Theorem wwlksnredwwlkn0 28841

Description: For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)

Hypothesis

Ref	Expression
wwlksnredwwlkn.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
wwlksnredwwlkn0	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑁   𝑦,𝑊   𝑦,𝑃

Proof of Theorem wwlksnredwwlkn0

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnredwwlkn.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2	1	wwlksnredwwlkn 28840	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
3	2	imp 407	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))
4		simpl 483	. . . . . . . . 9 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → (𝑊 prefix (𝑁 + 1)) = 𝑦)
5	4	adantl 482	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)) → (𝑊 prefix (𝑁 + 1)) = 𝑦)
6		fveq1 6841	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
7	6	eqcoms 2744	. . . . . . . . . . . . 13 ⊢ ((𝑊 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
8	7	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) → (𝑦‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
9		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 1	wwlknp 28788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
11		nn0p1nn 12452	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
12		peano2nn0 12453	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
13		nn0re 12422	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
14		lep1 11996	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℝ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
15	12, 13, 14	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
16		peano2nn0 12453	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
17	16	nn0zd 12525	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
18		fznn 13509	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 + 1) + 1) ∈ ℤ → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
19	12, 17, 18	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
20	11, 15, 19	mpbir2and 711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (1...(♯‘𝑊)) = (1...((𝑁 + 1) + 1)))
22	21	eleq2d 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
23	20, 22	syl5ibr 245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(♯‘𝑊))))
24	23	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(♯‘𝑊))))
25		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word (Vtx‘𝐺))
26	24, 25	jctild 526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
27	26	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
28	10, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
29	28	impcom 408	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
30	29	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
31	30	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
32	31	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
33		pfxfv0 14580	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
35		simprll 777	. . . . . . . . . . . 12 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) → (𝑊‘0) = 𝑃)
36	8, 34, 35	3eqtrd 2780	. . . . . . . . . . 11 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺))) → (𝑦‘0) = 𝑃)
37	36	ex 413	. . . . . . . . . 10 ⊢ ((𝑊 prefix (𝑁 + 1)) = 𝑦 → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (𝑦‘0) = 𝑃))
38	37	adantr 481	. . . . . . . . 9 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (𝑦‘0) = 𝑃))
39	38	impcom 408	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)) → (𝑦‘0) = 𝑃)
40		simpr 485	. . . . . . . . 9 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)
41	40	adantl 482	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)) → {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)
42	5, 39, 41	3jca 1128	. . . . . . 7 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) ∧ ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))
43	42	ex 413	. . . . . 6 ⊢ ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
44	43	reximdva 3165	. . . . 5 ⊢ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
45	44	ex 413	. . . 4 ⊢ ((𝑊‘0) = 𝑃 → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))))
46	45	com13 88	. . 3 ⊢ (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))))
47	3, 46	mpcom 38	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
48	29, 33	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
49	48	eqcomd 2742	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
50	49	adantl 482	. . . . . . 7 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
51		fveq1 6841	. . . . . . . . 9 ⊢ ((𝑊 prefix (𝑁 + 1)) = 𝑦 → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑦‘0))
52	51	adantr 481	. . . . . . . 8 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑦‘0))
53	52	adantr 481	. . . . . . 7 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑦‘0))
54		simpr 485	. . . . . . . 8 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (𝑦‘0) = 𝑃)
55	54	adantr 481	. . . . . . 7 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑦‘0) = 𝑃)
56	50, 53, 55	3eqtrd 2780	. . . . . 6 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊‘0) = 𝑃)
57	56	ex 413	. . . . 5 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = 𝑃))
58	57	3adant3 1132	. . . 4 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = 𝑃))
59	58	com12 32	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → (𝑊‘0) = 𝑃))
60	59	rexlimdvw 3157	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → (𝑊‘0) = 𝑃))
61	47, 60	impbid 211	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {cpr 4588   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   ≤ cle 11190  ℕcn 12153  ℕ₀cn0 12413  ℤcz 12499  ...cfz 13424  ..^cfzo 13567  ♯chash 14230  Word cword 14402  lastSclsw 14450   prefix cpfx 14558  Vtxcvtx 27947  Edgcedg 27998   WWalksN cwwlksn 28771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-substr 14529  df-pfx 14559  df-wwlks 28775  df-wwlksn 28776

This theorem is referenced by:  rusgrnumwwlks  28919