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Theorem clwlksfclwwlk 26942
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 3192 . . . . 5 (𝑐𝐶 ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁))
3 fusgrusgr 26195 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
4 usgrupgr 26058 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
53, 4syl 17 . . . . . . . . . . 11 (𝐺 ∈ FinUSGraph → 𝐺 ∈ UPGraph )
65adantr 481 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ UPGraph )
7 eqid 2620 . . . . . . . . . . 11 (Vtx‘𝐺) = (Vtx‘𝐺)
8 eqid 2620 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
9 clwlksfclwwlk.1 . . . . . . . . . . 11 𝐴 = (1st𝑐)
10 clwlksfclwwlk.2 . . . . . . . . . . 11 𝐵 = (2nd𝑐)
117, 8, 9, 10upgrclwlkcompim 26658 . . . . . . . . . 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 488 . . . . . . . . 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 13307 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word dom (iEdg‘𝐺) → (#‘𝐴) ∈ ℕ0)
14 clwlksfclwwlk.f . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
159, 10, 1, 14clwlksfclwwlk2wrd 26938 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐𝐶𝐵 ∈ Word (Vtx‘𝐺))
1615ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . 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 13401 . . . . . . . . . . . . . . . . . . . . . . 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 13315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 ∈ Word (Vtx‘𝐺))
20193ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝐵 ∈ Word (Vtx‘𝐺))
21 prmnn 15369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
2221adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ)
23223ad2ant3 1082 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ∈ ℕ)
24 oveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝐴) = 𝑁 → (0...(#‘𝐴)) = (0...𝑁))
2524feq2d 6018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ↔ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)))
2622nnnn0d 11336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℕ0)
27 ffz0hash 13214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 ∈ ℕ0𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
2826, 27sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝐵:(0...𝑁)⟶(Vtx‘𝐺)) → (#‘𝐵) = (𝑁 + 1))
2928ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → (#‘𝐵) = (𝑁 + 1)))
3021nnred 11020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 ∈ ℙ → 𝑁 ∈ ℝ)
3130adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ∈ ℝ)
3231lep1d 10940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (𝑁 + 1))
33 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝐵) = (𝑁 + 1) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
3433adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → (𝑁 ≤ (#‘𝐵) ↔ 𝑁 ≤ (𝑁 + 1)))
3532, 34mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 ∈ ℙ ∧ (#‘𝐵) = (𝑁 + 1)) → 𝑁 ≤ (#‘𝐵))
3635ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℙ → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
3736adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((#‘𝐵) = (𝑁 + 1) → 𝑁 ≤ (#‘𝐵)))
3829, 37syldc 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵:(0...𝑁)⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵)))
3925, 38syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ≤ (#‘𝐵))))
40393imp21 1275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → 𝑁 ≤ (#‘𝐵))
41 swrdn0 13412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (#‘𝐵)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
4220, 23, 40, 41syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅)
43 opeq2 4394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐴) = 𝑁 → ⟨0, (#‘𝐴)⟩ = ⟨0, 𝑁⟩)
4443oveq2d 6651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = (𝐵 substr ⟨0, 𝑁⟩))
4544neeq1d 2850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐴) = 𝑁 → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
46453ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅ ↔ (𝐵 substr ⟨0, 𝑁⟩) ≠ ∅))
4742, 46mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) ∧ (#‘𝐴) = 𝑁 ∧ (𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ)) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)
48473exp 1262 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → ((#‘𝐴) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅)))
4948ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅)))
5049imp 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) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ≠ ∅))
5150adantr 481 . . . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅))
5251imp 445 . . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅)
5318, 52jca 554 . . . . . . . . . . . . . . . . . . . . 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)⟩) ≠ ∅))
54 simp-5r 808 . . . . . . . . . . . . . . . . . . . . . . . . 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))
553adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USGraph )
5654, 55anim12ci 590 . . . . . . . . . . . . . . . . . . . . . . . 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)))
57 simp-5r 808 . . . . . . . . . . . . . . . . . . . . . . . 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))
58 prmuz2 15389 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ‘2))
59 ffz0hash 13214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ ℕ0𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
6059adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐵) = ((#‘𝐴) + 1))
61 eluz2 11678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝐴) ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)))
