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Theorem wwlksnext 26664
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
Hypotheses
Ref Expression
wwlksnext.v 𝑉 = (Vtx‘𝐺)
wwlksnext.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksnext ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))

Proof of Theorem wwlksnext
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnext.v . . . 4 𝑉 = (Vtx‘𝐺)
21wwlknbp 26609 . . 3 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑇 ∈ Word 𝑉))
3 wwlksnext.e . . . . . . . . . . . 12 𝐸 = (Edg‘𝐺)
41, 3wwlknp 26610 . . . . . . . . . . 11 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
5 simp1 1059 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → 𝑇 ∈ Word 𝑉)
6 simprl 793 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑆𝑉)
7 cats1un 13416 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Word 𝑉𝑆𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
85, 6, 7syl2an 494 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
9 opex 4895 . . . . . . . . . . . . . . . . . . . 20 ⟨(#‘𝑇), 𝑆⟩ ∈ V
109snnz 4281 . . . . . . . . . . . . . . . . . . 19 {⟨(#‘𝑇), 𝑆⟩} ≠ ∅
1110neii 2792 . . . . . . . . . . . . . . . . . 18 ¬ {⟨(#‘𝑇), 𝑆⟩} = ∅
1211intnan 959 . . . . . . . . . . . . . . . . 17 ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅)
13 df-ne 2791 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
14 un00 3985 . . . . . . . . . . . . . . . . . 18 ((𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅) ↔ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
1513, 14xchbinxr 325 . . . . . . . . . . . . . . . . 17 ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅))
1612, 15mpbir 221 . . . . . . . . . . . . . . . 16 (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅
1716a1i 11 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅)
188, 17eqnetrd 2857 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ≠ ∅)
19 s1cl 13324 . . . . . . . . . . . . . . . 16 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
2019ad2antrl 763 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
21 ccatcl 13301 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
225, 20, 21syl2an 494 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
23 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑇 ∈ Word 𝑉)
24 simpll 789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆𝑉)
2524adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆𝑉)
26 fzossfzop1 12489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
2726sseld 3583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 ∈ ℕ0 → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2827ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2928imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
30 oveq2 6615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝑇) = (𝑁 + 1) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
3130eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3231adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3332ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3429, 33mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(#‘𝑇)))
35 ccats1val1 13344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑖 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3623, 25, 34, 35syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3736eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖))
38 fzonn0p1p1 12490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3938adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
4030adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
4140ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
4239, 41eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑇)))
43 ccats1val1 13344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4423, 25, 42, 43syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4544eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)))
4637, 45preq12d 4248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
4746exp41 637 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑆𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
4847adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
4948impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})))
5049impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑖 ∈ (0..^𝑁) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))
5150imp 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
5251eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
5352ralbidva 2979 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
5453biimpd 219 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
5554ex 450 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)))
5655com23 86 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)))
57563impia 1258 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
5857imp 445 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
59 oveq1 6614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝑇) = (𝑁 + 1) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
6059ad2antll 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
61 nn0cn 11249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
62 ax-1cn 9941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ ℂ
63 pncan 10234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
6461, 62, 63sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
6564ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁)
6660, 65eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = 𝑁)
6766fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑇‘((#‘𝑇) − 1)) = (𝑇𝑁))
68 lsw 13293 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑇 ∈ Word 𝑉 → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
6968ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
70 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑇 ∈ Word 𝑉)
71 fzonn0p1 12488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
7271ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
7330eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝑇) = (𝑁 + 1) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
7473ad2antll 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
7572, 74mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(#‘𝑇)))
76 ccats1val1 13344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑁 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7770, 24, 75, 76syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7867, 69, 773eqtr4d 2665 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
79 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (#‘𝑇) = (𝑁 + 1))
8079eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑁 + 1) = (#‘𝑇))
8180adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (#‘𝑇))
82 ccats1val2 13345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑁 + 1) = (#‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8370, 24, 81, 82syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8483eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
