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Theorem clwlkfclwwlk 26371

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 (in an undirected simple graph). (Contributed by Alexander van der Vekens, 25-Jun-2018.)

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

**Assertion**
Ref	Expression
clwlkfclwwlk	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

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

Proof of Theorem clwlkfclwwlk Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkfclwwlk.c	. . . . . 6 ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
2	1	rabeq2i 3170	. . . . 5 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝐴) = 𝑁))
3		clwlkfclwwlk.1	. . . . . . . 8 ⊢ 𝐴 = (1^st ‘𝑐)
4		clwlkfclwwlk.2	. . . . . . . 8 ⊢ 𝐵 = (2^nd ‘𝑐)
5	3, 4	clwlkcompim 26292	. . . . . . 7 ⊢ (𝑐 ∈ (𝑉 ClWalks 𝐸) → ((𝐴 ∈ Word dom 𝐸 ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))))
6		lencl 13179	. . . . . . . . 9 ⊢ (𝐴 ∈ Word dom 𝐸 → (#‘𝐴) ∈ ℕ₀)
7		clwlkfclwwlk.f	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
8	3, 4, 1, 7	clwlkfclwwlk2wrd 26367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ 𝐶 → 𝐵 ∈ Word 𝑉)
9	8	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word 𝑉)
10		swrdcl 13271	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉)
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉)
12		simp-5r 805	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 𝐴 ∈ Word dom 𝐸)
13		simp1 1054	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑉 USGrph 𝐸)
14	12, 13	anim12ci 589	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝑉 USGrph 𝐸 ∧ 𝐴 ∈ Word dom 𝐸))
15		simp-5r 805	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → 𝐵:(0...(#‘𝐴))⟶𝑉)
16		prmuz2 15246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
17		ffz0hash 13088	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) → (#‘𝐵) = ((#‘𝐴) + 1))
18	17	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) → (#‘𝐵) = ((#‘𝐴) + 1))
19		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)))
20		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 2 ∈ ℝ
21	20	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ ℤ → 2 ∈ ℝ)
22		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℝ)
23		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐴) ∈ ℝ → ((#‘𝐴) + 1) ∈ ℝ)
24	22, 23	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) + 1) ∈ ℝ)
25	21, 22, 24	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℤ → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
27		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ (#‘𝐴))
28	22	lep1d 10834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ≤ ((#‘𝐴) + 1))
29	28	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + 1))
30		letr 10010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) → ((2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1)) → 2 ≤ ((#‘𝐴) + 1)))
31	30	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) ∧ (2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1))) → 2 ≤ ((#‘𝐴) + 1))
32	26, 27, 29, 31	syl12anc 1316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
33	32	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
34	19, 33	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1))
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → ((#‘𝐴) ∈ (ℤ_≥‘2) → 2 ≤ ((#‘𝐴) + 1)))
36		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
37	36	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) ↔ (#‘𝐴) ∈ (ℤ_≥‘2)))
39		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
40	39	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
41	35, 38, 40	3imtr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
42	41	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
43	18, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
44	43	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵))))
45	44	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
46	45	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ (#‘𝐵)))
47	16, 46	syl5com 31	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 2 ≤ (#‘𝐵)))
48	47	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → 2 ≤ (#‘𝐵)))
49	48	impcom 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → 2 ≤ (#‘𝐵))
50		simp-4r 803	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
51		clwlkisclwwlklem1 26315	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝐴 ∈ Word dom 𝐸) ∧ (𝐵:(0...(#‘𝐴))⟶𝑉 ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(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 𝐸 ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran 𝐸))
52	14, 15, 49, 50, 51	syl121anc 1323	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran 𝐸))
53	3, 4, 1, 7	clwlkfclwwlk1hash 26369	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
54		simp2 1055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝐵 ∈ Word 𝑉)
55		simp1 1054	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
56		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (#‘𝐴) ∈ ℤ)
57		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ∈ ℤ)
58		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℤ)
59	22	lem1d 10836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ≤ (#‘𝐴))
60		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) ↔ (((#‘𝐴) − 1) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴)))
61	57, 58, 59, 60	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)))
62		fzoss2 12365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) ∈ (ℤ_≥‘((#‘𝐴) − 1)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
63	56, 61, 62	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
64	63	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
65	64	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
66		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
67	54, 55, 65, 66	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵‘𝑖))
68	67	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘𝑖) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖))
69		elfzom1elp1fzo 12402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝐴) ∈ ℤ ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
70	56, 69	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
71	70	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
72		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)) = (𝐵‘(𝑖 + 1)))
73	72	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
74	54, 55, 71, 73	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
75	68, 74	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
76	75	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (𝐵 ∈ Word 𝑉 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})))
77	53, 8, 76	sylc 63	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝐶 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))}))
78	77	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
79	78	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ({(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
80	79	ralbidva 2968	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ 𝐶 → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
81	80	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
82	3, 4, 1, 7	clwlkfclwwlk2sswd 26370	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
83	82	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
84	83	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
85	84	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (0..^((#‘𝐴) − 1)) = (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)))
86	85	raleqdv 3121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
87	81, 86	bitrd 267	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
88		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ ↔ (#‘𝐴) ∈ ℙ))
89	88	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
90	89	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
