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

Theorem wwlknext 26252
 Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.)
Assertion
Ref Expression
wwlknext ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))

Proof of Theorem wwlknext
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlknprop 26214 . . 3 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑇 ∈ Word 𝑉)))
2 wwlknimp 26215 . . . . . . . . . . . 12 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸))
3 simp1 1054 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) → 𝑇 ∈ Word 𝑉)
4 simprl 790 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑆𝑉)
5 cats1un 13327 . . . . . . . . . . . . . . . . 17 ((𝑇 ∈ Word 𝑉𝑆𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
63, 4, 5syl2an 493 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
7 opex 4859 . . . . . . . . . . . . . . . . . . . . 21 ⟨(#‘𝑇), 𝑆⟩ ∈ V
87snnz 4252 . . . . . . . . . . . . . . . . . . . 20 {⟨(#‘𝑇), 𝑆⟩} ≠ ∅
98neii 2784 . . . . . . . . . . . . . . . . . . 19 ¬ {⟨(#‘𝑇), 𝑆⟩} = ∅
109intnan 951 . . . . . . . . . . . . . . . . . 18 ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅)
11 df-ne 2782 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
12 un00 3963 . . . . . . . . . . . . . . . . . . 19 ((𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅) ↔ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
1311, 12xchbinxr 324 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅))
1410, 13mpbir 220 . . . . . . . . . . . . . . . . 17 (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅
1514a1i 11 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅)
166, 15eqnetrd 2849 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ≠ ∅)
17 s1cl 13235 . . . . . . . . . . . . . . . . 17 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
1817ad2antrl 760 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
19 ccatcl 13212 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
203, 18, 19syl2an 493 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
21 simplrl 796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑇 ∈ Word 𝑉)
22 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆𝑉)
2322adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆𝑉)
24 fzossfzop1 12412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
2524sseld 3567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 ∈ ℕ0 → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2625ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2726imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
28 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘𝑇) = (𝑁 + 1) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
2928eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3029adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3130ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3227, 31mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(#‘𝑇)))
33 ccats1val1 13255 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑖 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3421, 23, 32, 33syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3534eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖))
36 fzonn0p1p1 12413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3828adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
3938ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
4037, 39eleqtrrd 2691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑇)))
41 ccats1val1 13255 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4221, 23, 40, 41syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4342eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)))
4435, 43preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
4544exp41 636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑆𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
4645adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
4746impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})))
4847impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))
4948imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
5049eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
5150ralbidva 2968 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
5251biimpd 218 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
5352ex 449 . . . . . . . . . . . . . . . . . . . . 21 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
5453com23 84 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
55543impia 1253 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
5655imp 444 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
57 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝑇) = (𝑁 + 1) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
5857ad2antll 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
59 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
60 ax-1cn 9873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ ℂ
61 pncan 10166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
6259, 60, 61sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
6362ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁)
6458, 63eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = 𝑁)
6564fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑇‘((#‘𝑇) − 1)) = (𝑇𝑁))
66 lsw 13204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑇 ∈ Word 𝑉 → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
6766ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
68 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑇 ∈ Word 𝑉)
69 fzonn0p1 12411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
7069ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
7128eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝑇) = (𝑁 + 1) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
7271ad2antll 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
7370, 72mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(#‘𝑇)))
74 ccats1val1 13255 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑁 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7568, 22, 73, 74syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7665, 67, 753eqtr4d 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
77 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (#‘𝑇) = (𝑁 + 1))
7877eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑁 + 1) = (#‘𝑇))
7978adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (#‘𝑇))
80 ccats1val2 13256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑁 + 1) = (#‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8168, 22, 79, 80syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8281eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
8376, 82preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {( lastS ‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
8483eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
8584biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
8685exp4c 634 . . . . . . . . . . . . . . . . . . . . . . . 24 ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑆𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))))
8786impcom 445 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸)))
8887impcom 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
8988com12 32 . . . . . . . . . . . . . . . . . . . . 21 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
90893adant3 1074 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
9190imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸)
92 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
93 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
9493fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
9592, 94preq12d 4220 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
9695eleq1d 2672 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
9796ralsng 4165 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
9897ad2antrl 760 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
9991, 98mpbird 246 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
100 ralunb 3756 . . . . . . . . . . . . . . . . . 18 (∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ (∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ ∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
10156, 99, 100sylanbrc 695 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
102 elnn0uz 11601 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
103 eluzfz2 12220 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
104102, 103sylbi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
105 fzelp1 12263 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
106 fzosplit 12370 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (0...(𝑁 + 1)) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
107104, 105, 1063syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
108 nn0z 11277 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
109 fzosn 12405 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
110108, 109syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁..^(𝑁 + 1)) = {𝑁})
111110uneq2d 3729 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
112107, 111eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
113112ad2antrl 760 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
114113raleqdv 3121 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
115101, 114mpbird 246 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
116 ccatlen 13213 . . . . . . . . . . . . . . . . . . . . 21 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1173, 18, 116syl2an 493 . . . . . . . . . . . . . . . . . . . 20 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
118117oveq1d 6564 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (((#‘𝑇) + (#‘⟨“𝑆”⟩)) − 1))
119 simpl2 1058 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (#‘𝑇) = (𝑁 + 1))
120 s1len 13238 . . . . . . . . . . . . . . . . . . . . . 22 (#‘⟨“𝑆”⟩) = 1
121120a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (#‘⟨“𝑆”⟩) = 1)
122119, 121oveq12d 6567 . . . . . . . . . . . . . . . . . . . 20 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
123122oveq1d 6564 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (((#‘𝑇) + (#‘⟨“𝑆”⟩)) − 1) = (((𝑁 + 1) + 1) − 1))
124 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
125124nn0cnd 11230 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
126 pncan 10166 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
127125, 60, 126sylancl 693 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
128127ad2antrl 760 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
129118, 123, 1283eqtrd 2648 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (𝑁 + 1))
130129oveq2d 6565 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)) = (0..^(𝑁 + 1)))
131130raleqdv 3121 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
132115, 131mpbird 246 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
13316, 20, 1323jca 1235 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
134117, 122eqtrd 2644 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
135133, 134jca 553 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸))) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
136135ex 449 . . . . . . . . . . . 12 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
1372, 136syl 17 . . . . . . . . . . 11 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
138137expd 451 . . . . . . . . . 10 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ0 → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
139138com12 32 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
1401393ad2ant3 1077 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
141140imp 444 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
142 iswwlkn 26212 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑁 + 1) ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
143124, 142syl3an3 1353 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
144 iswwlk 26211 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ (𝑉 WWalks 𝐸) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
145144anbi1d 737 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (((𝑇 ++ ⟨“𝑆”⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
1461453adant3 1074 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (((𝑇 ++ ⟨“𝑆”⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
147143, 146bitrd 267 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
148147adantr 480 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
149141, 148sylibrd 248 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
150149ex 449 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
1511503expa 1257 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
152151adantrr 749 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑇 ∈ Word 𝑉)) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
1531, 152mpcom 37 . 2 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
1541533impib 1254 1 ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   WWalks cwwlk 26205   WWalksN cwwlkn 26206 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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-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-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wwlknextbi  26253  wwlkextsur  26259
 Copyright terms: Public domain W3C validator