Theorem wwlkext2clwwlk 26331

Description: If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

Assertion

Ref	Expression
wwlkext2clwwlk	⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))

Proof of Theorem wwlkext2clwwlk

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknprop 26214	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)))
2		simplrr 797	. . . . . . . 8 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → 𝑊 ∈ Word 𝑉)
3		s1cl 13235	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → ⟨“𝑍”⟩ ∈ Word 𝑉)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ⟨“𝑍”⟩ ∈ Word 𝑉)
5		ccatcl 13212	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
6	2, 4, 5	syl2an 493	. . . . . . 7 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
7	6	adantr 480	. . . . . 6 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉)
8		wwlknimp 26215	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
9		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑊 ∈ Word 𝑉)
10	9	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑊 ∈ Word 𝑉)
11	4	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ⟨“𝑍”⟩ ∈ Word 𝑉)
12		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 ∈ (0..^𝑁) ↔ (𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
13		simp1 1054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ ℕ0)
14		peano2nn 10909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
15	14	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → (𝑁 + 1) ∈ ℕ)
16		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ)
17	16	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ ℝ)
18		nnre 10904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
19	18	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑁 ∈ ℝ)
20		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
21	18, 20	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
22	21	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → (𝑁 + 1) ∈ ℝ)
23		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 < 𝑁)
24	18	ltp1d 10833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 1))
25	24	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑁 < (𝑁 + 1))
26	17, 19, 22, 23, 25	lttrd 10077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 < (𝑁 + 1))
27		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ∈ (0..^(𝑁 + 1)) ↔ (𝑖 ∈ ℕ0 ∧ (𝑁 + 1) ∈ ℕ ∧ 𝑖 < (𝑁 + 1)))
28	13, 15, 26, 27	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁) → 𝑖 ∈ (0..^(𝑁 + 1)))
29	12, 28	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1)))
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
31		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0..^(#‘𝑊)) = (0..^(𝑁 + 1)))
32	31	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (0..^(#‘𝑊)) = (0..^(𝑁 + 1)))
33	32	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
35	34	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(𝑁 + 1))))
36	30, 35	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(#‘𝑊)))
37		ccatval1 13214	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = (𝑊‘𝑖))
38	10, 11, 36, 37	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = (𝑊‘𝑖))
39	38	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊‘𝑖) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖))
40		fzonn0p1p1 12413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
41	40	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
42	31	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((𝑖 + 1) ∈ (0..^(#‘𝑊)) ↔ (𝑖 + 1) ∈ (0..^(𝑁 + 1))))
43	42	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑖 + 1) ∈ (0..^(#‘𝑊)) ↔ (𝑖 + 1) ∈ (0..^(𝑁 + 1))))
44	41, 43	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(#‘𝑊)))
45		ccatval1 13214	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ (𝑖 + 1) ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
46	10, 11, 44, 45	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
47	46	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊‘(𝑖 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)))
48	39, 47	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))})
49	48	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
50	49	ralbidva 2968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
51	50	biimpd 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
52	51	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
53	52	com23 84	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
54	53	3impia 1253	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
55	54	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
56	55	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
57	56	impcom 445	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
58		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
59	58	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
60		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
61	60	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
62		pncan1 10333	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
63	61, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑁 + 1) − 1) = 𝑁)
64	59, 63	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑁 = ((#‘𝑊) − 1))
65	64	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) = ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)))
66	4	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ⟨“𝑍”⟩ ∈ Word 𝑉)
67		nn0p1gt0 11199	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
68	67	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 0 < (𝑁 + 1))
69		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
70	69	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
71	68, 70	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 0 < (#‘𝑊))
72		hashneq0 13016	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
73	9, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
74	71, 73	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑊 ≠ ∅)
75		ccatval1lsw 13221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
76	9, 66, 74, 75	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
77	65, 76	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ( lastS ‘𝑊) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁))
78		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) = (#‘𝑊) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
79	78	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
80	79	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)))
81		ccatws1ls 13262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑍 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)) = 𝑍)
82	81	ad2ant2r 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑊 ++ ⟨“𝑍”⟩)‘(#‘𝑊)) = 𝑍)
83	80, 82	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑍 = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)))
84	77, 83	preq12d 4220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
85	84	3adantl3 1212	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {( lastS ‘𝑊), 𝑍} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
86	85	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
87	86	biimpd 218	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 → {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
88	87	impr 647	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸)
89		simprlr 799	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → 𝑁 ∈ ℕ0)
90		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → ((𝑊 ++ ⟨“𝑍”⟩)‘𝑖) = ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁))
91		oveq1 6556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
92	91	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1)) = ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1)))
93	90, 92	preq12d 4220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑁 → {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} = {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))})
94	93	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑁 → ({((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
95	94	ralsng 4165	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → (∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
96	89, 95	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑊 ++ ⟨“𝑍”⟩)‘𝑁), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑁 + 1))} ∈ ran 𝐸))
97	88, 96	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
98		ralunb 3756	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ (∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ ∀𝑖 ∈ {𝑁} {((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
99	57, 97, 98	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
100		elnn0uz 11601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
101	100	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
102	101	ad2antlr 759	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → 𝑁 ∈ (ℤ≥‘0))
103	102	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → 𝑁 ∈ (ℤ≥‘0))
104		fzosplitsn 12442	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
105	103, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
106	105	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ ((0..^𝑁) ∪ {𝑁}){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