62 2re 11075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 ∈ ℝ
6362a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → 2 ∈ ℝ)
64 zre 11366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℝ)
65 peano2re 10194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((#‘𝐴) ∈ ℝ → ((#‘𝐴) + 1) ∈ ℝ)
6664, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ ℤ → ((#‘𝐴) + 1) ∈ ℝ)
6763, 64, 663jca 1240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
6867adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ))
69 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ (#‘𝐴))
7064lep1d 10940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ≤ ((#‘𝐴) + 1))
7170adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + 1))
72 letr 10116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) → ((2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1)) → 2 ≤ ((#‘𝐴) + 1)))
7372imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((2 ∈ ℝ ∧ (#‘𝐴) ∈ ℝ ∧ ((#‘𝐴) + 1) ∈ ℝ) ∧ (2 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1))) → 2 ≤ ((#‘𝐴) + 1))
7468, 69, 71, 73syl12anc 1322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
75743adant1 1077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((2 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 2 ≤ (#‘𝐴)) → 2 ≤ ((#‘𝐴) + 1))
7661, 75sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐴) ∈ (ℤ‘2) → 2 ≤ ((#‘𝐴) + 1))
7776a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → ((#‘𝐴) ∈ (ℤ‘2) → 2 ≤ ((#‘𝐴) + 1)))
78 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = (#‘𝐴) → (𝑁 ∈ (ℤ‘2) ↔ (#‘𝐴) ∈ (ℤ‘2)))
7978eqcoms 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) ↔ (#‘𝐴) ∈ (ℤ‘2)))
8079adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ‘2) ↔ (#‘𝐴) ∈ (ℤ‘2)))
81 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((#‘𝐵) = ((#‘𝐴) + 1) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
8281adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (2 ≤ (#‘𝐵) ↔ 2 ≤ ((#‘𝐴) + 1)))
8377, 80, 823imtr4d 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐵) = ((#‘𝐴) + 1) ∧ (#‘𝐴) = 𝑁) → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵)))
8483ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8560, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((#‘𝐴) ∈ ℕ0𝐴 ∈ Word dom (iEdg‘𝐺)) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → ((#‘𝐴) = 𝑁 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8685adantr 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 → (𝑁 ∈ (ℤ‘2) → 2 ≤ (#‘𝐵))))
8786imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
8887adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
8958, 88syl5com 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)))
9089adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
9190impcom 446 . . . . . . . . . . . . . . . . . . . . . . . 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))
92 simp-4r 806 . . . . . . . . . . . . . . . . . . . . . . . 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))))
937, 8usgrf 26031 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
9493anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . 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 26882 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
9694, 95syl3an1 1357 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
97 biid 251 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( lastS ‘𝐵) = (𝐵‘0) ↔ ( lastS ‘𝐵) = (𝐵‘0))
98 edgval 25922 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Edg‘𝐺) = ran (iEdg‘𝐺)
9998eleq2i 2691 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
10099ralbii 2977 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
10198eleq2i 2691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ (Edg‘𝐺) ↔ {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} ∈ ran (iEdg‘𝐺))
10297, 100, 1013anbi123i 1249 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( 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)))
10396, 102sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . 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)))
10456, 57, 91, 92, 103syl121anc 1329 . . . . . . . . . . . . . . . . . . . . . . 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)))
1059, 10, 1, 14clwlksfclwwlk1hash 26940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
106 simp2 1060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝐵 ∈ Word (Vtx‘𝐺))
107 simp1 1059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
108 elfzelz 12327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ (0...(#‘𝐵)) → (#‘𝐴) ∈ ℤ)
109 peano2zm 11405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ∈ ℤ)
110 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ ℤ)
11164lem1d 10942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ ℤ → ((#‘𝐴) − 1) ≤ (#‘𝐴))
112 eluz2 11678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)) ↔ (((#‘𝐴) − 1) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴)))
113109, 110, 111, 112syl3anbrc 1244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ ℤ → (#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)))
114 fzoss2 12480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ (ℤ‘((#‘𝐴) − 1)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
115108, 113, 1143syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝐴) ∈ (0...(#‘𝐵)) → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
116115sselda 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
1171163adant2 1078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → 𝑖 ∈ (0..^(#‘𝐴)))
118 swrd0fv 13421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵𝑖))
