Theorem wwlksnext 26788

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 26733	. . 3 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑇 ∈ Word 𝑉))
3		wwlksnext.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (Edg‘𝐺)
4	1, 3	wwlknp 26734	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸))
5		simp1 1061	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸) → 𝑇 ∈ Word 𝑉)
6		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑆 ∈ 𝑉)
7		cats1un 13475	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) = (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}))
8	5, 6, 7	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ (((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 4932	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨(#‘𝑇), 𝑆⟩ ∈ V
10	9	snnz 4309	. . . . . . . . . . . . . . . . . . 19 ⊢ {⟨(#‘𝑇), 𝑆⟩} ≠ ∅
11	10	neii 2796	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ {⟨(#‘𝑇), 𝑆⟩} = ∅
12	11	intnan 960	. . . . . . . . . . . . . . . . 17 ⊢ ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅)
13		df-ne 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
14		un00 4011	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅) ↔ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) = ∅)
15	13, 14	xchbinxr 325	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅ ↔ ¬ (𝑇 = ∅ ∧ {⟨(#‘𝑇), 𝑆⟩} = ∅))
16	12, 15	mpbir 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∪ {⟨(#‘𝑇), 𝑆⟩}) ≠ ∅
17	16	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((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 2861	. . . . . . . . . . . . . 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”⟩) ≠ ∅)
19		s1cl 13382	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
20	19	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ⟨“𝑆”⟩ ∈ Word 𝑉)
21		ccatcl 13359	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉)
22	5, 20, 21	syl2an 494	. . . . . . . . . . . . . 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”⟩) ∈ Word 𝑉)
23		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑇 ∈ Word 𝑉)
24		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆 ∈ 𝑉)
25	24	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ 𝑉)
26		fzossfzop1 12545	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
27	26	sseld 3602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ0 → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
28	27	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
29	28	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
30		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((#‘𝑇) = (𝑁 + 1) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
31	30	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((#‘𝑇) = (𝑁 + 1) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
32	31	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
33	32	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(#‘𝑇)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
34	29, 33	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(#‘𝑇)))
35		ccats1val1 13403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝑖 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
36	23, 25, 34, 35	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = (𝑇‘𝑖))
37	36	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖))
38		fzonn0p1p1 12546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
39	38	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
40	30	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
41	40	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (0..^(#‘𝑇)) = (0..^(𝑁 + 1)))
42	39, 41	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑇)))
43		ccats1val1 13403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
44	23, 25, 42, 43	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = (𝑇‘(𝑖 + 1)))
45	44	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑇‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)))
46	37, 45	preq12d 4276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
47	46	exp41 638	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑆 ∈ 𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
48	47	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))))
49	48	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑖 ∈ (0..^𝑁) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})))
50	49	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) → (𝑖 ∈ (0..^𝑁) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))}))
51	50	imp 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))})
52	51	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸))) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑇‘𝑖), (𝑇‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
53	52	ralbidva 2985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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))
54	53	biimpd 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((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))
55	54	ex 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)))
56	55	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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)))
57	56	3impia 1261	. . . . . . . . . . . . . . . . . 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))
58	57	imp 445	. . . . . . . . . . . . . . . . 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)
59		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝑇) = (𝑁 + 1) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
60	59	ad2antll 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = ((𝑁 + 1) − 1))
61		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
62		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ ℂ
63		pncan 10287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
64	61, 62, 63	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
65	64	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁)
66	60, 65	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((#‘𝑇) − 1) = 𝑁)
67	66	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑇‘((#‘𝑇) − 1)) = (𝑇‘𝑁))
68		lsw 13351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑇 ∈ Word 𝑉 → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
69	68	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ( lastS ‘𝑇) = (𝑇‘((#‘𝑇) − 1)))
70		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑇 ∈ Word 𝑉)
71		fzonn0p1 12544	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
72	71	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
73	30	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝑇) = (𝑁 + 1) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
74	73	ad2antll 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 ∈ (0..^(#‘𝑇)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
75	72, 74	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑁 ∈ (0..^(#‘𝑇)))
76		ccats1val1 13403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ (0..^(#‘𝑇))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
77	70, 24, 75, 76	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁) = (𝑇‘𝑁))
78	67, 69, 77	3eqtr4d 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ 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))
80	79	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → (𝑁 + 1) = (#‘𝑇))
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → (𝑁 + 1) = (#‘𝑇))
82		ccats1val2 13404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ (𝑁 + 1) = (#‘𝑇)) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
83	70, 24, 81, 82	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)) = 𝑆)
84	83	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → 𝑆 = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
85	78, 84	preq12d 4276	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {( lastS ‘𝑇), 𝑆} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
