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Theorem vtxdginducedm1 29269

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 4381	. . . . . . . . . . 11 ⊢ (𝐽 ∪ 𝐼) = dom 𝐸
4	3	eqcomi 2733	. . . . . . . . . 10 ⊢ dom 𝐸 = (𝐽 ∪ 𝐼)
5	4	rabeqi 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)}
6		rabun2 4305	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
7	5, 6	eqtri 2752	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)} = ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∪ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
8	7	fveq2i 6884	. . . . . . 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 6895	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11	10	dmex 7895	. . . . . . . . 9 ⊢ dom 𝐸 ∈ V
12	1, 11	rab2ex 5325	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
13	2, 11	rab2ex 5325	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∈ V
14		ssrab2 4069	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐽
15		ssrab2 4069	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)} ⊆ 𝐼
16		ss2in 4228	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)} ∩ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) ⊆ (𝐽 ∩ 𝐼)
18	1, 2	elneldisj 4380	. . . . . . . . . . 11 ⊢ (𝐽 ∩ 𝐼) = ∅
19	18	sseq2i 4003	. . . . . . . . . 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 4390	. . . . . . . . . 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 216	. . . . . . . . 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 14343	. . . . . . . 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 1457	. . . . . . 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 2752	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
26	4	rabeqi 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}}
27		rabun2 4305	. . . . . . . . 9 ⊢ {𝑘 ∈ (𝐽 ∪ 𝐼) ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
28	26, 27	eqtri 2752	. . . . . . . 8 ⊢ {𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}} = ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
29	28	fveq2i 6884	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = (♯‘({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∪ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
30	1, 11	rab2ex 5325	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
31	2, 11	rab2ex 5325	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ∈ V
32		ssrab2 4069	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐽
33		ssrab2 4069	. . . . . . . . . 10 ⊢ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}} ⊆ 𝐼
34		ss2in 4228	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ ({𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}} ∩ {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}) ⊆ (𝐽 ∩ 𝐼)
36	18	sseq2i 4003	. . . . . . . . . 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 4390	. . . . . . . . . 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 216	. . . . . . . . 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 14343	. . . . . . . 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 1457	. . . . . . 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 2752	. . . . . 6 ⊢ (♯‘{𝑘 ∈ dom 𝐸 ∣ (𝐸‘𝑘) = {𝑣}}) = ((♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) +_𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}))
43	25, 42	oveq12i 7413	. . . . 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 14296	. . . . . . . . 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 14296	. . . . . . . . 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 14296	. . . . . . . . 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 14296	. . . . . . . . 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 13280	. . . . . 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 13211	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ₀^*
59		xnn0xr 12546	. . . . . . . . 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 13211	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ ((♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})) ∈ ℕ₀^*
63		xnn0xr 12546	. . . . . . . . 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 13216	. . . . . . . 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 689	. . . . . . 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 29268	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}}) = 0)
72	71	oveq2d 7417	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 0))
73		xnn0xr 12546	. . . . . . . . . . . 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 13217	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}))