119106, 107, 117, 118syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖) = (𝐵𝑖))
120119eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵𝑖) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖))
121 elfzom1elp1fzo 12518 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((#‘𝐴) ∈ ℤ ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
122108, 121sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
1231223adant2 1078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝑖 + 1) ∈ (0..^(#‘𝐴)))
124 swrd0fv 13421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)) = (𝐵‘(𝑖 + 1)))
125124eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (𝑖 + 1) ∈ (0..^(#‘𝐴))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
126106, 107, 123, 125syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ 𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝑖 ∈ (0..^((#‘𝐴) − 1))) → (𝐵‘(𝑖 + 1)) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1)))
127120, 126preq12d 4267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))})
1281273exp 1262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))})))
129105, 15, 128sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝐶 → (𝑖 ∈ (0..^((#‘𝐴) − 1)) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))}))
130129imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑐𝐶𝑖 ∈ (0..^((#‘𝐴) − 1))) → {(𝐵𝑖), (𝐵‘(𝑖 + 1))} = {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))})
131130eleq1d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑐𝐶𝑖 ∈ (0..^((#‘𝐴) − 1))) → ({(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {((𝐵 substr ⟨0, (#‘𝐴)⟩)‘𝑖), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
132131ralbidva 2982 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
133132ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
1349, 10, 1, 14clwlksfclwwlk2sswd 26941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
135134oveq1d 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐𝐶 → ((#‘𝐴) − 1) = ((#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) − 1))
136135ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
137136oveq2d 6651 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
138137raleqdv 3139 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
139133, 138bitrd 268 . . . . . . . . . . . . . . . . . . . . . . . 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)))
140 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ ↔ (#‘𝐴) ∈ ℙ))
141140biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 = (#‘𝐴) → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
142141eqcoms 2628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (#‘𝐴) ∈ ℙ))
143 prmnn 15369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘𝐴) ∈ ℙ → (#‘𝐴) ∈ ℕ)
144 elfz2nn0 12415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ ((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)))
145 1zzd 11393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → 1 ∈ ℤ)
146 nn0z 11385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((#‘𝐵) ∈ ℕ0 → (#‘𝐵) ∈ ℤ)
147146adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐵) ∈ ℤ)
148 nn0z 11385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℤ)
149148adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐴) ∈ ℤ)
150145, 147, 1493jca 1240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
1511503adant3 1079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
152151adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
153 simp3 1061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
154 nnge1 11031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((#‘𝐴) ∈ ℕ → 1 ≤ (#‘𝐴))
155153, 154anim12ci 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵)))
156152, 155jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
157144, 156sylanb 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
158 elfz2 12318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝐴) ∈ (1...(#‘𝐵)) ↔ ((1 ∈ ℤ ∧ (#‘𝐵) ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) ∧ (1 ≤ (#‘𝐴) ∧ (#‘𝐴) ≤ (#‘𝐵))))
159157, 158sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐴) ∈ (0...(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ) → (#‘𝐴) ∈ (1...(#‘𝐵)))
160 swrd0fvlsw 13425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = (𝐵‘((#‘𝐴) − 1)))
161160eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘((#‘𝐴) − 1)) = ( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))
162 swrd0fv0 13422 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0) = (𝐵‘0))
163162eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → (𝐵‘0) = ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0))
164161, 163preq12d 4267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐵 ∈ Word (Vtx‘𝐺) ∧ (#‘𝐴) ∈ (1...(#‘𝐵))) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})
165164expcom 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝐴) ∈ (1...(#‘𝐵)) → (𝐵 ∈ Word (Vtx‘𝐺) → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
166159, 165syl 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)}))
167166ex 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)})))
168167com23 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)})))
169105, 15, 168sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐𝐶 → ((#‘𝐴) ∈ ℕ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
170143, 169syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝐴) ∈ ℙ → (𝑐𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)}))
171142, 170syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐴) = 𝑁 → (𝑁 ∈ ℙ → (𝑐𝐶 → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