8578, 84preq12d 4248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {( lastS ‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
8685eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
8786biimpcd 239 . . . . . . . . . . . . . . . . . . . . . . . 24 ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
8887exp4c 635 . . . . . . . . . . . . . . . . . . . . . . 23 ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))))
8988impcom 446 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)))
9089impcom 446 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
9190com12 32 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
92913adant3 1079 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
9392imp 445 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)
94 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
95 oveq1 6614 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
9695fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
9794, 96preq12d 4248 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
9897eleq1d 2683 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
9998ralsng 4191 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
10099ad2antrl 763 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
10193, 100mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
102 ralunb 3774 . . . . . . . . . . . . . . . . 17 (∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ (∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ∧ ∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
10358, 101, 102sylanbrc 697 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
104 elnn0uz 11672 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
105 eluzfz2 12294 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
106104, 105sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
107 fzelp1 12338 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
108 fzosplit 12445 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (0...(𝑁 + 1)) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
109106, 107, 1083syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
110 nn0z 11347 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
111 fzosn 12482 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
112110, 111syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁..^(𝑁 + 1)) = {𝑁})
113112uneq2d 3747 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
114109, 113eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
115114ad2antrl 763 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
116115raleqdv 3133 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
117103, 116mpbird 247 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
118 ccatlen 13302 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1195, 20, 118syl2an 494 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
120119oveq1d 6622 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (((#‘𝑇) + (#‘⟨“𝑆”⟩)) − 1))
121 simpl2 1063 . . . . . . . . . . . . . . . . . . . 20 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (#‘𝑇) = (𝑁 + 1))
122 s1len 13327 . . . . . . . . . . . . . . . . . . . . 21 (#‘⟨“𝑆”⟩) = 1
123122a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (#‘⟨“𝑆”⟩) = 1)
124121, 123oveq12d 6625 . . . . . . . . . . . . . . . . . . 19 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
125124oveq1d 6622 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (((#‘𝑇) + (#‘⟨“𝑆”⟩)) − 1) = (((𝑁 + 1) + 1) − 1))
126 peano2nn0 11280 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
127126nn0cnd 11300 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
128 pncan 10234 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
129127, 62, 128sylancl 693 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
130129ad2antrl 763 . . . . . . . . . . . . . . . . . 18 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
131120, 125, 1303eqtrd 2659 . . . . . . . . . . . . . . . . 17 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (𝑁 + 1))
132131oveq2d 6623 . . . . . . . . . . . . . . . 16 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)) = (0..^(𝑁 + 1)))
133132raleqdv 3133 . . . . . . . . . . . . . . 15 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
134117, 133mpbird 247 . . . . . . . . . . . . . 14 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
13518, 22, 1343jca 1240 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
136119, 124eqtrd 2655 . . . . . . . . . . . . 13 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
137135, 136jca 554 . . . . . . . . . . . 12 (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
138137ex 450 . . . . . . . . . . 11 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
1394, 138syl 17 . . . . . . . . . 10 (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
140139expd 452 . . . . . . . . 9 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
141140com12 32 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
142141adantl 482 . . . . . . 7 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))))
143142imp 445 . . . . . 6 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
144 iswwlksn 26606 . . . . . . . . . 10 ((𝑁 + 1) ∈ ℕ0 → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
145126, 144syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
146145adantl 482 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
1471, 3iswwlks 26604 . . . . . . . . 9 ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
148147anbi1i 730 . . . . . . . 8 (((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
149146, 148syl6bb 276 . . . . . . 7 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
150149adantr 481 . . . . . 6 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
151143, 150sylibrd 249 . . . . 5 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
152151ex 450 . . . 4 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
1531523adant3 1079 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑇 ∈ Word 𝑉) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
1542, 153mpcom 38 . 2 (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
1551543impib 1259 1 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cun 3554  c0 3893  {csn 4150  {cpr 4152  cop 4156  cfv 5849  (class class class)co 6607  cc 9881  0cc0 9883  1c1 9884   + caddc 9886  cmin 10213  0cn0 11239  cz 11324  cuz 11634  ...cfz 12271  ..^cfzo 12409  #chash 13060  Word cword 13233   lastS clsw 13234   ++ cconcat 13235  ⟨“cs1 13236  Vtxcvtx 25781  Edgcedg 25846  WWalkscwwlks 26593   WWalksN cwwlksn 26594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-fzo 12410  df-hash 13061  df-word 13241  df-lsw 13242  df-concat 13243  df-s1 13244  df-wwlks 26598  df-wwlksn 26599
This theorem is referenced by:  wwlksnextbi  26665  wwlksnextsur  26671
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