Theorem vtxdginducedm1 29576

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 4399	. . . . . . . . . . 11 ⊢ (𝐽 ∪ 𝐼) = dom 𝐸
4	3	eqcomi 2744	. . . . . . . . . 10 ⊢ dom 𝐸 = (𝐽 ∪ 𝐼)
5	4	rabeqi 3447	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)}
6		rabun2 4330	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
7	5, 6	eqtri 2763	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
8	7	fveq2i 6910	. . . . . . 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 6921	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11	10	dmex 7932	. . . . . . . . 9 ⊢ dom 𝐸 ∈ V
12	1, 11	rab2ex 5348	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
13	2, 11	rab2ex 5348	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
14		ssrab2 4090	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐽
15		ssrab2 4090	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐼
16		ss2in 4253	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼)
18	1, 2	elneldisj 4398	. . . . . . . . . . 11 ⊢ (𝐽 ∩ 𝐼) = ∅
19	18	sseq2i 4025	. . . . . . . . . 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 4408	. . . . . . . . . 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 217	. . . . . . . . 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 14422	. . . . . . . 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 1460	. . . . . . 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 2763	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
26	4	rabeqi 3447	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}}
27		rabun2 4330	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
28	26, 27	eqtri 2763	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
29	28	fveq2i 6910	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
30	1, 11	rab2ex 5348	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
31	2, 11	rab2ex 5348	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
32		ssrab2 4090	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐽
33		ssrab2 4090	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐼
34		ss2in 4253	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼)
36	18	sseq2i 4025	. . . . . . . . . 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 4408	. . . . . . . . . 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 217	. . . . . . . . 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 14422	. . . . . . . 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 1460	. . . . . . 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 2763	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
43	25, 42	oveq12i 7443	. . . . 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 14375	. . . . . . . . 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 14375	. . . . . . . . 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 14375	. . . . . . . . 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 14375	. . . . . . . . 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 13343	. . . . . 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 13274	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
59		xnn0xr 12602	. . . . . . . . 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 13274	. . . . . . . . . 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 692	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ0*
63		xnn0xr 12602	. . . . . . . . 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 13279	. . . . . . . 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 692	. . . . . . 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 29575	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) = 0)
72	71	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 0))
73		xnn0xr 12602	. . . . . . . . . . . 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 13280	. . . . . . . . . . 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 2791	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
78		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝐸‘𝑘) = (𝐸‘𝑙))
79	78	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑣 ∈ (𝐸‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑙)))
80	79	cbvrabv 3444	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}
81	80	fveq2i 6910	. . . . . . . . 9 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})
82	77, 81	eqtrdi 2791	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
83	82	oveq2d 7447	. . . . . . 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 2787	. . . . . 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 2775	. . . . 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 2787	. . . 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 29573	. . . . . . . . . 10 ⊢ dom (iEdg‘𝑆) = 𝐼
88	87	rabeqi 3447	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}
89	67, 9, 68, 2, 69, 70	vtxdginducedm1lem3 29574	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐼 → ((iEdg‘𝑆)‘𝑘) = (𝐸‘𝑘))
90	89	eleq2d 2825	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (𝑣 ∈ ((iEdg‘𝑆)‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑘)))
91	90	rabbiia 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
92	88, 91	eqtri 2763	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
93	92	fveq2i 6910	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) = (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
94	87	rabeqi 3447	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}
95	89	eqeq1d 2737	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (((iEdg‘𝑆)‘𝑘) = {𝑣} ↔ (𝐸‘𝑘) = {𝑣}))
96	95	rabbiia 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
97	94, 96	eqtri 2763	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
98	97	fveq2i 6910	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}) = (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
99	93, 98	oveq12i 7443	. . . . . 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 2744	. . . . 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 7441	. . . 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 2791	. . 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 4141	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
104		eqid 2735	. . . . 5 ⊢ dom 𝐸 = dom 𝐸
105	67, 9, 104	vtxdgval 29501	. . . 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 6910	. . . . . . . 8 ⊢ (Vtx‘𝑆) = (Vtx‘⟨𝐾, 𝑃⟩)
108	67	fvexi 6921	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
109		difexg 5335	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
110	68, 109	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → 𝐾 ∈ V)
111	108, 110	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ V
112		resexg 6047	. . . . . . . . . . 11 ⊢ (𝐸 ∈ V → (𝐸 ↾ 𝐼) ∈ V)
113	69, 112	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → 𝑃 ∈ V)
114	10, 113	ax-mp 5	. . . . . . . . 9 ⊢ 𝑃 ∈ V
115	111, 114	opvtxfvi 29041	. . . . . . . 8 ⊢ (Vtx‘⟨𝐾, 𝑃⟩) = 𝐾
116	107, 115	eqtri 2763	. . . . . . 7 ⊢ (Vtx‘𝑆) = 𝐾
117	116	eleq2i 2831	. . . . . 6 ⊢ (𝑣 ∈ (Vtx‘𝑆) ↔ 𝑣 ∈ 𝐾)
118	68	eleq2i 2831	. . . . . 6 ⊢ (𝑣 ∈ 𝐾 ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))
119	117, 118	sylbbr 236	. . . . 5 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ (Vtx‘𝑆))
120		eqid 2735	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
121		eqid 2735	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
122		eqid 2735	. . . . . 6 ⊢ dom (iEdg‘𝑆) = dom (iEdg‘𝑆)
123	120, 121, 122	vtxdgval 29501	. . . . 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 7446	. . 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 2785	. 2 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
127	126	rgen 3061	1 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))

Syntax hints:   = wceq 1537   ∈ wcel 2106   ∉ wnel 3044  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631  ⟨cop 4637  dom cdm 5689   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431  0cc0 11153  ℝ^*cxr 11292  ℕ₀^*cxnn0 12597   +_𝑒 cxad 13150  ♯chash 14366  Vtxcvtx 29028  iEdgciedg 29029  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-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

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-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-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-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-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-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  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-nn 12265  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-xadd 13153  df-fz 13545  df-hash 14367  df-vtx 29030  df-iedg 29031  df-vtxdg 29499

This theorem is referenced by:  vtxdginducedm1fi  29577