Theorem reuccatpfxs1 40297

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 13332. (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 13233	. . . . 5 ⊢ (𝑣 = 𝑢 → ⟨“𝑣”⟩ = ⟨“𝑢”⟩)
2	1	oveq2d 6565	. . . 4 ⊢ (𝑣 = 𝑢 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
3	2	eleq1d 2672	. . 3 ⊢ (𝑣 = 𝑢 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
4	3	reu8 3369	. 2 ⊢ (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)))
5		simprl 790	. . . . 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 472	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
7	6	ad2antrr 758	. . . . . . . . 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 590	. . . . . . . 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 797	. . . . . . . 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 803	. . . . . . . 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 40296	. . . . . . . 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 1318	. . . . . . 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 590	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉))
14	13	adantr 480	. . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . 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 13263	. . . . . . . . . . . . . 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 2616	. . . . . . . . . . . 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 13234	. . . . . . . . . . 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 6565	. . . . . . . . . 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 6103	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ( lastS ‘𝑤) = ( lastS ‘(𝑊 ++ ⟨“𝑣”⟩)))
23	22	s1eqd 13234	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ⟨“( lastS ‘𝑤)”⟩ = ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)
24	23	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩))
25	21, 24	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
26	25	adantl 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 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ↔ (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“( lastS ‘(𝑊 ++ ⟨“𝑣”⟩))”⟩)))
27	20, 26	mpbird 246	. . . . . . . . 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 2676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ Word 𝑉 ↔ 𝑤 ∈ Word 𝑉))
29		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (#‘𝑥) = (#‘𝑤))
30	29	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → ((#‘𝑥) = ((#‘𝑊) + 1) ↔ (#‘𝑤) = ((#‘𝑊) + 1)))
31	28, 30	anbi12d 743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
32	31	rspcva 3280	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)))
33		3anass 1035	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1)) ↔ (𝑊 ∈ Word 𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((#‘𝑊) + 1))))
34	33	simplbi2com 655	. . . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . 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 444	. . . . . . . . . . . . 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 758	. . . . . . . . . . . 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 444	. . . . . . . . . . 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 480	. . . . . . . . . 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 40294	. . . . . . . . . 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 246	. . . . . . . 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 449	. . . . . . 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 201	. . . . . 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 2949	. . . . 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 3364	. . . . 5 ⊢ (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 (𝑊 = (𝑤 prefix (#‘𝑊)) ↔ 𝑤 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑤 ∈ 𝑋 𝑊 = (𝑤 prefix (#‘𝑊)))
49	5, 47, 48	syl2anc 691	. . . 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 449	. . 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 3013	. 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 231	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   prefix cpfx 40244

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-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-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-substr 13158  df-pfx 40245

This theorem is referenced by: (None)