MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlksext2clwwlk Structured version   Visualization version   GIF version

Theorem wwlksext2clwwlk 27021
Description: If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Revised by AV, 14-Mar-2022.)
Hypotheses
Ref Expression
clwwlkext2edg.v 𝑉 = (Vtx‘𝐺)
clwwlkext2edg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksext2clwwlk ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍𝑉) → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))

Proof of Theorem wwlksext2clwwlk
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlknbp1 26792 . . 3 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
2 clwwlkext2edg.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
32eqcomi 2660 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
43wrdeqi 13360 . . . . . . . . . . . 12 Word (Vtx‘𝐺) = Word 𝑉
54eleq2i 2722 . . . . . . . . . . 11 (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
65biimpi 206 . . . . . . . . . 10 (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
763ad2ant2 1103 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → 𝑊 ∈ Word 𝑉)
87adantl 481 . . . . . . . 8 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑊 ∈ Word 𝑉)
9 s1cl 13418 . . . . . . . 8 (𝑍𝑉 → ⟨“𝑍”⟩ ∈ Word 𝑉)
10 ccatcl 13392 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
118, 9, 10syl2an 493 . . . . . . 7 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
1211adantr 480 . . . . . 6 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
13 clwwlkext2edg.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
142, 13wwlknp 26791 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
15 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 𝑊 ∈ Word 𝑉)
1615adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑊 ∈ Word 𝑉)
179adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑍𝑉𝑁 ∈ ℕ0) → ⟨“𝑍”⟩ ∈ Word 𝑉)
1817ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ⟨“𝑍”⟩ ∈ Word 𝑉)
19 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 ∈ (0..^𝑁) ↔ (𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
20 simp1 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ ℕ0)
21 peano2nn 11070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
22213ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → (𝑁 + 1) ∈ ℕ)
23 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
24233ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ ℝ)
25 nnre 11065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
26253ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑁 ∈ ℝ)
27 peano2re 10247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
2825, 27syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
29283ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → (𝑁 + 1) ∈ ℝ)
30 simp3 1083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 < 𝑁)
3125ltp1d 10992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 1))
32313ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑁 < (𝑁 + 1))
3324, 26, 29, 30, 32lttrd 10236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 < (𝑁 + 1))
34 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 ∈ (0..^(𝑁 + 1)) ↔ (𝑖 ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ ∧ 𝑖 < (𝑁 + 1)))
3520, 22, 33, 34syl3anbrc 1265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑖 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ (0..^(𝑁 + 1)))
3619, 35sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1)))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
38 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝑊) = (𝑁 + 1) → (0..^(#‘𝑊)) = (0..^(𝑁 + 1)))
3938adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (0..^(#‘𝑊)) = (0..^(𝑁 + 1)))
4039eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
4140ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
4237, 41mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(#‘𝑊)))
43 ccatval1 13395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = (𝑊𝑖))
4416, 18, 42, 43syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = (𝑊𝑖))
4544eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊𝑖) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖))
46 fzonn0p1p1 12586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
4746adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
4838eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘𝑊) = (𝑁 + 1) → ((𝑖 + 1) ∈ (0..^(#‘𝑊)) ↔ (𝑖 + 1) ∈ (0..^(𝑁 + 1))))
4948ad3antlr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑖 + 1) ∈ (0..^(#‘𝑊)) ↔ (𝑖 + 1) ∈ (0..^(𝑁 + 1))))
5047, 49mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑊)))
51 ccatval1 13395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
5216, 18, 50, 51syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
5352eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊‘(𝑖 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)))
5445, 53preq12d 4308 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))})
5554ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))}))
5655expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑍𝑉𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))})))
5756expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ0 → (𝑍𝑉 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))}))))
58573ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑍𝑉 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))}))))
5958imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))})))
6059expdcom 454 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = (𝑁 + 1) → (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → (𝑖 ∈ (0..^𝑁) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))}))))
61603imp1 1302 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))})
6261eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
6362ralbidva 3014 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
6463biimpd 219 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
65643exp 1283 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = (𝑁 + 1) → (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → (∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))))
6665com34 91 . . . . . . . . . . . . . . . 16 (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = (𝑁 + 1) → (∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))))
