Theorem wwlknredwwlkn0 26255

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.)

Assertion

Ref	Expression
wwlknredwwlkn0	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

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

Proof of Theorem wwlknredwwlkn0

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknredwwlkn 26254	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
2	1	imp 444	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸))
3		simpl 472	. . . . . . . . 9 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦)
4	3	adantl 481	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦)
5		fveq1 6102	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
6	5	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
8		wwlknimp 26215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
9		nn0p1nn 11209	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
10		peano2nn0 11210	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
11		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
12		lep1 10741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℝ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
13	10, 11, 12	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
14		peano2nn0 11210	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
15	14	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
16		fznn 12278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 + 1) + 1) ∈ ℤ → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
17	10, 15, 16	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
18	9, 13, 17	mpbir2and 959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
19		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (1...(#‘𝑊)) = (1...((𝑁 + 1) + 1)))
20	19	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
21	18, 20	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊))))
22	21	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊))))
23		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word 𝑉)
24	22, 23	jctild 564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
25	24	3adant3 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
26	8, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
27	26	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
28	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
29	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
30	29	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
31		swrd0fv0 13292	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
33		simprll 798	. . . . . . . . . . . 12 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑊‘0) = 𝑃)
34	7, 32, 33	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑦‘0) = 𝑃)
35	34	ex 449	. . . . . . . . . 10 ⊢ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → (𝑦‘0) = 𝑃))
36	35	adantr 480	. . . . . . . . 9 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → (𝑦‘0) = 𝑃))
37	36	impcom 445	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)) → (𝑦‘0) = 𝑃)
38		simpr 476	. . . . . . . . 9 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)
39	38	adantl 481	. . . . . . . 8 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)) → {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)
40	4, 37, 39	3jca 1235	. . . . . . 7 ⊢ (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸))
41	40	ex 449	. . . . . 6 ⊢ ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
42	41	reximdva 3000	. . . . 5 ⊢ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
43	42	ex 449	. . . 4 ⊢ ((𝑊‘0) = 𝑃 → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸))))
44	43	com13 86	. . 3 ⊢ (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸))))
45	2, 44	mpcom 37	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
46	27, 31	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
47	46	eqcomd 2616	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
48	47	adantl 481	. . . . . . 7 ⊢ ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → (𝑊‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
49		fveq1 6102	. . . . . . . . 9 ⊢ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
50	49	adantr 480	. . . . . . . 8 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
51	50	adantr 480	. . . . . . 7 ⊢ ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
52		simpr 476	. . . . . . . 8 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (𝑦‘0) = 𝑃)
53	52	adantr 480	. . . . . . 7 ⊢ ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → (𝑦‘0) = 𝑃)
54	48, 51, 53	3eqtrd 2648	. . . . . 6 ⊢ ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))) → (𝑊‘0) = 𝑃)
55	54	ex 449	. . . . 5 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊‘0) = 𝑃))
56	55	3adant3 1074	. . . 4 ⊢ (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊‘0) = 𝑃))
57	56	com12 32	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → (𝑊‘0) = 𝑃))
58	57	rexlimdvw 3016	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) → (𝑊‘0) = 𝑃))
59	45, 58	impbid 201	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-substr 13158  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  rusgranumwlks  26483