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Theorem clwlksfclwwlk 26962

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 3197	. . . . 5 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁))
3		fusgrusgr 26214	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
4		usgrupgr 26077	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ UPGraph )
6	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ UPGraph )
7		eqid 2622	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8		eqid 2622	. . . . . . . . . . 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 26677	. . . . . . . . . 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 488	. . . . . . . . 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 13324	. . . . . . . . . . . . . . 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 26958	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ 𝐶 → 𝐵 ∈ Word (Vtx‘𝐺))
16	15	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 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 13419	. . . . . . . . . . . . . . . . . . . . . . 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 13332	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 ∈ Word (Vtx‘𝐺))
20	19	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word (Vtx‘𝐺))
21		prmnn 15388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
22	21	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
23	22	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ∈ ℕ)
24		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) = 𝑁 → (0...(#‘𝐴)) = (0...𝑁))
25	24	feq2d 6031	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ↔ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)))
26	22	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ₀)
27		ffz0hash 13231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
28	26, 27	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
29	28	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → (#‘𝐵) = (𝑁 + 1)))
30	21	nnred 11035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℝ)
31	30	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ∈ ℝ)
32	31	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (𝑁 + 1))
33		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((#‘𝐵) = (𝑁 + 1) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
34	33	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
35	32, 34	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (#‘𝐵))
36	35	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℙ → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
37	36	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
38	29, 37	syldc 48	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵)))
39	25, 38	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵))))
40	39	3imp21 1277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ≤ (#‘𝐵))
41		swrdn0 13430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (#‘𝐵)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
42	20, 23, 40, 41	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
43		opeq2 4403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → ⟨0, (#‘𝐴)⟩ = ⟨0, 𝑁⟩)
44	43	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = (𝐵 substr ⟨0, 𝑁⟩))
45	44	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐴) = 𝑁 → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
46	45	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
47	42, 46	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
48	47	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
49	48	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅)))
50	49	imp 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) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
51	50	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅))
52	51	imp 445	. . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅)
53	18, 52	jca 554	. . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅))
54		simp-5r 809	. . . . . . . . . . . . . . . . . . . . . . . . 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))
55	3	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USGraph )
56	54, 55	anim12ci 591	. . . . . . . . . . . . . . . . . . . . . . . 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)))
57		simp-5r 809	. . . . . . . . . . . . . . . . . . . . . . . 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))
58		prmuz2 15408	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
59		ffz0hash 13231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
60	59	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
61		eluz2 11693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)))
62		2re 11090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 2 ∈ ℝ
63	62	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → 2 ∈ ℝ)
64		zre 11381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℝ)
65		peano2re 10209	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((#‘𝐴) ∈ ℝ → ((#‘𝐴) + 1) ∈ ℝ)
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) + 1) ∈ ℝ)
67	63, 64, 66	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
68	67	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
69		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ (#‘𝐴))
70	64	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ≤ ((#‘𝐴) + 1))
71	70	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + 1))
72		letr 10131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) → ((2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1)) → 2 ≤ ((#‘𝐴) + 1)))
73	72	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) ∧ (2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1))) → 2 ≤ ((#‘𝐴) + 1))
74	68, 69, 71, 73	syl12anc 1324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
75	74	3adant1 1079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
76	61, 75	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1))
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1)))
78		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
79	78	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
80	79	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
81		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
82	81	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
83	77, 80, 82	3imtr4d 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
84	83	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
85	60, 84	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
86	85	adantr 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 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
87	86	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
88	87	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
89	58, 88	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)))
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
91	90	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . 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))
92		simp-4r 807	. . . . . . . . . . . . . . . . . . . . . . . 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))))
93	7, 8	usgrf 26050	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
94	93	anim1i 592	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
95		clwlkclwwlklem2 26901	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
96	94, 95	syl3an1 1359	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
97		biid 251	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( lastS ‘𝐵) = (𝐵‘0) ↔ ( lastS ‘𝐵) = (𝐵‘0))
98		edgval 25941	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
99	98	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
100	99	ralbii 2980	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀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	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))
102	97, 100, 101	3anbi123i 1251	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( 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	96, 102	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . 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)))
104	56, 57, 91, 92, 103	syl121anc 1331	. . . . . . . . . . . . . . . . . . . . . . 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)))
105	9, 10, 1, 14	clwlksfclwwlk1hash 26960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
106		simp2 1062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝐵 ∈ Word (Vtx‘𝐺))
107		simp1 1061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
108		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (#‘𝐴) ∈ ℤ)
109		peano2zm 11420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ∈ ℤ)
110		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℤ)
111	64	lem1d 10957	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ≤ (#‘𝐴))
112		eluz2 11693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) ↔ (((#‘𝐴) − 1) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴)))
113	109, 110, 111, 112	syl3anbrc 1246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)))
114		fzoss2 12496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
115	108, 113, 114	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
116	115	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
117	116	3adant2 1080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
