clwlksfclwwlk - Mathbox for Alexander van der Vekens

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Theorem clwlksfclwwlk 41269

Description: There is a function between the set of closed walks (defined as words) of length n and the set of closed walks of length n. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)

**Hypotheses**
Ref	Expression
clwlksfclwwlk.1	⊢ 𝐴 = (1^st ‘𝑐)
clwlksfclwwlk.2	⊢ 𝐵 = (2^nd ‘𝑐)
clwlksfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))

**Assertion**
Ref	Expression
clwlksfclwwlk	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺))

Distinct variable groups: 𝐺,𝑐 𝑁,𝑐 𝐶,𝑐 Allowed substitution hints: 𝐴(𝑐) 𝐵(𝑐) 𝐹(𝑐)

Proof of Theorem clwlksfclwwlk Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.c	. . . . . 6 ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
2	1	rabeq2i 3170	. . . . 5 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁))
3		fusgrusgr 40541	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
4		usgrupgr 40412	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ UPGraph )
6	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ UPGraph )
7		eqid 2610	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8		eqid 2610	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
9		clwlksfclwwlk.1	. . . . . . . . . . 11 ⊢ 𝐴 = (1^st ‘𝑐)
10		clwlksfclwwlk.2	. . . . . . . . . . 11 ⊢ 𝐵 = (2^nd ‘𝑐)
11	7, 8, 9, 10	upgrclwlkcompim 40988	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
12	6, 11	sylan 487	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
13		lencl 13179	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word dom (iEdg‘𝐺) → (#‘𝐴) ∈ ℕ₀)
14		clwlksfclwwlk.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
15	9, 10, 1, 14	clwlksfclwwlk2wrd 41265	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ 𝐶 → 𝐵 ∈ Word (Vtx‘𝐺))
16	15	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word (Vtx‘𝐺))
17		swrdcl 13271	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐵 ∈ Word (Vtx‘𝐺) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺))
19		ffz0iswrd 13187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 ∈ Word (Vtx‘𝐺))
20	19	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word (Vtx‘𝐺))
21		prmnn 15226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
22	21	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
23	22	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ∈ ℕ)
24		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) = 𝑁 → (0...(#‘𝐴)) = (0...𝑁))
25	24	feq2d 5944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ↔ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)))
26	22	nnnn0d 11228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ₀)
27		ffz0hash 13088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
28	26, 27	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
29	28	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → (#‘𝐵) = (𝑁 + 1)))
30	21	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℝ)
31	30	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ∈ ℝ)
32	31	lep1d 10834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (𝑁 + 1))
33		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐵) = (𝑁 + 1) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
34	33	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
35	32, 34	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (#‘𝐵))
36	35	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ ℙ → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
38	29, 37	syld 46	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → 𝑁 ≤ (#‘𝐵)))
39	38	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵)))
40	25, 39	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵))))
41	40	3imp21 1269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ≤ (#‘𝐵))
42		swrdn0 13282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (#‘𝐵)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
43	20, 23, 41, 42	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
44		opeq2 4341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → ⟨0, (#‘𝐴)⟩ = ⟨0, 𝑁⟩)
45	44	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = (𝐵 substr ⟨0, 𝑁⟩))
46	45	neeq1d 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐴) = 𝑁 → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
47	46	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
48	43, 47	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
49	48	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
50	49	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
51	50	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
52	51	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
53	52	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
54	18, 53	jca 553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
55		simp-5r 805	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 𝐴 ∈ Word dom (iEdg‘𝐺))
56	3	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USGraph )
57	55, 56	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)))
58		simp-5r 805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺))
59		prmuz2 15246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
60		ffz0hash 13088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
61	60	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
62		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)))
63		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ∈ ℝ
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → 2 ∈ ℝ)
65		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℝ)
66		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((#‘𝐴) ∈ ℝ → ((#‘𝐴) + 1) ∈ ℝ)
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) + 1) ∈ ℝ)
68	64, 65, 67	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
69	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
70		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ (#‘𝐴))
71	65	lep1d 10834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ≤ ((#‘𝐴) + 1))
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + 1))
73		letr 10010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) → ((2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1)) → 2 ≤ ((#‘𝐴) + 1)))
74	73	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) ∧ (2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1))) → 2 ≤ ((#‘𝐴) + 1))
75	69, 70, 72, 74	syl12anc 1316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
76	75	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
77	62, 76	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1))
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1)))
79		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
80	79	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
81	80	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
82		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
83	82	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
84	78, 81, 83	3imtr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
85	84	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
86	61, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
87	86	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
88	87	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
89	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
90	59, 89	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 2 ≤ (#‘𝐵)))
91	90	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 2 ≤ (#‘𝐵)))
92	91	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 2 ≤ (#‘𝐵))
93		simp-4r 803	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
94	7, 8	usgrf 40385	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
95	94	anim1i 590	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2} ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)))
96		clwlkclwwlklem2 41209	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2} ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺)))
97	95, 96	syl3an1 1351	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺)))
98		edgaval 25794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
99	98	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐺 ∈ USGraph → ({(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
100	99	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐺 ∈ USGraph → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
101	98	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐺 ∈ USGraph → ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺)))
102	100, 101	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ USGraph → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
104	103	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
105	97, 104	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)))
106	57, 58, 92, 93, 105	syl121anc 1323	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)))
107	9, 10, 1, 14	clwlksfclwwlk1hash 41267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
108		simp2 1055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝐵 ∈ Word (Vtx‘𝐺))
109		simp1 1054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
110		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (#‘𝐴) ∈ ℤ)
111		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ∈ ℤ)
112		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℤ)
