Theorem reuccatpfxs1 41434

Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 13480. (Contributed by AV, 10-May-2020.)

Assertion

Ref	Expression
reuccatpfxs1	⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))

Distinct variable groups:   𝑣,𝑉,𝑤,𝑥   𝑣,𝑊,𝑤,𝑥   𝑣,𝑋,𝑤,𝑥

Proof of Theorem reuccatpfxs1

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		s1eq 13380	. . . . 5 ⊢ (𝑣 = 𝑢 → ⟨“𝑣”⟩ = ⟨“𝑢”⟩)
2	1	oveq2d 6666	. . . 4 ⊢ (𝑣 = 𝑢 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
3	2	eleq1d 2686	. . 3 ⊢ (𝑣 = 𝑢 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
4	3	reu8 3402	. 2 ⊢ (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)))
5		simprl 794	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
6		simpl 473	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
7	6	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → 𝑊 ∈ Word 𝑉)
8	7	anim1i 592	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ 𝑋))
9		simplrr 801	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))
10		simp-4r 807	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)))
11		reuccatpfxs1lem 41433	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑊 = (𝑤 prefix (#‘𝑊)) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
12	8, 9, 10, 11	syl3anc 1326	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑊 = (𝑤 prefix (#‘𝑊)) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
13	6	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉))
14	13	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉))
15	14	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉))
16		lswccats1 13411	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)) = 𝑣)
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)) = 𝑣)
18	17	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑣 = ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)))
19	18	s1eqd 13381	. . . . . . . . . . 11 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → ⟨“𝑣”⟩ = ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)
20	19	oveq2d 6666	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩))
21		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑤 = (𝑊 ++ ⟨“𝑣”⟩))
22		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ( lastS ‘𝑤) = ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)))
23	22	s1eqd 13381	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ⟨“( lastS ‘𝑤)”⟩ = ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)
24	23	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩))
25	21, 24	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
26	25	adantl 482	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
27	20, 26	mpbird 247	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩))
28		eleq1 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ Word 𝑉 ↔ 𝑤 ∈ Word 𝑉))
29		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (#‘𝑥) = (#‘𝑤))
30	29	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → ((#‘𝑥) = ((#‘𝑊) + 1) ↔ (#‘𝑤) = ((#‘𝑊) + 1)))
31	28, 30	anbi12d 747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
32	31	rspcva 3307	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
33		3anass 1042	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
34	33	simplbi2com 657	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
35	32, 34	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
36	35	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))))
37	36	com13 88	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑉 → (∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) → (𝑤 ∈ 𝑋 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))))
38	37	imp 445	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑤 ∈ 𝑋 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
39	38	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑤 ∈ 𝑋 → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
40	39	imp 445	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
41	40	adantr 481	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
42		ccats1pfxeqbi 41431	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩)))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩)))
44	27, 43	mpbird 247	. . . . . . . 8 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑊 = (𝑤 prefix (#‘𝑊)))
45	44	ex 450	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑊 = (𝑤 prefix (#‘𝑊))))
46	12, 45	impbid 202	. . . . . 6 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
47	46	ralrimiva 2966	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∀𝑤 ∈ 𝑋 (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩)))
48		reu6i 3397	. . . . 5 ⊢ (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊)))
49	5, 47, 48	syl2anc 693	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊)))
50	49	ex 450	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
51	50	rexlimdva 3031	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))
52	4, 51	syl5bi 232	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294   prefix cpfx 41381

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-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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-pfx 41382

This theorem is referenced by: (None)