  df-fz 12743  df-hash 13541  df-vtx 26466  df-iedg 26467  df-vtxdg 26931

This theorem is referenced by:  vtxdginducedm1fi  27009