Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwlksfclwwlk Structured version   Visualization version   GIF version

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 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (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.
StepHypRef Expression
1 clwlksfclwwlk.c . . . . . 6 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
21rabeq2i 3170 . . . . 5 (𝑐𝐶 ↔ (𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁))
3 fusgrusgr 40541 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
4 usgrupgr 40412 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
53, 4syl 17 . . . . . . . . . . 11 (𝐺 ∈ FinUSGraph → 𝐺 ∈ UPGraph )
65adantr 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 = (1st𝑐)
10 clwlksfclwwlk.2 . . . . . . . . . . 11 𝐵 = (2nd𝑐)
117, 8, 9, 10upgrclwlkcompim 40988 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
126, 11sylan 487 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
13 lencl 13179 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word dom (iEdg‘𝐺) → (#‘𝐴) ∈ ℕ0)
14 clwlksfclwwlk.f . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
159, 10, 1, 14clwlksfclwwlk2wrd 41265 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐𝐶𝐵 ∈ Word (Vtx‘𝐺))
1615ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word (Vtx‘𝐺))
17 swrdcl 13271 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵 ∈ Word (Vtx‘𝐺) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺))
1816, 17syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺))
19 ffz0iswrd 13187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 ∈ Word (Vtx‘𝐺))
20193ad2ant1 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)
2221adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
23223ad2ant3 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))
2524feq2d 5944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ↔ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)))
2622nnnn0d 11228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
27 ffz0hash 13088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 ∈ ℕ0𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
2826, 27sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
2928ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → (#‘𝐵) = (𝑁 + 1)))
3021nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑁 ∈ ℙ → 𝑁 ∈ ℝ)
3130adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ∈ ℝ)
3231lep1d 10834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (𝑁 + 1))
33 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐵) = (𝑁 + 1) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
3433adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
3532, 34mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (#‘𝐵))
3635ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 ∈ ℙ → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
3829, 37syld 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → 𝑁 ≤ (#‘𝐵)))
3938com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵)))
4025, 39syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵))))
41403imp21 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⟩) ≠ ∅)
4320, 23, 41, 42syl3anc 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⟩)
4544oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = (𝐵 substr ⟨0, 𝑁⟩))
4645neeq1d 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐴) = 𝑁 → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
47463ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
4843, 47mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
49483exp 1256 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
5049ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
5150imp 444 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
5251adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
5352imp 444 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
5418, 53jca 553 . . . . . . . . . . . . . . . . . . . . 21 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ Word (Vtx‘𝐺) ∧ (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
55 simp-5r 805 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → 𝐴 ∈ Word dom (iEdg‘𝐺))
563adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USGraph )
5755, 56anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)))
58 simp-5r 805 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺))
59 prmuz2 15246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ‘2))
60 ffz0hash 13088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ ℕ0𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
6160adantlr 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((#‘𝐴) ∈ ℕ0𝐴 ∈ 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
6463a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → 2 ∈ ℝ)
65 zre 11258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℝ)
66 peano2re 10088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((#‘𝐴) ∈ ℝ → ((#‘𝐴) + 1) ∈ ℝ)
6765, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → ((#‘𝐴) + 1) ∈ ℝ)
6864, 65, 673jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
70 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ (#‘𝐴))
7165lep1d 10834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ≤ ((#‘𝐴) + 1))
7271adantr 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)))
7473imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) ∧ (2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1))) → 2 ≤ ((#‘𝐴) + 1))
7569, 70, 72, 74syl12anc 1316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
76753adant1 1072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
7762, 76sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐴) ∈ (ℤ‘2) → 2 ≤ ((#‘𝐴) + 1))
7877a1i 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)))
8079eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) ↔ (#‘𝐴) ∈ (ℤ‘2)))
8180adantl 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)))
8382adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
8478, 81, 833imtr4d 282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵)))
8584ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8661, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8786adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8887imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵)))
8988adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵)))
9059, 89syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → 2 ≤ (#‘𝐵)))
9190adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → 2 ≤ (#‘𝐵)))
9291impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 2 ≤ (#‘𝐵))
93 simp-4r 803 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))
947, 8usgrf 40385 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
9594anim1i 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)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺)))
9795, 96syl3an1 1351 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (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))
9998eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐺 ∈ USGraph → ({(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
10099ralbidv 2969 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐺 ∈ USGraph → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
10198eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐺 ∈ USGraph → ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺)))
102100, 1013anbi23d 1394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ USGraph → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
103102adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
1041033ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)) ↔ (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))))
10597, 104mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺 ∈ USGraph ∧ 𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ 2 ≤ (#‘𝐵)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)))
10657, 58, 92, 93, 105syl121anc 1323 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺)))
1079, 10, 1, 14clwlksfclwwlk1hash 41267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐𝐶 → (#‘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)
11365lem1d 10836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ≤ (#‘𝐴))
114 eluz2 11569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)) ↔ (((#‘𝐴) − 1) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴)))
115111, 112, 113, 114syl3anbrc 1239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)))
116 fzoss2 12365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