67663imp1 1302 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
6867adantr 480 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
69 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
7069ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
71 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
7271ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
73 pncan1 10492 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((𝑁 + 1) − 1) = 𝑁)
7570, 74eqtr2d 2686 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 𝑁 = ((#‘𝑊) − 1))
7675fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) = ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)))
7717adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ⟨“𝑍”⟩ ∈ Word 𝑉)
78 nn0p1gt0 11360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
7978ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 0 < (𝑁 + 1))
80 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝑊) = (𝑁 + 1) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
8180ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
8279, 81mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 0 < (#‘𝑊))
83 hashneq0 13193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
8483ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
8582, 84mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 𝑊 ≠ ∅)
86 ccatval1lsw 13402 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
8715, 77, 85, 86syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
8876, 87eqtr2d 2686 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ( lastS ‘𝑊) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁))
89 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 + 1) = (#‘𝑊) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
9089eqcoms 2659 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
9190ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
92 ccatws1ls 13455 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑊 ∈ Word 𝑉𝑍𝑉) → ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)) = 𝑍)
9392ad2ant2r 798 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)) = 𝑍)
9491, 93eqtr2d 2686 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → 𝑍 = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)))
9588, 94preq12d 4308 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍𝑉𝑁 ∈ ℕ0)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
9695expcom 450 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑍𝑉𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))}))
9796expcom 450 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑍𝑉 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})))
98973ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑍𝑉 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})))
9998imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))}))
10099com12 32 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))}))
1011003adant3 1101 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))}))
102101imp 444 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
103102eleq1d 2715 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ 𝐸))
104103biimpa 500 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ 𝐸)
105 simprl1 1126 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → 𝑁 ∈ ℕ0)
106105adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → 𝑁 ∈ ℕ0)
107 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑁 → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁))
108 oveq1 6697 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
109108fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑁 → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)))
110107, 109preq12d 4308 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑁 → {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
111110eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑁 → ({((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ 𝐸))
112111ralsng 4250 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ 𝐸))
113106, 112syl 17 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ 𝐸))
114104, 113mpbird 247 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
115 ralunb 3827 . . . . . . . . . . . . . 14 (∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ (∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ ∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
11668, 114, 115sylanbrc 699 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
117 elnn0uz 11763 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
118105, 117sylib 208 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → 𝑁 ∈ (ℤ‘0))
119118adantr 480 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → 𝑁 ∈ (ℤ‘0))
120 fzosplitsn 12616 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
121119, 120syl 17 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
122121raleqdv 3174 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
123116, 122mpbird 247 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
124 ccatws1len 13437 . . . . . . . . . . . . . . . . . 18 (𝑊 ∈ Word 𝑉 → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
1251243ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
126125ad2antrr 762 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
127126oveq1d 6705 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1) = (((#‘𝑊) + 1) − 1))
128 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) + 1) = ((𝑁 + 1) + 1))
129128oveq1d 6705 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑊) = (𝑁 + 1) → (((#‘𝑊) + 1) − 1) = (((𝑁 + 1) + 1) − 1))
130129adantr 480 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (((#‘𝑊) + 1) − 1) = (((𝑁 + 1) + 1) − 1))
131713ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → 𝑁 ∈ ℂ)
132131adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → 𝑁 ∈ ℂ)
133132adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → 𝑁 ∈ ℂ)
134 1cnd 10094 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → 1 ∈ ℂ)
135133, 134addcld 10097 . . . . . . . . . . . . . . . . . . . 20 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (𝑁 + 1) ∈ ℂ)
136135, 134pncand 10431 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