172171com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐴) = 𝑁 → (𝑐𝐶 → (𝑁 ∈ ℙ → {(𝐵‘((#‘𝐴) − 1)), (𝐵‘0)} = {( lastS ‘(𝐵 substr ⟨0, (#‘𝐴)⟩)), ((𝐵 substr ⟨0, (#‘𝐴)⟩)‘0)})))
173172adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)})))
174173imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}))
175174com12 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)}))
176175adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 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)}))
177176impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . 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)})
178177eleq1d 2684 . . . . . . . . . . . . . . . . . . . . . . . 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)))
179139, 1783anbi23d 1400 . . . . . . . . . . . . . . . . . . . . . . 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))))
180104, 179mpbid 222 . . . . . . . . . . . . . . . . . . . . . 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)))
181 3simpc 1058 . . . . . . . . . . . . . . . . . . . . . 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)))
182180, 181syl 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)))
183 3anass 1040 . . . . . . . . . . . . . . . . . . . . 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))))
18453, 182, 183sylanbrc 697 . . . . . . . . . . . . . . . . . . . 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)))
185 eqid 2620 . . . . . . . . . . . . . . . . . . . . 21 (Edg‘𝐺) = (Edg‘𝐺)
1867, 185isclwwlks 26861 . . . . . . . . . . . . . . . . . . . 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)))
187184, 186sylibr 224 . . . . . . . . . . . . . . . . . . 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))
188134eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐𝐶 → ((#‘𝐴) = 𝑁 ↔ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
189188biimpcd 239 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝐴) = 𝑁 → (𝑐𝐶 → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁))
190189adantl 482 . . . . . . . . . . . . . . . . . . . . 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))
191190imp 445 . . . . . . . . . . . . . . . . . . . 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)
192191adantr 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)) → (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)
193187, 192jca 554 . . . . . . . . . . . . . . . . . 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))
19422adantl 482 . . . . . . . . . . . . . . . . . . 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)
195 isclwwlksn 26863 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)) = 𝑁)))
196194, 195syl 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)))
197193, 196mpbird 247 . . . . . . . . . . . . . . . . 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))
198197exp31 629 . . . . . . . . . . . . . . . 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))))
199198exp41 637 . . . . . . . . . . . . . . 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)))))))
20013, 199mpancom 702 . . . . . . . . . . . . . 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)))))))
201200imp 445 . . . . . . . . . . . . 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))))))
2022013impib 1260 . . . . . . . . . . . 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)))))
203202com12 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)))))
204203com14 96 . . . . . . . . . 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)))))
205204adantr 481 . . . . . . . . 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)))))
20612, 205mpd 15 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
207206expcom 451 . . . . . . 7 (𝑐 ∈ (ClWalks‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑐𝐶 → ((#‘𝐴) = 𝑁 → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))))
208207com24 95 . . . . . 6 (𝑐 ∈ (ClWalks‘𝐺) → ((#‘𝐴) = 𝑁 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))))
209208imp 445 . . . . 5 ((𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁) → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
2102, 209sylbi 207 . . . 4 (𝑐𝐶 → (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))))
211210pm2.43i 52 . . 3 (𝑐𝐶 → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺)))
212211impcom 446 . 2 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑐𝐶) → (𝐵 substr ⟨0, (#‘𝐴)⟩) ∈ (𝑁 ClWWalksN 𝐺))
213212, 14fmptd 6371 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 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  {crab 2913  cdif 3564  wss 3567  c0 3907  𝒫 cpw 4149  {csn 4168  {cpr 4170  cop 4174   class class class wbr 4644  cmpt 4720  dom cdm 5104  ran crn 5105  wf 5872  1-1wf1 5873  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  cr 9920  0cc0 9921  1c1 9922   + caddc 9924  cle 10060  cmin 10251  cn 11005  2c2 11055  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  ..^cfzo 12449  #chash 13100  Word cword 13274   lastS clsw 13275   substr csubstr 13278  cprime 15366  Vtxcvtx 25855  iEdgciedg 25856  Edgcedg 25920   UPGraph cupgr 25956   USGraph cusgr 26025   FinUSGraph cfusgr 26189  ClWalkscclwlks 26647  ClWWalkscclwwlks 26856   ClWWalksN cclwwlksn 26857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-word 13282  df-lsw 13283  df-substr 13286  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-dvds 14965  df-prm 15367  df-edg 25921  df-uhgr 25934  df-upgr 25958  df-uspgr 26026  df-usgr 26027  df-fusgr 26190  df-wlks 26476  df-clwlks 26648  df-clwwlks 26858  df-clwwlksn 26859
This theorem is referenced by:  clwlksfoclwwlk  26943  clwlksf1clwwlk  26949
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