113	65	lem1d 10836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ≤ (#‘𝐴))
114		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) ↔ (((#‘𝐴) − 1) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴)))
115	111, 112, 113, 114	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)))
116		fzoss2 12365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
117	110, 115, 116	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
118	117	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
119	118	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
120		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
121	108, 109, 119, 120	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
122	121	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘𝑖) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖))
123		elfzom1elp1fzo 12402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ ℤ ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
124	110, 123	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
125	124	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
126		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)) = (𝐵‘(𝑖 + 1)))
127	126	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
128	108, 109, 125, 127	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
129	122, 128	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
130	129	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})))
131	107, 15, 130	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ 𝐶 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))}))
132	131	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
133	132	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ({(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
134	133	ralbidva 2968	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 ∈ 𝐶 → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
135	134	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
136	9, 10, 1, 14	clwlksfclwwlk2sswd 41268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
137	136	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
138	137	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
139	138	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (0..^((#‘𝐴) − 1)) = (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)))
140	139	raleqdv 3121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
141	135, 140	bitrd 267	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
142		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ ↔ (#‘𝐴) ∈ ℙ))
143	142	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
144	143	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
145		prmnn 15226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ ℙ → (#‘𝐴) ∈ ℕ)
146		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ ((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)))
147		1zzd 11285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → 1 ∈ ℤ)
148		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((#‘𝐵) ∈ ℕ₀ → (#‘𝐵) ∈ ℤ)
149	148	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐵) ∈ ℤ)
150		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℤ)
151	150	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐴) ∈ ℤ)
152	147, 149, 151	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
153	152	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
154	153	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
155		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
156		nnge1 10923	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((#‘𝐴) ∈ ℕ → 1 ≤ (#‘𝐴))
157	155, 156	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵)))
158	154, 157	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
159	146, 158	sylanb 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
160		elfz2 12204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) ↔ ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
161	159, 160	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (#‘𝐴) ∈ (1...(#‘𝐵)))
162		swrd0fvlsw 13295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = (𝐵‘((#‘𝐴) − 1)))
163	162	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘((#‘𝐴) − 1)) = ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
164		swrd0fv0 13292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0) = (𝐵‘0))
165	164	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘0) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0))
166	163, 165	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
167	166	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
168	161, 167	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
169	168	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → ((#‘𝐴) ∈ ℕ → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
170	169	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
171	107, 15, 170	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
172	145, 171	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
173	144, 172	syl6 34	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
174	173	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
175	174	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
176	175	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
177	176	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
178	177	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
179	178	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
180	179	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
181	141, 180	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺))))
182	106, 181	mpbid 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
183		3simpc 1053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)) → (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
184	182, 183	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
185		3anass 1035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)) ↔ (((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅) ∧ (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺))))
186	54, 184, 185	sylanbrc 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
187		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
188	7, 187	isclwwlks 41188	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ↔ (((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
189	186, 188	sylibr 223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺))
190	136	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 ↔ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
191	190	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
192	191	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
193	192	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
194	193	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
195	189, 194	jca 553	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
196	22	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ∈ ℕ)
197		isclwwlksn 41190	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
198	196, 197	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
199	195, 198	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))
200	199	exp31 628	. . . . . . . . . . . . . . . 16 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
201	200	exp41 636	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))))
202	13, 201	mpancom 700	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word dom (iEdg‘𝐺) → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))))
203	202	imp 444	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))))
204	203	3impib 1254	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
205	204	com12 32	. . . . . . . . . . 11 ⊢ ((#‘𝐴) = 𝑁 → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
206	205	com14 94	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
207	206	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
208	12, 207	mpd 15	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
209	208	expcom 450	. . . . . . 7 ⊢ (𝑐 ∈ (ClWalkS‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
210	209	com24 93	. . . . . 6 ⊢ (𝑐 ∈ (ClWalkS‘𝐺) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
211	210	imp 444	. . . . 5 ⊢ ((𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
212	2, 211	sylbi 206	. . . 4 ⊢ (𝑐 ∈ 𝐶 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
213	212	pm2.43i 50	. . 3 ⊢ (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))
214	213	impcom 445	. 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ 𝐶) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))
215	214, 14	fmptd 6292	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 substr csubstr 13150 ℙcprime 15223 Vtxcvtx 25673 iEdgciedg 25674 UPGraph cupgr 25747 Edgcedga 25792 USGraph cusgr 40379 FinUSGraph cfusgr 40535 ClWalkScclwlks 40976 ClWWalkScclwwlks 41183 ClWWalkSN cclwwlksn 41184

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-ifp 1007 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-cda 8873 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-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-substr 13158 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-fusgr 40536 df-1wlks 40800 df-wlks 40801 df-clwlks 40977 df-clwwlks 41185 df-clwwlksn 41186

This theorem is referenced by: clwlksfoclwwlk 41270 clwlksf1clwwlk 41276

W3C validator