107	99, 106	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
108		simp1 1054	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → 𝑊 ∈ Word 𝑉)
109		simpll 786	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → 𝑍 ∈ 𝑉)
110		ccatws1len 13251	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑍 ∈ 𝑉) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
111	108, 109, 110	syl2an 493	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
112	111	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1) = (((#‘𝑊) + 1) − 1))
113		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) + 1) = ((𝑁 + 1) + 1))
114	113	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑊) = (𝑁 + 1) → (((#‘𝑊) + 1) − 1) = (((𝑁 + 1) + 1) − 1))
115		ax-1cn 9873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
116		addcl 9897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) ∈ ℂ)
117		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 1 ∈ ℂ)
118	116, 117	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
119	60, 115, 118	sylancl 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
120	114, 119	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1))
121	120	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((#‘𝑊) = (𝑁 + 1) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
122	121	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → ((#‘𝑊) = (𝑁 + 1) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
123	122	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑊) = (𝑁 + 1) → (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
124	123	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1)))
125	124	imp 444	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (((#‘𝑊) + 1) − 1) = (𝑁 + 1))
126	112, 125	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1) = (𝑁 + 1))
127	126	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)) = (0..^(𝑁 + 1)))
128	127	raleqdv 3121	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → (∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
129	107, 128	mpbird 246	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ∧ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ {( lastS ‘𝑊), 𝑍} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
130	129	exp32 629	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
131	8, 130	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
132	131	adantl 481	. . . . . . . . 9 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
133	132	imp 444	. . . . . . . 8 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
134	133	adantrd 483	. . . . . . 7 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
135	134	imp 444	. . . . . 6 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
136		wwlknimpb 26232	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))
137		simpll 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) → 𝑊 ∈ Word 𝑉)
138		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑍 ∈ 𝑉)
139		lswccats1 13263	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑍 ∈ 𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)) = 𝑍)
140	137, 138, 139	syl2an 493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)) = 𝑍)
141	140	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑍 = ( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)))
142	137	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 𝑊 ∈ Word 𝑉)
143	4	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ⟨“𝑍”⟩ ∈ Word 𝑉)
144	67	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 1))
145	69	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
146	144, 145	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) → 0 < (#‘𝑊))
147	146	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → 0 < (#‘𝑊))
148		ccatfv0 13220	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0))
149	148	eqcomd 2616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑍”⟩ ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊‘0) = ((𝑊 ++ ⟨“𝑍”⟩)‘0))
150	142, 143, 147, 149	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑊‘0) = ((𝑊 ++ ⟨“𝑍”⟩)‘0))
151	141, 150	preq12d 4220	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ0) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})
152	151	exp31 628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ0 → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})))
153	152	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})))
154	153	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})))
155	154	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})))
156	136, 155	syl5com 31	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})))
157	156	impcom 445	. . . . . . . . . . 11 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)}))
158	157	imp 444	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {𝑍, (𝑊‘0)} = {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)})
159	158	eleq1d 2672	. . . . . . . . 9 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ({𝑍, (𝑊‘0)} ∈ ran 𝐸 ↔ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸))
160	159	biimpcd 238	. . . . . . . 8 ⊢ ({𝑍, (𝑊‘0)} ∈ ran 𝐸 → (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸))
161	160	adantl 481	. . . . . . 7 ⊢ (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸))
162	161	impcom 445	. . . . . 6 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸)
163	7, 135, 162	3jca 1235	. . . . 5 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸))
164	110	ad2ant2r 779	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = ((#‘𝑊) + 1))
165	113	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) + 1) = ((𝑁 + 1) + 1))
166	115	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
167	60, 166, 166	addassd 9941	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
168		1p1e2 11011	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) = 2
169	168	oveq2i 6560	. . . . . . . . . . . . . 14 ⊢ (𝑁 + (1 + 1)) = (𝑁 + 2)
170	167, 169	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
171	170	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 1) + 1) = (𝑁 + 2))
172	165, 171	sylan9eq 2664	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((#‘𝑊) + 1) = (𝑁 + 2))
173	164, 172	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))
174	173	ex 449	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2)))
175	136, 174	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2)))
176	175	adantl 481	. . . . . . 7 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2)))
177	176	imp 444	. . . . . 6 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))
178	177	adantr 480	. . . . 5 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))
179		id 22	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
180		2nn0 11186	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ0
181	180	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ0)
182	179, 181	nn0addcld 11232	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 2) ∈ ℕ0)
183	182	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑁 + 2) ∈ ℕ0)
184	183	anim2i 591	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 + 2) ∈ ℕ0))
185		df-3an 1033	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑁 + 2) ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 + 2) ∈ ℕ0))
186	184, 185	sylibr 223	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑁 + 2) ∈ ℕ0))
187		isclwwlkn 26297	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑁 + 2) ∈ ℕ0) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)) ↔ ((𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
188	186, 187	syl 17	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)) ↔ ((𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
189		isclwwlk 26296	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸)))
190	189	adantr 480	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸)))
191	190	anbi1d 737	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (((𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2)) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
192	188, 191	bitrd 267	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
193	192	ad3antrrr 762	. . . . 5 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → ((𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)) ↔ (((𝑊 ++ ⟨“𝑍”⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 ++ ⟨“𝑍”⟩)) − 1)){((𝑊 ++ ⟨“𝑍”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑍”⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 ++ ⟨“𝑍”⟩)), ((𝑊 ++ ⟨“𝑍”⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑊 ++ ⟨“𝑍”⟩)) = (𝑁 + 2))))
194	163, 178, 193	mpbir2and 959	. . . 4 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) ∧ (𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) ∧ ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
195	194	exp31 628	. . 3 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))))
196	1, 195	mpancom 700	. 2 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))))
197	196	3impib 1254	1 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑍 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤ_≥cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   WWalksN cwwlkn 26206   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  numclwwlk2lem1  26629