117110, 115, 1163syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝐴) ∈ (0...(#‘𝐵)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
118117sselda 3568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
1191183adant2 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) = (𝐵𝑖))
121108, 109, 119, 120syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵𝑖))
122121eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵𝑖) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖))
123 elfzom1elp1fzo 12402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((#‘𝐴) ∈ ℤ ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
124110, 123sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
1251243adant2 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)))
127126eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
128108, 109, 125, 127syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
129122, 128preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
1301293exp 1256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) ∈ (0...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})))
131107, 15, 130sylc 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝐶 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))}))
132131imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐𝐶𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
133132eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐𝐶𝑖 ∈ (0..^((#‘𝐴) − 1))) → ({(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
134133ralbidva 2968 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐𝐶 → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
135134ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
1369, 10, 1, 14clwlksfclwwlk2sswd 41268 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
137136oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐𝐶 → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
138137ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
139138oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (0..^((#‘𝐴) − 1)) = (0..^((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1)))
140139raleqdv 3121 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (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)))
141135, 140bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (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))
143142biimpd 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
144143eqcoms 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)
149148adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐵) ∈ ℤ)
150 nn0z 11277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℤ)
151150adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐴) ∈ ℤ)
152147, 149, 1513jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
1531523adant3 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
154153adantr 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))
157155, 156anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵)))
158154, 157jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
159146, 158sylanb 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))))
161159, 160sylibr 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)))
163162eqcomd 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))
165164eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘0) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0))
166163, 165preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
167166expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ (1...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
168161, 167syl 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)}))
169168ex 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)})))
170169com23 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)})))
171107, 15, 170sylc 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐𝐶 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
172145, 171syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝐴) ∈ ℙ → (𝑐𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
173144, 172syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (𝑐𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
174173com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝑐𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
175174adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
176175imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
177176com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℙ → (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
178177adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
179178impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
180179eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)} ∈ (Edg‘𝐺)))
181141, 1803anbi23d 1394 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((( lastS ‘𝐵) = (𝐵‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (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))))
182106, 181mpbid 221 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (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)))
184182, 183syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (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))))
18654, 184, 185sylanbrc 695 . . . . . . . . . . . . . . . . . . . 20 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (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)
1887, 187isclwwlks 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)))
189186, 188sylibr 223 . . . . . . . . . . . . . . . . . . 19 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺))
190136eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐𝐶 → ((#‘𝐴) = 𝑁 ↔ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
191190biimpcd 238 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝐴) = 𝑁 → (𝑐𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
192191adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
193192imp 444 . . . . . . . . . . . . . . . . . . . 20 (((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
194193adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
195189, 194jca 553 . . . . . . . . . . . . . . . . . 18 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
19622adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (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)))
198196, 197syl 17 . . . . . . . . . . . . . . . . . 18 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalkS‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
199195, 198mpbird 246 . . . . . . . . . . . . . . . . 17 ((((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) ∧ 𝑐𝐶) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))
200199exp31 628 . . . . . . . . . . . . . . . 16 ((((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) ∧ (#‘𝐴) = 𝑁) → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
201200exp41 636 . . . . . . . . . . . . . . 15 (((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))))
20213, 201mpancom 700 . . . . . . . . . . . . . 14 (𝐴 ∈ Word dom (iEdg‘𝐺) → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))))
203202imp 444 . . . . . . . . . . . . 13 ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))))
2042033impib 1254 . . . . . . . . . . . 12 (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
205204com12 32 . . . . . . . . . . 11 ((#‘𝐴) = 𝑁 → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
206205com14 94 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
207206adantr 480 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ ∀𝑖 ∈ (0..^(#‘𝐴))((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
20812, 207mpd 15 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalkS‘𝐺)) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
209208expcom 450 . . . . . . 7 (𝑐 ∈ (ClWalkS‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
210209com24 93 . . . . . 6 (𝑐 ∈ (ClWalkS‘𝐺) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))))
211210imp 444 . . . . 5 ((𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁) → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
2122, 211sylbi 206 . . . 4 (𝑐𝐶 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))))
213212pm2.43i 50 . . 3 (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺)))
214213impcom 445 . 2 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐𝐶) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalkSN 𝐺))
215214, 14fmptd 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  1st c1st 7057  2nd 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
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