Theorem wwlksnredwwlkn0 29982

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 29981	. . . 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 406	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))
4		simpl 482	. . . . . . . . 9 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → (𝑊 prefix (𝑁 + 1)) = 𝑦)
5	4	adantl 481	. . . . . . . 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 6834	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
7	6	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ ((𝑊 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
8	7	adantr 480	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 1	wwlknp 29929	. . . . . . . . . . . . . . . . . 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 12470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
12		peano2nn0 12471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
13		nn0re 12440	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
14		lep1 11990	. . . . . . . . . . . . . . . . . . . . . . . 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 12471	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
17	16	nn0zd 12543	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
18		fznn 13540	. . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21		oveq2 7369	. . . . . . . . . . . . . . . . . . . . . . 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	imbitrrid 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(♯‘𝑊))))
24	23	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(♯‘𝑊))))
25		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word (Vtx‘𝐺))
26	24, 25	jctild 525	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
27	26	3adant3 1133	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
30	29	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
31	30	adantr 480	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 14648	. . . . . . . . . . . . 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 779	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 412	. . . . . . . . . 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 480	. . . . . . . . 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 407	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)
41	40	adantl 481	. . . . . . . 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 1129	. . . . . . 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 412	. . . . . 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 3151	. . . . 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 412	. . . 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 2743	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = ((𝑊 prefix (𝑁 + 1))‘0))
50	49	adantl 481	. . . . . . 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 6834	. . . . . . . . 9 ⊢ ((𝑊 prefix (𝑁 + 1)) = 𝑦 → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑦‘0))
52	51	adantr 480	. . . . . . . 8 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑦‘0))
53	52	adantr 480	. . . . . . 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 484	. . . . . . . 8 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (𝑦‘0) = 𝑃)
55	54	adantr 480	. . . . . . 7 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑦‘0) = 𝑃)
56	50, 53, 55	3eqtrd 2776	. . . . . 6 ⊢ ((((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊‘0) = 𝑃)
57	56	ex 412	. . . . 5 ⊢ (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = 𝑃))
58	57	3adant3 1133	. . . 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 3144	. 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 212	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {cpr 4570   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  ℝcr 11031  0cc0 11032  1c1 11033   + caddc 11035   ≤ cle 11174  ℕcn 12168  ℕ₀cn0 12431  ℤcz 12518  ...cfz 13455  ..^cfzo 13602  ♯chash 14286  Word cword 14469  lastSclsw 14518   prefix cpfx 14627  Vtxcvtx 29082  Edgcedg 29133   WWalksN cwwlksn 29912

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-n0 12432  df-z 12519  df-uz 12783  df-fz 13456  df-fzo 13603  df-hash 14287  df-word 14470  df-lsw 14519  df-substr 14598  df-pfx 14628  df-wwlks 29916  df-wwlksn 29917

This theorem is referenced by:  rusgrnumwwlks  30063