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Theorem wwlksnext 29748

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) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))

Proof of Theorem wwlksnext Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnext.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wwlknbp 29697	. . 3 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑇 ∈ Word 𝑉))
3		wwlksnext.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
4	1, 3	wwlknp 29698	. . . . . . . . . 10 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
5		simp1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → 𝑇 ∈ Word 𝑉)
6		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → 𝑆 ∈ 𝑉)
7		cats1un 14703	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}))
8	5, 6, 7	syl2an 594	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}))
9		opex 5460	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨(♯‘𝑇), 𝑆⟩ ∈ V
10	9	snnz 4776	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨(♯‘𝑇), 𝑆⟩} ≠ ∅
11	10	neii 2932	. . . . . . . . . . . . . . . . 17 ⊢ ¬ {⟨(♯‘𝑇), 𝑆⟩} = ∅
12	11	intnan 485	. . . . . . . . . . . . . . . 16 ⊢ ¬ (𝑇 = ∅ ∧ {⟨(♯‘𝑇), 𝑆⟩} = ∅)
13		df-ne 2931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) = ∅)
14		un00 4438	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 = ∅ ∧ {⟨(♯‘𝑇), 𝑆⟩} = ∅) ↔ (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) = ∅)
15	13, 14	xchbinxr 334	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 = ∅ ∧ {⟨(♯‘𝑇), 𝑆⟩} = ∅))
16	12, 15	mpbir 230	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) ≠ ∅
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) ≠ ∅)
18	8, 17	eqnetrd 2998	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ≠ ∅)
19		s1cl 14584	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
20	19	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
21		ccatcl 14556	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
22	5, 20, 21	syl2an 594	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
23		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑇 ∈ Word 𝑉)
24		fzossfzop1 13742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ℕ₀ → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
25	24	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
26	25	sselda 3972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
27		oveq2 7424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑇) = (𝑁 + 1) → (0..^(♯‘𝑇)) = (0..^(𝑁 + 1)))
28	27	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(♯‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
29	28	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^(♯‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
30	29	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(♯‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
31	26, 30	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(♯‘𝑇)))
32		ccats1val1 14608	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
33	23, 31, 32	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
34		fzonn0p1p1 13743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
35	34	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
36	27	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → (0..^(♯‘𝑇)) = (0..^(𝑁 + 1)))
37	36	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (0..^(♯‘𝑇)) = (0..^(𝑁 + 1)))
38	35, 37	eleqtrrd 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(♯‘𝑇)))
39		ccats1val1 14608	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (𝑖 + 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
40	23, 38, 39	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
41	33, 40	preq12d 4741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))})
42	41	exp31 418	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) → ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))})))
43	42	adantrr 715	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))})))
44	43	impcom 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (𝑖 ∈ (0..^𝑁) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))}))
45	44	imp 405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))})
46	45	eleq1d 2810	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
47	46	ralbidva 3166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
48	47	exbiri 809	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)))
49	48	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 → ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)))
50	49	3impia 1114	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
51	50	imp 405	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^𝑁){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
52		oveq1 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝑇) = (𝑁 + 1) → ((♯‘𝑇) − 1) = ((𝑁 + 1) − 1))
53	52	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → ((♯‘𝑇) − 1) = ((𝑁 + 1) − 1))
54		nn0cn 12512	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
55		1cnd 11239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
56	54, 55	pncand 11602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
57	56	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑁 + 1) − 1) = 𝑁)
58	53, 57	sylan9eqr 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → ((♯‘𝑇) − 1) = 𝑁)
59	58	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (𝑇‘((♯‘𝑇) − 1)) = (𝑇‘𝑁))
60		lsw 14546	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑇 ∈ Word 𝑉 → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
61	60	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
62		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑇 ∈ Word 𝑉)
63		fzonn0p1 13741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(𝑁 + 1)))
64	63	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
65	27	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝑇) = (𝑁 + 1) → (𝑁 ∈ (0..^(♯‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
66	65	ad2antll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (𝑁 ∈ (0..^(♯‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
67	64, 66	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(♯‘𝑇)))
68		ccats1val1 14608	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
69	62, 67, 68	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
70	59, 61, 69	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (lastS‘𝑇) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
71		simpll 765	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑆 ∈ 𝑉)
72		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (♯‘𝑇) = (𝑁 + 1))
73	72	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (♯‘𝑇))
74		ccats1val2 14609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (♯‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
