Theorem clwlkfoclwwlk 26372

Description: There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 30-Jun-2018.)

Hypotheses

Ref	Expression
clwlkfclwwlk.1	⊢ 𝐴 = (1st ‘𝑐)
clwlkfclwwlk.2	⊢ 𝐵 = (2nd ‘𝑐)
clwlkfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))

Assertion

Ref	Expression
clwlkfoclwwlk	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐   𝐶,𝑐   𝐹,𝑐

Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlkfoclwwlk

Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkfclwwlk.1	. . 3 ⊢ 𝐴 = (1st ‘𝑐)
2		clwlkfclwwlk.2	. . 3 ⊢ 𝐵 = (2nd ‘𝑐)
3		clwlkfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
4		clwlkfclwwlk.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
5	1, 2, 3, 4	clwlkfclwwlk 26371	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
6		clwwlknprop 26300	. . . . 5 ⊢ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
7	6	adantl 481	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
8		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → 𝑁 ∈ ℕ0)
9	8	anim2i 591	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
10		df-3an 1033	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
11	9, 10	sylibr 223	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
12	11	3adant2 1073	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
13	12	adantl 481	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
14		isclwwlkn 26297	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁)))
15	13, 14	syl 17	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁)))
16		simpl1 1057	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 𝑉 USGrph 𝐸)
17		simpr2 1061	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 𝑤 ∈ Word 𝑉)
18		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑤) = 𝑁 → ((#‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
19		prmnn 15226	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑤) ∈ ℙ → (#‘𝑤) ∈ ℕ)
20	19	nnge1d 10940	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑤) ∈ ℙ → 1 ≤ (#‘𝑤))
21	18, 20	syl6bir 243	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
22	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
23	22	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
24	23	com12 32	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℙ → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤)))
25	24	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤)))
26	25	imp 444	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ (#‘𝑤))
27		clwlkisclwwlk2 26318	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (𝑉 ClWWalks 𝐸)))
28	16, 17, 26, 27	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (𝑉 ClWWalks 𝐸)))
29	28	bicomd 212	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ (𝑉 ClWWalks 𝐸) ↔ ∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
30	29	anbi1d 737	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁) ↔ (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
31	15, 30	bitrd 267	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
32		df-br 4584	. . . . . . . . . . 11 ⊢ (𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
33		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
34		prmnn 15226	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
35	34	nnge1d 10940	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℙ → 1 ≤ 𝑁)
36	35	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 1 ≤ 𝑁)
37	36	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ 𝑁)
38		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑤) = 𝑁 → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
39	38	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
40	39	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
41	40	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
42	37, 41	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ (#‘𝑤))
43	17, 42	jca 553	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
44	43	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
45		clwlkswlks 26286	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸))
46		wlklenvclwlk 26366	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝑓) = (#‘𝑤)))
47	44, 45, 46	syl2im 39	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
48	47	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘𝑓) = (#‘𝑤))
49		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑓 ∈ V
50		ovex 6577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
51	49, 50	op1st 7067	. . . . . . . . . . . . . . . . . . 19 ⊢ (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
52	51	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
53	52	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
54	53	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
55		eqcom 2617	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑤) = 𝑁 ↔ 𝑁 = (#‘𝑤))
56	55	biimpi 205	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑤) = 𝑁 → 𝑁 = (#‘𝑤))
57	56	ad2antll 761	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → 𝑁 = (#‘𝑤))
58	48, 54, 57	3eqtr4d 2654	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
59	1	fveq2i 6106	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘𝐴) = (#‘(1st ‘𝑐))
60	59	eqeq1i 2615	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘𝑐)) = 𝑁)
61		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
62	61	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘𝑐)) = (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
63	62	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st ‘𝑐)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
64	60, 63	syl5bb 271	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
65	64, 3	elrab2 3333	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
66	33, 58, 65	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
67	43, 45, 46	syl2im 39	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
68	67	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
69	68	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (#‘𝑓) = (#‘𝑤))
70	69	opeq2d 4347	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨0, (#‘𝑓)⟩ = ⟨0, (#‘𝑤)⟩)
71	70	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
72		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
73		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (#‘𝑤) → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
74	73	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑤) = 𝑁 → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
75	74	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤))))
76	75	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤))))
77	68, 76	mpbird 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁))
78	77	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (#‘𝑓) = 𝑁)
79	51	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
80	79	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
81	62, 80	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘𝑐)) = (#‘𝑓))
82	81	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st ‘𝑐)) = 𝑁 ↔ (#‘𝑓) = 𝑁))
83	60, 82	syl5bb 271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘𝑓) = 𝑁))
84	83, 3	elrab2 3333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝑓) = 𝑁))
85	72, 78, 84	sylanbrc 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
86		ovex 6577	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V
87	59	opeq2i 4344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st ‘𝑐))⟩
88	2, 87	oveq12i 6561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd ‘𝑐) substr ⟨0, (#‘(1st ‘𝑐))⟩)
89		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd ‘𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
90	62	opeq2d 4347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (#‘(1st ‘𝑐))⟩ = ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
91	89, 90	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) substr ⟨0, (#‘(1st ‘𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
92	49, 50	op2nd 7068	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
93	51	fveq2i 6106	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓)
94	93	opeq2i 4344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (#‘𝑓)⟩
95	92, 94	oveq12i 6561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩)
96	91, 95	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) substr ⟨0, (#‘(1st ‘𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
97	88, 96	syl5eq 2656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
98	97, 4	fvmptg 6189	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
99	85, 86, 98	sylancl 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
100	43	ad2antrl 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
101	100	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
102		simpl 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → 𝑤 ∈ Word 𝑉)
103		wrdsymb1 13197	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (𝑤‘0) ∈ 𝑉)
104	103	s1cld 13236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word 𝑉)
105		eqidd 2611	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (#‘𝑤) = (#‘𝑤))
106		swrdccatid 13348	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ⟨“(𝑤‘0)”⟩ ∈ Word 𝑉 ∧ (#‘𝑤) = (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
107	102, 104, 105, 106	syl3anc 1318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
108	107	eqcomd 2616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
109	101, 108	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
110	71, 99, 109	3eqtr4rd 2655	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
111	110	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
112	111	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
113		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
114	113	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹‘𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
115	114	imbi2d 329	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
116	115	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
117	112, 116	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘𝑐)))
118	66, 117	rspcimedv 3284	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
119	118	ex 449	. . . . . . . . . . . 12 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
120	119	pm2.43b 53	. . . . . . . . . . 11 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
121	32, 120	syl5bi 231	. . . . . . . . . 10 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
122	121	exlimdv 1848	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
123	122	ex 449	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((#‘𝑤) = 𝑁 → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
124	123	com23 84	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ((#‘𝑤) = 𝑁 → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
125	124	impd 446	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
126	31, 125	sylbid 229	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
127	126	impancom 455	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
128	7, 127	mpd 15	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
129	128	ralrimiva 2949	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
130		dffo3 6282	. 2 ⊢ (𝐹:𝐶–onto→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
131	5, 129, 130	sylanbrc 695	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Fincfn 7841  0cc0 9815  1c1 9816   ≤ cle 9954  ℕ₀cn0 11169  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  ℙcprime 15223   USGrph cusg 25859   Walks cwalk 26026   ClWalks cclwlk 26275   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277

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  ax-pre-sup 9893

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-rmo 2904  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-2o 7448  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-sup 8231  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-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-usgra 25862  df-wlk 26036  df-clwlk 26278  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  clwlkf1oclwwlk  26378