78		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝐸‘𝑘) = (𝐸‘𝑙))
79	78	eleq2d 2811	. . . . . . . . . . 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 6884	. . . . . . . . 9 ⊢ (♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})
82	77, 81	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((♯‘{𝑘 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑘)}) +_𝑒 (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑣}})) = (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
83	82	oveq2d 7417	. . . . . . 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 2776	. . . . . 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 2764	. . . . 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 2776	. . . 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 29266	. . . . . . . . . 10 ⊢ dom (iEdg‘𝑆) = 𝐼
88	87	rabeqi 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}
89	67, 9, 68, 2, 69, 70	vtxdginducedm1lem3 29267	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐼 → ((iEdg‘𝑆)‘𝑘) = (𝐸‘𝑘))
90	89	eleq2d 2811	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (𝑣 ∈ ((iEdg‘𝑆)‘𝑘) ↔ 𝑣 ∈ (𝐸‘𝑘)))
91	90	rabbiia 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
92	88, 91	eqtri 2752	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)} = {𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)}
93	92	fveq2i 6884	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ 𝑣 ∈ ((iEdg‘𝑆)‘𝑘)}) = (♯‘{𝑘 ∈ 𝐼 ∣ 𝑣 ∈ (𝐸‘𝑘)})
94	87	rabeqi 3437	. . . . . . . . 9 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}
95	89	eqeq1d 2726	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐼 → (((iEdg‘𝑆)‘𝑘) = {𝑣} ↔ (𝐸‘𝑘) = {𝑣}))
96	95	rabbiia 3428	. . . . . . . . 9 ⊢ {𝑘 ∈ 𝐼 ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
97	94, 96	eqtri 2752	. . . . . . . 8 ⊢ {𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}} = {𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}}
98	97	fveq2i 6884	. . . . . . 7 ⊢ (♯‘{𝑘 ∈ dom (iEdg‘𝑆) ∣ ((iEdg‘𝑆)‘𝑘) = {𝑣}}) = (♯‘{𝑘 ∈ 𝐼 ∣ (𝐸‘𝑘) = {𝑣}})
99	93, 98	oveq12i 7413	. . . . . 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 2733	. . . . 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 7411	. . . 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 2780	. . 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 4118	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
104		eqid 2724	. . . . 5 ⊢ dom 𝐸 = dom 𝐸
105	67, 9, 104	vtxdgval 29194	. . . 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 6884	. . . . . . . 8 ⊢ (Vtx‘𝑆) = (Vtx‘⟨𝐾, 𝑃⟩)
108	67	fvexi 6895	. . . . . . . . . 10 ⊢ 𝑉 ∈ V
109		difexg 5317	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
110	68, 109	eqeltrid 2829	. . . . . . . . . 10 ⊢ (𝑉 ∈ V → 𝐾 ∈ V)
111	108, 110	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ V
112		resexg 6017	. . . . . . . . . . 11 ⊢ (𝐸 ∈ V → (𝐸 ↾ 𝐼) ∈ V)
113	69, 112	eqeltrid 2829	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → 𝑃 ∈ V)
114	10, 113	ax-mp 5	. . . . . . . . 9 ⊢ 𝑃 ∈ V
115	111, 114	opvtxfvi 28738	. . . . . . . 8 ⊢ (Vtx‘⟨𝐾, 𝑃⟩) = 𝐾
116	107, 115	eqtri 2752	. . . . . . 7 ⊢ (Vtx‘𝑆) = 𝐾
117	116	eleq2i 2817	. . . . . 6 ⊢ (𝑣 ∈ (Vtx‘𝑆) ↔ 𝑣 ∈ 𝐾)
118	68	eleq2i 2817	. . . . . 6 ⊢ (𝑣 ∈ 𝐾 ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))
119	117, 118	sylbbr 235	. . . . 5 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ (Vtx‘𝑆))
120		eqid 2724	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
121		eqid 2724	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
122		eqid 2724	. . . . . 6 ⊢ dom (iEdg‘𝑆) = dom (iEdg‘𝑆)
123	120, 121, 122	vtxdgval 29194	. . . . 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 7416	. . 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 2774	. 2 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +_𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
127	126	rgen 3055	1 ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +_𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ∉ wnel 3038 ∀wral 3053 {crab 3424 Vcvv 3466 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 ⟨cop 4626 dom cdm 5666 ↾ cres 5668 ‘cfv 6533 (class class class)co 7401 0cc0 11106 ℝ^*cxr 11244 ℕ₀^*cxnn0 12541 +_𝑒 cxad 13087 ♯chash 14287 Vtxcvtx 28725 iEdgciedg 28726 VtxDegcvtxdg 29191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-xnn0 12542 df-z 12556 df-uz 12820 df-xadd 13090 df-fz 13482 df-hash 14288 df-vtx 28727 df-iedg 28728 df-vtxdg 29192

This theorem is referenced by: vtxdginducedm1fi 29270

W3C validator