75	74	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (♯‘𝑇)) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
76	62, 71, 73, 75	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
77	70, 76	preq12d 4741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → {(lastS‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
78	77	eleq1d 2810	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
79	78	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
80	79	exp4c 431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑁 ∈ ℕ₀ → ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))))
81	80	impcom 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ₀ → ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)))
82	81	impcom 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
83	82	impcom 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)
84	83	3adantl3 1165	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)
85		fveq2 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
86		fvoveq1 7439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
87	85, 86	preq12d 4741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
88	87	eleq1d 2810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
89	88	ralsng 4673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
90	89	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
91	84, 90	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
92		ralunb 4185	. . . . . . . . . . . . . . . 16 ⊢ (∀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))
93	51, 91, 92	sylanbrc 581	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
94		elnn0uz 12897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
95		eluzfz2 13541	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
96	94, 95	sylbi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...𝑁))
97		fzelp1 13585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
98		fzosplit 13697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (0...(𝑁 + 1)) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
99	96, 97, 98	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
100		nn0z 12613	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
101		fzosn 13735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁..^(𝑁 + 1)) = {𝑁})
103	102	uneq2d 4156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
104	99, 103	eqtrd 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
105	104	ad2antrl 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
106	105	raleqdv 3315	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(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))
107	93, 106	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
108		ccatlen 14557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)))
109	5, 20, 108	syl2an 594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)))
110	109	oveq1d 7431	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) − 1))
111		simpl2 1189	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (♯‘𝑇) = (𝑁 + 1))
112		s1len 14588	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘⟨“𝑆”⟩) = 1
113	112	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (♯‘⟨“𝑆”⟩) = 1)
114	111, 113	oveq12d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
115	114	oveq1d 7431	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) − 1) = (((𝑁 + 1) + 1) − 1))
116		peano2nn0 12542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
117	116	nn0cnd 12564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
118	117, 55	pncand 11602	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
119	118	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
120	110, 115, 119	3eqtrd 2769	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1) = (𝑁 + 1))
121	120	oveq2d 7432	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (0..^((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)) = (0..^(𝑁 + 1)))
122	121	raleqdv 3315	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(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))
123	107, 122	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ∀𝑖 ∈ (0..^((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
124	18, 22, 123	3jca 1125	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
125	109, 114	eqtrd 2765	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸))) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
126	124, 125	jca 510	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) ∧ (𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(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)))
127	126	ex 411	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(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))))
128	4, 127	syl 17	. . . . . . . . 9 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(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))))
129	128	expd 414	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ₀ → ((𝑆 ∈ 𝑉 ∧ {(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)))))
130	129	impcom 406	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆 ∈ 𝑉 ∧ {(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))))
131	130	adantll 712	. . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆 ∈ 𝑉 ∧ {(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))))
132		iswwlksn 29693	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
133	116, 132	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
134	133	adantl 480	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
135	1, 3	iswwlks 29691	. . . . . . . . 9 ⊢ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
136	135	anbi1i 622	. . . . . . . 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)))
137	134, 136	bitrdi 286	. . . . . . 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))))
138	137	adantr 479	. . . . . 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))))
139	131, 138	sylibrd 258	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
140	139	ex 411	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
141	140	3adant3 1129	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
142	2, 141	mpcom 38	. 2 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
143	142	3impib 1113	1 ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 Vcvv 3463 ∪ cun 3937 ⊆ wss 3939 ∅c0 4318 {csn 4624 {cpr 4626 ⟨cop 4630 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 − cmin 11474 ℕ₀cn0 12502 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13516 ..^cfzo 13659 ♯chash 14321 Word cword 14496 lastSclsw 14544 ++ cconcat 14552 ⟨“cs1 14577 Vtxcvtx 28853 Edgcedg 28904 WWalkscwwlks 29680 WWalksN cwwlksn 29681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-wwlks 29685 df-wwlksn 29686

This theorem is referenced by: wwlksnextbi 29749 wwlksnextsurj 29755

W3C validator