137130, 136eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1))
138137a1d 25 . . . . . . . . . . . . . . . . 17 (((#‘𝑊) = (𝑁 + 1) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
1391383ad2antl2 1244 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
140139imp 444 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1))
141127, 140eqtrd 2685 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1) = (𝑁 + 1))
142141oveq2d 6706 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)) = (0..^(𝑁 + 1)))
143142raleqdv 3174 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → (∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
144123, 143mpbird 247 . . . . . . . . . . 11 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉)) ∧ {( lastS ‘𝑊), 𝑍} ∈ 𝐸) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
145144exp42 638 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑍𝑉 → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))))
14614, 145syl 17 . . . . . . . . 9 (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑍𝑉 → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))))
147146imp31 447 . . . . . . . 8 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
148147adantrd 483 . . . . . . 7 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸))
149148imp 444 . . . . . 6 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸)
150 lswccats1 13456 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉𝑍𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)) = 𝑍)
1518, 150sylan 487 . . . . . . . . . . . 12 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)) = 𝑍)
152151eqcomd 2657 . . . . . . . . . . 11 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → 𝑍 = ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)))
1537ad2antlr 763 . . . . . . . . . . . 12 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → 𝑊 ∈ Word 𝑉)
1549adantl 481 . . . . . . . . . . . 12 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → ⟨“𝑍”⟩ ∈ Word 𝑉)
155783ad2ant1 1102 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → 0 < (𝑁 + 1))
156803ad2ant3 1104 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
157155, 156mpbird 247 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → 0 < (#‘𝑊))
158157ad2antlr 763 . . . . . . . . . . . 12 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → 0 < (#‘𝑊))
159 ccatfv0 13401 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0))
160159eqcomd 2657 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊‘0) = ((𝑊 ++ ⟨“𝑍”⟩)‘0))
161153, 154, 158, 160syl3anc 1366 . . . . . . . . . . 11 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → (𝑊‘0) = ((𝑊 ++ ⟨“𝑍”⟩)‘0))
162152, 161preq12d 4308 . . . . . . . . . 10 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})
163162eleq1d 2715 . . . . . . . . 9 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → ({𝑍, (𝑊‘0)} ∈ 𝐸 ↔ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸))
164163biimpcd 239 . . . . . . . 8 ({𝑍, (𝑊‘0)} ∈ 𝐸 → (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸))
165164adantl 481 . . . . . . 7 (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸))
166165impcom 445 . . . . . 6 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸)
16712, 149, 1663jca 1261 . . . . 5 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸))
1687, 124syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
169168adantr 480 . . . . . . 7 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
1701283ad2ant3 1104 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) + 1) = ((𝑁 + 1) + 1))
171 1cnd 10094 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
17271, 171, 171addassd 10100 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
173 1p1e2 11172 . . . . . . . . . . . 12 (1 + 1) = 2
174173oveq2i 6701 . . . . . . . . . . 11 (𝑁 + (1 + 1)) = (𝑁 + 2)
175172, 174syl6eq 2701 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
1761753ad2ant1 1102 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑁 + 1) + 1) = (𝑁 + 2))
177170, 176eqtrd 2685 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) + 1) = (𝑁 + 2))
178177adantr 480 . . . . . . 7 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → ((#‘𝑊) + 1) = (𝑁 + 2))
179169, 178eqtrd 2685 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑍𝑉) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))
180179ad4ant23 1325 . . . . 5 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))
181 2nn 11223 . . . . . . . . 9 2 ∈ ℕ
182 nn0nnaddcl 11362 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℕ) → (𝑁 + 2) ∈ ℕ)
183181, 182mpan2 707 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 2) ∈ ℕ)
1841833ad2ant1 1102 . . . . . . 7 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 + 2) ∈ ℕ)
1852, 13isclwwlknx 26998 . . . . . . 7 ((𝑁 + 2) ∈ ℕ → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
186184, 185syl 17 . . . . . 6 ((𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
187186ad3antlr 767 . . . . 5 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
188167, 180, 187mpbir2and 977 . . . 4 ((((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) ∧ 𝑍𝑉) ∧ ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
189188exp31 629 . . 3 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑍𝑉 → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))))
1901, 189mpdan 703 . 2 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑍𝑉 → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))))
191190imp 444 1 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍𝑉) → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cun 3605  c0 3948  {csn 4210  {cpr 4212   class class class wbr 4685  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cmin 10304  cn 11058  2c2 11108  0cn0 11330  cuz 11725  ..^cfzo 12504  #chash 13157  Word cword 13323   lastS clsw 13324   ++ cconcat 13325  ⟨“cs1 13326  Vtxcvtx 25919  Edgcedg 25984   WWalksN cwwlksn 26774   ClWWalksN cclwwlkn 26981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-wwlks 26778  df-wwlksn 26779  df-clwwlk 26950  df-clwwlkn 26983
This theorem is referenced by:  numclwwlk2lem1  27356  numclwwlk2lem1OLD  27363
  Copyright terms: Public domain W3C validator