118		swrd0fv 13439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
119	106, 107, 117, 118	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
120	119	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘𝑖) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖))
121		elfzom1elp1fzo 12534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ ℤ ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
122	108, 121	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
123	122	3adant2 1080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
124		swrd0fv 13439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)) = (𝐵‘(𝑖 + 1)))
125	124	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
126	106, 107, 123, 125	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
127	120, 126	preq12d 4276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))})
128	127	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))})))
129	105, 15, 128	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ 𝐶 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))}))
130	129	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
131	130	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
132	131	ralbidva 2985	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
133	132	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
134	9, 10, 1, 14	clwlksfclwwlk2sswd 26961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
135	134	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
136	135	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
137	136	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
138	137	raleqdv 3144	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
139	133, 138	bitrd 268	. . . . . . . . . . . . . . . . . . . . . . . 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)))
140		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ ↔ (#‘𝐴) ∈ ℙ))
141	140	biimpd 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
142	141	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
143		prmnn 15388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ ℙ → (#‘𝐴) ∈ ℕ)
144		elfz2nn0 12431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ ((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)))
145		1zzd 11408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → 1 ∈ ℤ)
146		nn0z 11400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((#‘𝐵) ∈ ℕ₀ → (#‘𝐵) ∈ ℤ)
147	146	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐵) ∈ ℤ)
148		nn0z 11400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℤ)
149	148	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐴) ∈ ℤ)
150	145, 147, 149	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
151	150	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
152	151	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
153		simp3 1063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
154		nnge1 11046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((#‘𝐴) ∈ ℕ → 1 ≤ (#‘𝐴))
155	153, 154	anim12ci 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵)))
156	152, 155	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
157	144, 156	sylanb 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
158		elfz2 12333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) ↔ ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
159	157, 158	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (#‘𝐴) ∈ (1...(#‘𝐵)))
160		swrd0fvlsw 13443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = (𝐵‘((#‘𝐴) − 1)))
161	160	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘((#‘𝐴) − 1)) = ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
162		swrd0fv0 13440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0) = (𝐵‘0))
163	162	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘0) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0))
164	161, 163	preq12d 4276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
165	164	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
166	159, 165	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)}))
167	166	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)})))
168	167	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)})))
169	105, 15, 168	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
170	143, 169	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
171	142, 170	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
172	171	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
173	172	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)})))
174	173	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}))
175	174	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)}))
176	175	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 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)}))
177	176	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . 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)})
178	177	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . 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)))
179	139, 178	3anbi23d 1402	. . . . . . . . . . . . . . . . . . . . . . 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))))
180	104, 179	mpbid 222	. . . . . . . . . . . . . . . . . . . . . 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)))
181		3simpc 1060	. . . . . . . . . . . . . . . . . . . . . 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)))
182	180, 181	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)))
183		3anass 1042	. . . . . . . . . . . . . . . . . . . . 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))))
184	53, 182, 183	sylanbrc 698	. . . . . . . . . . . . . . . . . . . 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)))
185		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
186	7, 185	isclwwlks 26880	. . . . . . . . . . . . . . . . . . . 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)))
187	184, 186	sylibr 224	. . . . . . . . . . . . . . . . . . 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))
188	134	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 ↔ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
189	188	biimpcd 239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
190	189	adantl 482	. . . . . . . . . . . . . . . . . . . . 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))
191	190	imp 445	. . . . . . . . . . . . . . . . . . . 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)
192	191	adantr 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)) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
193	187, 192	jca 554	. . . . . . . . . . . . . . . . . 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))
194	22	adantl 482	. . . . . . . . . . . . . . . . . . 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)
195		isclwwlksn 26882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
196	194, 195	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)))
197	193, 196	mpbird 247	. . . . . . . . . . . . . . . . 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))
198	197	exp31 630	. . . . . . . . . . . . . . . 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))))
199	198	exp41 638	. . . . . . . . . . . . . . 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)))))))
200	13, 199	mpancom 703	. . . . . . . . . . . . . 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)))))))
201	200	imp 445	. . . . . . . . . . . . 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))))))
202	201	3impib 1262	. . . . . . . . . . . 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)))))
203	202	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)))))
204	203	com14 96	. . . . . . . . . 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)))))
205	204	adantr 481	. . . . . . . . 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)))))
206	12, 205	mpd 15	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
207	206	expcom 451	. . . . . . 7 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))))
208	207	com24 95	. . . . . 6 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))))
209	208	imp 445	. . . . 5 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
210	2, 209	sylbi 207	. . . 4 ⊢ (𝑐 ∈ 𝐶 → (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
211	210	pm2.43i 52	. . 3 ⊢ (𝑐 ∈ 𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))
212	211	impcom 446	. 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ 𝐶) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))
213	212, 14	fmptd 6385	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 − cmin 10266 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 substr csubstr 13295 ℙcprime 15385 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 UPGraph cupgr 25975 USGraph cusgr 26044 FinUSGraph cfusgr 26208 ClWalkscclwlks 26666 ClWWalkscclwwlks 26875 ClWWalksN cclwwlksn 26876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-substr 13303 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-uspgr 26045 df-usgr 26046 df-fusgr 26209 df-wlks 26495 df-clwlks 26667 df-clwwlks 26877 df-clwwlksn 26878

This theorem is referenced by: clwlksfoclwwlk 26963 clwlksf1clwwlk 26969

W3C validator