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Theorem wwlksnext 26856
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 26790 . . 3 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑇 ∈ Word 𝑉))
3 wwlksnext.e . . . . . . . . . . . 12 𝐸 = (Edg‘𝐺)
41, 3wwlknp 26791 . . . . . . . . . . 11 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
5 simp1 1081 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → 𝑇 ∈ Word 𝑉)
6 simprl 809 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑆𝑉)
7 cats1un 13521 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Word 𝑉𝑆𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
85, 6, 7syl2an 493 . . . . . . . . . . . . . . 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 4962 . . . . . . . . . . . . . . . . . . . 20 ⟨(#‘𝑇), 𝑆⟩ ∈ V
109snnz 4340 . . . . . . . . . . . . . . . . . . 19 {⟨(#‘𝑇), 𝑆⟩} ≠ ∅
1110neii 2825 . . . . . . . . . . . . . . . . . 18 ¬ {⟨(#‘𝑇), 𝑆⟩} = ∅
1211intnan 980 . . . . . . . . . . . . . . . . 17 ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅)
13 df-ne 2824 . . . . . . . . . . . . . . . . . 18 ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
14 un00 4044 . . . . . . . . . . . . . . . . . 18 ((𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅) ↔ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
1513, 14xchbinxr 324 . . . . . . . . . . . . . . . . 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 2890 . . . . . . . . . . . . . 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 13418 . . . . . . . . . . . . . . . 16 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
2019ad2antrl 764 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
21 ccatcl 13392 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
225, 20, 21syl2an 493 . . . . . . . . . . . . . 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 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑇 ∈ Word 𝑉)
24 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆𝑉)
2524adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆𝑉)
26 fzossfzop1 12585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
2726sseld 3635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 ∈ ℕ0 → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2827ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
2928imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
30 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((#‘𝑇) = (𝑁 + 1) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
3130eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((#‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3231adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
3332ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑖 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3623, 25, 34, 35syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇𝑖))
3736eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖))
38 fzonn0p1p1 12586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
3938adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
4030adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
4140ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
4239, 41eleqtrrd 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑇)))
43 ccats1val1 13446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4423, 25, 42, 43syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
4544eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)))
4637, 45preq12d 4308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . . . . . . . . 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 3014 . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . 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 1280 . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝑇) = (𝑁 + 1) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
6059ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
61 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
62 ax-1cn 10032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ ℂ
63 pncan 10325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
6461, 62, 63sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
6564ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁)
6660, 65eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = 𝑁)
6766fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑇‘((#‘𝑇) − 1)) = (𝑇𝑁))
68 lsw 13384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑇 ∈ Word 𝑉 → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
6968ad2antrl 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
70 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑇 ∈ Word 𝑉)
71 fzonn0p1 12584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
7271ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
7330eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝑇) = (𝑁 + 1) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
7473ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑇 ∈ Word 𝑉𝑆𝑉𝑁 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7770, 24, 75, 76syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇𝑁))
7867, 69, 773eqtr4d 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
79 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (#‘𝑇) = (𝑁 + 1))
8079eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑁 + 1) = (#‘𝑇))
8180adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (#‘𝑇))
82 ccats1val2 13447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑇 ∈ Word 𝑉𝑆𝑉 ∧ (𝑁 + 1) = (#‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8370, 24, 81, 82syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
8483eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
8578, 84preq12d 4308 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆𝑉𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {( lastS ‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
8685eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)))
9089impcom 445 . . . . . . . . . . . . . . . . . . . . 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 1101 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
95 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
9695fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
9794, 96preq12d 4308 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
9897eleq1d 2715 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
9998ralsng 4250 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
10099ad2antrl 764 . . . . . . . . . . . . . . . . . 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 3827 . . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . 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 11763 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
105 eluzfz2 12387 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
106104, 105sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
107 fzelp1 12431 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
108 fzosplit 12540 . . . . . . . . . . . . . . . . . . . 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 11438 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
111 fzosn 12578 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
112110, 111syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁..^(𝑁 + 1)) = {𝑁})
113112uneq2d 3800 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
114109, 113eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
115114ad2antrl 764 . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . . 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 13393 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1195, 20, 118syl2an 493 . . . . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . . . 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 1085 . . . . . . . . . . . . . . . . . . . 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 13422 . . . . . . . . . . . . . . . . . . . . 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 6708 . . . . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . . . 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 11371 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
127126nn0cnd 11391 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
128 pncan 10325 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
129127, 62, 128sylancl 695 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
130129ad2antrl 764 . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . 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 6706 . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . 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 1261 . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . 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 553 . . . . . . . . . . . 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 449 . . . . . . . . . . 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 451 . . . . . . . . 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 481 . . . . . . 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 444 . . . . . 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 26786 . . . . . . . . . 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 481 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
1471, 3iswwlks 26784 . . . . . . . . 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 731 . . . . . . . 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 480 . . . . . 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 449 . . . 4 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
1531523adant3 1101 . . 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 1281 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cun 3605  c0 3948  {csn 4210  {cpr 4212  cop 4216  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974  1c1 9975   + caddc 9977  cmin 10304  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323   lastS clsw 13324   ++ cconcat 13325  ⟨“cs1 13326  Vtxcvtx 25919  Edgcedg 25984  WWalkscwwlks 26773   WWalksN cwwlksn 26774
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-n0 11331  df-z 11416  df-uz 11726  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
This theorem is referenced by:  wwlksnextbi  26857  wwlksnextsur  26863
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