91		prmnn 15226	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝐴) ∈ ℙ → (#‘𝐴) ∈ ℕ)
92		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ ((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)))
93		1zzd 11285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → 1 ∈ ℤ)
94		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐵) ∈ ℕ₀ → (#‘𝐵) ∈ ℤ)
95	94	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐵) ∈ ℤ)
96		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℤ)
97	96	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (#‘𝐴) ∈ ℤ)
98	93, 95, 97	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
99	98	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
100	99	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
101		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
102		nnge1 10923	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((#‘𝐴) ∈ ℕ → 1 ≤ (#‘𝐴))
103	101, 102	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵)))
104	100, 103	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
105	92, 104	sylanb 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
106		elfz2 12204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) ↔ ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
107	105, 106	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (#‘𝐴) ∈ (1...(#‘𝐵)))
108		swrd0fvlsw 13295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = (𝐵‘((#‘𝐴) − 1)))
109	108	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘((#‘𝐴) − 1)) = ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
110		swrd0fv0 13292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0) = (𝐵‘0))
111	110	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘0) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0))
112	109, 111	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
113	112	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐴) ∈ (1...(#‘𝐵)) → (𝐵 ∈ Word 𝑉 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
114	107, 113	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (𝐵 ∈ Word 𝑉 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
115	114	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → ((#‘𝐴) ∈ ℕ → (𝐵 ∈ Word 𝑉 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
116	115	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐵)) → (𝐵 ∈ Word 𝑉 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
117	53, 8, 116	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
118	91, 117	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝐴) ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
119	90, 118	syl6 34	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (𝑐 ∈ 𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
120	119	com23 84	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
121	120	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(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)})))
122	121	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(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)}))
123	122	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(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)}))
124	123	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(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)}))
125	124	impcom 445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
126	125	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran 𝐸 ↔ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸))
127	87, 126	3anbi23d 1394	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran 𝐸) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸)))
128	52, 127	mpbid 221	. . . . . . . . . . . . . . . 16 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸))
129		3simpc 1053	. . . . . . . . . . . . . . . 16 ⊢ ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸) → (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸))
130	128, 129	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸))
131		3anass 1035	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸)))
132	11, 130, 131	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸))
133		usgrav 25867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
134	133	3ad2ant1 1075	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
135	134	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
136		isclwwlk 26296	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸)))
137	135, 136	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ ran 𝐸)))
138	132, 137	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸))
139	82	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ 𝐶 → ((#‘𝐴) = 𝑁 ↔ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
140	139	biimpcd 238	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
141	140	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
142	141	imp 444	. . . . . . . . . . . . . 14 ⊢ (((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
143	142	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
144	138, 143	jca 553	. . . . . . . . . . . 12 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
145	133	simpld 474	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → 𝑉 ∈ V)
146	145	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → 𝑉 ∈ V)
147	133	simprd 478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → 𝐸 ∈ V)
148	147	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → 𝐸 ∈ V)
149		prmnn 15226	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
150	149	nnnn0d 11228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ₀)
151	150	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ₀)
152	146, 148, 151	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀))
153	152	3adant2 1073	. . . . . . . . . . . . . 14 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀))
154	153	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀))
155		isclwwlkn 26297	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
156	154, 155	syl 17	. . . . . . . . . . . 12 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
157	144, 156	mpbird 246	. . . . . . . . . . 11 ⊢ ((((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐 ∈ 𝐶) ∧ (𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
158	157	exp31 628	. . . . . . . . . 10 ⊢ ((((((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
159	158	exp41 636	. . . . . . . . 9 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ 𝐴 ∈ Word dom 𝐸) → (𝐵:(0...(#‘𝐴))⟶𝑉 → ((∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))))))
160	6, 159	mpancom 700	. . . . . . . 8 ⊢ (𝐴 ∈ Word dom 𝐸 → (𝐵:(0...(#‘𝐴))⟶𝑉 → ((∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))))))
161	160	imp31 447	. . . . . . 7 ⊢ (((𝐴 ∈ Word dom 𝐸 ∧ 𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴‘𝑖)) = {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))))
162	5, 161	syl 17	. . . . . 6 ⊢ (𝑐 ∈ (𝑉 ClWalks 𝐸) → ((#‘𝐴) = 𝑁 → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))))
163	162	imp 444	. . . . 5 ⊢ ((𝑐 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝐴) = 𝑁) → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
164	2, 163	sylbi 206	. . . 4 ⊢ (𝑐 ∈ 𝐶 → (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
165	164	pm2.43i 50	. . 3 ⊢ (𝑐 ∈ 𝐶 → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
166	165	impcom 445	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ 𝐶) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
167	166, 7	fmptd 6292	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Fincfn 7841 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 USGrph cusg 25859 ClWalks cclwlk 26275 ClWWalks cclwwlk 26276 ClWWalksN cclwwlkn 26277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-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-usgra 25862 df-wlk 26036 df-clwlk 26278 df-clwwlk 26279 df-clwwlkn 26280

This theorem is referenced by: clwlkfoclwwlk 26372 clwlkf1clwwlk 26377

W3C validator