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Theorem wwlksnext 28838

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 28787	. . 3 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑇 ∈ Word 𝑉))
3		wwlksnext.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
4	1, 3	wwlknp 28788	. . . . . . . . . 10 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
5		simp1 1136	. . . . . . . . . . . . . . 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 14609	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}))
8	5, 6, 7	syl2an 596	. . . . . . . . . . . . . 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 5421	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨(♯‘𝑇), 𝑆⟩ ∈ V
10	9	snnz 4737	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨(♯‘𝑇), 𝑆⟩} ≠ ∅
11	10	neii 2945	. . . . . . . . . . . . . . . . 17 ⊢ ¬ {⟨(♯‘𝑇), 𝑆⟩} = ∅
12	11	intnan 487	. . . . . . . . . . . . . . . 16 ⊢ ¬ (𝑇 = ∅ ∧ {⟨(♯‘𝑇), 𝑆⟩} = ∅)
13		df-ne 2944	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(♯‘𝑇), 𝑆⟩}) = ∅)
14		un00 4402	. . . . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . 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 14490	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
20	19	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
21		ccatcl 14462	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
22	5, 20, 21	syl2an 596	. . . . . . . . . . . . 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 13650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
27		oveq2 7365	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑇) = (𝑁 + 1) → (0..^(♯‘𝑇)) = (0..^(𝑁 + 1)))
28	27	eleq2d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(♯‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
29	28	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14514	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
33	23, 31, 32	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
34		fzonn0p1p1 13651	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
35	34	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
36	27	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(♯‘𝑇)))
39		ccats1val1 14514	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (𝑖 + 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
40	23, 38, 39	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝑇) = (𝑁 + 1) → ((♯‘𝑇) − 1) = ((𝑁 + 1) − 1))
53	52	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1)) → ((♯‘𝑇) − 1) = ((𝑁 + 1) − 1))
54		nn0cn 12423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
55		1cnd 11150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
56	54, 55	pncand 11513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
57	56	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑁 + 1) − 1) = 𝑁)
58	53, 57	sylan9eqr 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → ((♯‘𝑇) − 1) = 𝑁)
59	58	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (𝑇‘((♯‘𝑇) − 1)) = (𝑇‘𝑁))
60		lsw 14452	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13649	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14514	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
69	62, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
70	59, 61, 69	3eqtr4d 2786	. . . . . . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (♯‘𝑇))
74		ccats1val2 14515	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (♯‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
75	74	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (♯‘𝑇)) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
76	62, 71, 73, 75	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
77	70, 76	preq12d 4702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ (𝑇 ∈ Word 𝑉 ∧ (♯‘𝑇) = (𝑁 + 1))) → {(lastS‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
78	77	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
86		fvoveq1 7380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
87	85, 86	preq12d 4702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
88	87	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
89	88	ralsng 4634	. . . . . . . . . . . . . . . . . 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 4151	. . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . 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 12808	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
95		eluzfz2 13449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
96	94, 95	sylbi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...𝑁))
97		fzelp1 13493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
98		fzosplit 13605	. . . . . . . . . . . . . . . . . . 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 12524	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
101		fzosn 13643	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁..^(𝑁 + 1)) = {𝑁})
103	102	uneq2d 4123	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
104	99, 103	eqtrd 2776	. . . . . . . . . . . . . . . . 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 3313	. . . . . . . . . . . . . . 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 14463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)))
109	5, 20, 108	syl2an 596	. . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . . 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 14494	. . . . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . 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 12453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
117	116	nn0cnd 12475	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
118	117, 55	pncand 11513	. . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . 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 3313	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 512	. . . . . . . . . . 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 413	. . . . . . . . . 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 416	. . . . . . . 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 408	. . . . . . 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 28783	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
135	1, 3	iswwlks 28781	. . . . . . . . 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 624	. . . . . . . 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 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))))
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 413	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
141	140	3adant3 1132	. . 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 1116	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {csn 4586 {cpr 4588 ⟨cop 4592 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 + caddc 11054 − cmin 11385 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ..^cfzo 13567 ♯chash 14230 Word cword 14402 lastSclsw 14450 ++ cconcat 14458 ⟨“cs1 14483 Vtxcvtx 27947 Edgcedg 27998 WWalkscwwlks 28770 WWalksN cwwlksn 28771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-lsw 14451 df-concat 14459 df-s1 14484 df-wwlks 28775 df-wwlksn 28776

This theorem is referenced by: wwlksnextbi 28839 wwlksnextsurj 28845

W3C validator