86	85	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
87	86	biimpcd 239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1))) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
88	87	exp4c 636	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))))
89	88	impcom 446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑁 ∈ ℕ0 → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)))
90	89	impcom 446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
91	90	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
92	91	3adant3 1081	. . . . . . . . . . . . . . . . . . 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 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
93	92	imp 445	. . . . . . . . . . . . . . . . . 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), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸)
94		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘𝑖) = ((𝑇 ++ ⟨“𝑆”⟩)‘𝑁))
95		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
96	95	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑁 → ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1)))
97	94, 96	preq12d 4276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} = {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))})
98	97	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → ({((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
99	98	ralsng 4218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
100	99	ad2antrl 764	. . . . . . . . . . . . . . . . . 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 ∈ {𝑁} {((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ {((𝑇 ++ ⟨“𝑆”⟩)‘𝑁), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑁 + 1))} ∈ 𝐸))
101	93, 100	mpbird 247	. . . . . . . . . . . . . . . . 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)
102		ralunb 3794	. . . . . . . . . . . . . . . . 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))
103	58, 101, 102	sylanbrc 698	. . . . . . . . . . . . . . . 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) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸)
104		elnn0uz 11725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
105		eluzfz2 12349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘0) → 𝑁 ∈ (0...𝑁))
106	104, 105	sylbi 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁))
107		fzelp1 12393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ (0...(𝑁 + 1)))
108		fzosplit 12501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0...(𝑁 + 1)) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
109	106, 107, 108	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))))
110		nn0z 11400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
111		fzosn 12538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℤ → (𝑁..^(𝑁 + 1)) = {𝑁})
112	110, 111	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁..^(𝑁 + 1)) = {𝑁})
113	112	uneq2d 3767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → ((0..^𝑁) ∪ (𝑁..^(𝑁 + 1))) = ((0..^𝑁) ∪ {𝑁}))
114	109, 113	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
115	114	ad2antrl 764	. . . . . . . . . . . . . . . . 17 ⊢ (((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}))
116	115	raleqdv 3144	. . . . . . . . . . . . . . . 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 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
117	103, 116	mpbird 247	. . . . . . . . . . . . . . 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)
118		ccatlen 13360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
119	5, 20, 118	syl2an 494	. . . . . . . . . . . . . . . . . . 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 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
120	119	oveq1d 6665	. . . . . . . . . . . . . . . . . 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”⟩)) − 1) = (((#‘𝑇) + (#‘⟨“𝑆”⟩)) − 1))
121		simpl2 1065	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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))
122		s1len 13385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (#‘⟨“𝑆”⟩) = 1
123	122	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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)
124	121, 123	oveq12d 6668	. . . . . . . . . . . . . . . . . . 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) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))
125	124	oveq1d 6665	. . . . . . . . . . . . . . . . . 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”⟩)) − 1) = (((𝑁 + 1) + 1) − 1))
126		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
127	126	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
128		pncan 10287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 + 1) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
129	127, 62, 128	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
130	129	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
131	120, 125, 130	3eqtrd 2660	. . . . . . . . . . . . . . . . 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))
132	131	oveq2d 6666	. . . . . . . . . . . . . . . 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..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)) = (0..^(𝑁 + 1)))
133	132	raleqdv 3144	. . . . . . . . . . . . . . 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..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
134	117, 133	mpbird 247	. . . . . . . . . . . . . 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)
135	18, 22, 134	3jca 1242	. . . . . . . . . . . . 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”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
136	119, 124	eqtrd 2656	. . . . . . . . . . . . 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”⟩)) = ((𝑁 + 1) + 1))
137	135, 136	jca 554	. . . . . . . . . . . 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) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
138	137	ex 450	. . . . . . . . . . 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))))
139	4, 138	syl 17	. . . . . . . . . 10 ⊢ (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))))
140	139	expd 452	. . . . . . . . 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)))))
141	140	com12 32	. . . . . . . 8 ⊢ (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)))))
142	141	adantl 482	. . . . . . 7 ⊢ ((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)))))
143	142	imp 445	. . . . . 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))))
144		iswwlksn 26730	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
145	126, 144	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
146	145	adantl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ∧ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1))))
147	1, 3	iswwlks 26728	. . . . . . . . 9 ⊢ ((𝑇 ++ ⟨“𝑆”⟩) ∈ (WWalks‘𝐺) ↔ ((𝑇 ++ ⟨“𝑆”⟩) ≠ ∅ ∧ (𝑇 ++ ⟨“𝑆”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑇 ++ ⟨“𝑆”⟩)) − 1)){((𝑇 ++ ⟨“𝑆”⟩)‘𝑖), ((𝑇 ++ ⟨“𝑆”⟩)‘(𝑖 + 1))} ∈ 𝐸))
148	147	anbi1i 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)))
149	146, 148	syl6bb 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))))
150	149	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))))
151	143, 150	sylibrd 249	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑇 ∈ (𝑁 WWalksN 𝐺)) → ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
152	151	ex 450	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
153	152	3adant3 1081	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))))
154	2, 153	mpcom 38	. 2 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ((𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
155	154	3impib 1262	1 ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   − cmin 10266  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294  Vtxcvtx 25874  Edgcedg 25939  WWalkscwwlks 26717   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnextbi  26789  wwlksnextsur  26795