Theorem rusgrnumwwlkl1 29955

Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypothesis

Ref	Expression
rusgrnumwwlkl1.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
rusgrnumwwlkl1	⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Distinct variable groups:   𝑤,𝐺   𝑤,𝐾   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgrnumwwlkl1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn0 12522	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
2		iswwlksn 29825	. . . . . . . . 9 ⊢ (1 ∈ ℕ0 → (𝑤 ∈ (1 WWalksN 𝐺) ↔ (𝑤 ∈ (WWalks‘𝐺) ∧ (♯‘𝑤) = (1 + 1))))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑤 ∈ (1 WWalksN 𝐺) ↔ (𝑤 ∈ (WWalks‘𝐺) ∧ (♯‘𝑤) = (1 + 1)))
4		rusgrnumwwlkl1.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2736	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	iswwlks 29823	. . . . . . . . 9 ⊢ (𝑤 ∈ (WWalks‘𝐺) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
7	6	anbi1i 624	. . . . . . . 8 ⊢ ((𝑤 ∈ (WWalks‘𝐺) ∧ (♯‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)))
8	3, 7	bitri 275	. . . . . . 7 ⊢ (𝑤 ∈ (1 WWalksN 𝐺) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)))
9	8	a1i 11	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (1 WWalksN 𝐺) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1))))
10	9	anbi1d 631	. . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ (1 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃)))
11		1p1e2 12370	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
12	11	eqeq2i 2749	. . . . . . . . . 10 ⊢ ((♯‘𝑤) = (1 + 1) ↔ (♯‘𝑤) = 2)
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((♯‘𝑤) = (1 + 1) ↔ (♯‘𝑤) = 2))
14	13	anbi2d 630	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2)))
15		3anass 1094	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
16	15	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
17		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
18		hash0 14390	. . . . . . . . . . . . . . . 16 ⊢ (♯‘∅) = 0
19	17, 18	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0)
20		2ne0 12349	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
21	20	nesymi 2990	. . . . . . . . . . . . . . . 16 ⊢ ¬ 0 = 2
22		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑤) = 0 → ((♯‘𝑤) = 2 ↔ 0 = 2))
23	21, 22	mtbiri 327	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑤) = 0 → ¬ (♯‘𝑤) = 2)
24	19, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ¬ (♯‘𝑤) = 2)
25	24	necon2ai 2962	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑤) = 2 → 𝑤 ≠ ∅)
26	25	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → 𝑤 ≠ ∅)
27	26	biantrurd 532	. . . . . . . . . . 11 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
28		oveq1 7417	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑤) = 2 → ((♯‘𝑤) − 1) = (2 − 1))
29		2m1e1 12371	. . . . . . . . . . . . . . . . 17 ⊢ (2 − 1) = 1
30	28, 29	eqtrdi 2787	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑤) = 2 → ((♯‘𝑤) − 1) = 1)
31	30	oveq2d 7426	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑤) = 2 → (0..^((♯‘𝑤) − 1)) = (0..^1))
32	31	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → (0..^((♯‘𝑤) − 1)) = (0..^1))
33	32	raleqdv 3309	. . . . . . . . . . . . 13 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → (∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
34		fzo01 13768	. . . . . . . . . . . . . . 15 ⊢ (0..^1) = {0}
35	34	raleqi 3307	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ {0} {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺))
36		c0ex 11234	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
37		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑤‘𝑖) = (𝑤‘0))
38		fv0p1e1 12368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑤‘(𝑖 + 1)) = (𝑤‘1))
39	37, 38	preq12d 4722	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} = {(𝑤‘0), (𝑤‘1)})
40	39	eleq1d 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
41	36, 40	ralsn 4662	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ {0} {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
42	35, 41	bitri 275	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
43	33, 42	bitrdi 287	. . . . . . . . . . . 12 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → (∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
44	43	anbi2d 630	. . . . . . . . . . 11 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
45	16, 27, 44	3bitr2d 307	. . . . . . . . . 10 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (♯‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
46	45	ex 412	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((♯‘𝑤) = 2 → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
47	46	pm5.32rd 578	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2)))
48	14, 47	bitrd 279	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2)))
49	48	anbi1d 631	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃)))
50		anass 468	. . . . . 6 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)))
51	49, 50	bitrdi 287	. . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
52		anass 468	. . . . . . 7 ⊢ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
53		ancom 460	. . . . . . . . 9 ⊢ (({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
54		df-3an 1088	. . . . . . . . 9 ⊢ (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
55	53, 54	bitr4i 278	. . . . . . . 8 ⊢ (({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
56	55	anbi2i 623	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
57	52, 56	bitri 275	. . . . . 6 ⊢ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
58	57	a1i 11	. . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
59	10, 51, 58	3bitrd 305	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ (1 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
60	59	rabbidva2 3422	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))})
61	60	fveq2d 6885	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}))
62	4	rusgrnumwrdl2 29571	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
63	61, 62	eqtrd 2771	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  {crab 3420  ∅c0 4313  {csn 4606  {cpr 4608   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  0cc0 11134  1c1 11135   + caddc 11137   − cmin 11471  2c2 12300  ℕ₀cn0 12506  ..^cfzo 13676  ♯chash 14353  Word cword 14536  Vtxcvtx 28980  Edgcedg 29031   RegUSGraph crusgr 29541  WWalkscwwlks 29812   WWalksN cwwlksn 29813

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9920  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-xadd 13134  df-fz 13530  df-fzo 13677  df-hash 14354  df-word 14537  df-edg 29032  df-uhgr 29042  df-ushgr 29043  df-upgr 29066  df-umgr 29067  df-uspgr 29134  df-usgr 29135  df-nbgr 29317  df-vtxdg 29451  df-rgr 29542  df-rusgr 29543  df-wwlks 29817  df-wwlksn 29818

This theorem is referenced by:  rusgrnumwwlkb1  29959