Theorem vtxdginducedm1 30136

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 4345	. . . . . . . . . . 11 ⊢ (𝐽 ∪ 𝐼) = dom 𝐸
4	3	eqcomi 2772	. . . . . . . . . 10 ⊢ dom 𝐸 = (𝐽 ∪ 𝐼)
5	4	rabeqi 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)}
6		rabun2 4271	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
7	5, 6	eqtri 2786	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
8	7	fveq2i 6896	. . . . . . 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 6907	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11	10	dmex 7928	. . . . . . . . 9 ⊢ dom 𝐸 ∈ V
12	1, 11	rab2ex 5306	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
13	2, 11	rab2ex 5306	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
14		ssrab2 4007	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐽
15		ssrab2 4007	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐼
16		ss2in 4143	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼)
18	1, 2	elneldisj 4344	. . . . . . . . . . 11 ⊢ (𝐽 ∩ 𝐼) = ∅
19	18	sseq2i 3941	. . . . . . . . . 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 4354	. . . . . . . . . 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 220	. . . . . . . . 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 14540	. . . . . . . 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 1490	. . . . . . 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 2786	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
26	4	rabeqi 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}}
27		rabun2 4271	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
28	26, 27	eqtri 2786	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
29	28	fveq2i 6896	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
30	1, 11	rab2ex 5306	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
31	2, 11	rab2ex 5306	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
32		ssrab2 4007	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐽
33		ssrab2 4007	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐼
34		ss2in 4143	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼)
36	18	sseq2i 3941	. . . . . . . . . 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 4354	. . . . . . . . . 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 220	. . . . . . . . 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 14540	. . . . . . . 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 1490	. . . . . . 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 2786	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +e (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
43	25, 42	oveq12i 7439	. . . . 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 14493	. . . . . . . . 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 14493	. . . . . . . . 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 14493	. . . . . . . . 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 14493	. . . . . . . . 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 13444	. . . . . 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 13375	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
59		xnn0xr 12694	. . . . . . . . 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 13375	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
63		xnn0xr 12694	. . . . . . . . 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 13380	. . . . . . . 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 705	. . . . . . 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 30135	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) = 0)
72	71	oveq2d 7443	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e 0))
73		xnn0xr 12694	. . . . . . . . . . . 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		xaddrid 13381	. . . . . . . . . . 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	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
78		fveq2 6893	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝐸‘𝑘) = (𝐸‘𝑙))
79	78	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑣 ∈ (𝐸‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑙)))
80	79	cbvrabv 3425	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}
81	80	fveq2i 6896	. . . . . . . . 9 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})
82	77, 81	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
83	82	oveq2d 7443	. . . . . . 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	eqtrid 2810	. . . . . 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 2798	. . . . 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	eqtrid 2810	. . . 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 30133	. . . . . . . . . 10 ⊢ dom (iEdg‘𝑆) = 𝐼
88	87	rabeqi 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}
89	67, 9, 68, 2, 69, 70	vtxdginducedm1lem3 30134	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐼 → ((iEdg‘𝑆)‘𝑘) = (𝐸‘𝑘))
90	89	eleq2d 2849	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (𝑣 ∈ ((iEdg‘𝑆)‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑘)))
91	90	rabbiia 3419	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
92	88, 91	eqtri 2786	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
93	92	fveq2i 6896	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) = (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
94	87	rabeqi 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}
95	89	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (((iEdg‘𝑆)‘𝑘) = {𝑣} ↔ (𝐸‘𝑘) = {𝑣}))
96	95	rabbiia 3419	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
97	94, 96	eqtri 2786	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
98	97	fveq2i 6896	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}) = (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
99	93, 98	oveq12i 7439	. . . . . 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 2772	. . . . 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 7437	. . . 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	eqtrdi 2814	. . 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 4069	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
104		eqid 2763	. . . . 5 ⊢ dom 𝐸 = dom 𝐸
105	67, 9, 104	vtxdgval 30061	. . . 4 ⊢ (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) = ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})))
106	103, 105	syl 18	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = ((♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +e (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}})))
107	70	fveq2i 6896	. . . . . . . 8 ⊢ (Vtx‘𝑆) = (Vtx‘⟨𝐾, 𝑃⟩)
108	67	fvexi 6907	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
109		difexg 5294	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
110	68, 109	eqeltrid 2867	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → 𝐾 ∈ V)
111	108, 110	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ V
112		resexg 6021	. . . . . . . . . . 11 ⊢ (𝐸 ∈ V → (𝐸 ↾ 𝐼) ∈ V)
113	69, 112	eqeltrid 2867	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → 𝑃 ∈ V)
114	10, 113	ax-mp 5	. . . . . . . . 9 ⊢ 𝑃 ∈ V
115	111, 114	opvtxfvi 29599	. . . . . . . 8 ⊢ (Vtx‘⟨𝐾, 𝑃⟩) = 𝐾
116	107, 115	eqtri 2786	. . . . . . 7 ⊢ (Vtx‘𝑆) = 𝐾
117	116	eleq2i 2855	. . . . . 6 ⊢ (𝑣 ∈ (Vtx‘𝑆) ↔ 𝑣 ∈ 𝐾)
118	68	eleq2i 2855	. . . . . 6 ⊢ (𝑣 ∈ 𝐾 ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))
119	117, 118	sylbbr 239	. . . . 5 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ (Vtx‘𝑆))
120		eqid 2763	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
121		eqid 2763	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
122		eqid 2763	. . . . . 6 ⊢ dom (iEdg‘𝑆) = dom (iEdg‘𝑆)
123	120, 121, 122	vtxdgval 30061	. . . . 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 18	. . . 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 7442	. . 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 2808	. 2 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +e (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
127	126	rgen 3081	1 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +e (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146   ∉ wnel 3064  ∀wral 3079  {crab 3415  Vcvv 3453   ⊆ wss 3899   ∖ cdif 4047   ∩ cin 4091   ∪ cun 4151  ∅c0 4280  {csn 4587  ⟨cop 4837  dom cdm 5654   ↾ cres 5656  ‘cfv 6546  (class class class)co 7427  0cc0 11210  ℝ^*cxr 11352  ℕ₀^*cxnn0 12689   +_e cxad 13249  ♯chash 14484  Vtxcvtx 29586  iEdgciedg 29587  VtxDegcvtxdg 30058

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9992  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-n0 12617  df-xnn0 12690  df-z 12704  df-uz 12976  df-xadd 13252  df-fz 13650  df-hash 14485  df-vtx 29588  df-iedg 29589  df-vtxdg 30059

This theorem is used by:  vtxdginducedm1fi  30137