Theorem 2swrd2eqwrdeq 13742

Description: Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)

Assertion

Ref	Expression
2swrd2eqwrdeq	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Proof of Theorem 2swrd2eqwrdeq

Step	Hyp	Ref	Expression
1		lencl 13356	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2		1z 11445	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
3		nn0z 11438	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
4		zltp1le 11465	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ) → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
5	2, 3, 4	sylancr 696	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
6		1p1e2 11172	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 + 1) = 2)
8	7	breq1d 4695	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → ((1 + 1) ≤ (#‘𝑊) ↔ 2 ≤ (#‘𝑊)))
9	8	biimpd 219	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → ((1 + 1) ≤ (#‘𝑊) → 2 ≤ (#‘𝑊)))
10	5, 9	sylbid 230	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) → 2 ≤ (#‘𝑊)))
11	10	imp 444	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → 2 ≤ (#‘𝑊))
12		2nn0 11347	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13		simpl 472	. . . . . . . 8 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
14		nn0sub 11381	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (2 ≤ (#‘𝑊) ↔ ((#‘𝑊) − 2) ∈ ℕ0))
15	12, 13, 14	sylancr 696	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (2 ≤ (#‘𝑊) ↔ ((#‘𝑊) − 2) ∈ ℕ0))
16	11, 15	mpbid 222	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ ℕ0)
17	3	adantr 480	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℤ)
18		0red 10079	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 0 ∈ ℝ)
19		1red 10093	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 1 ∈ ℝ)
20		nn0re 11339	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ)
21	18, 19, 20	3jca 1261	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
22		0lt1 10588	. . . . . . . . 9 ⊢ 0 < 1
23		lttr 10152	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((0 < 1 ∧ 1 < (#‘𝑊)) → 0 < (#‘𝑊)))
24	23	expd 451	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → (0 < 1 → (1 < (#‘𝑊) → 0 < (#‘𝑊))))
25	21, 22, 24	mpisyl 21	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) → 0 < (#‘𝑊)))
26	25	imp 444	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → 0 < (#‘𝑊))
27		elnnz 11425	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
28	17, 26, 27	sylanbrc 699	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
29		2pos 11150	. . . . . . . 8 ⊢ 0 < 2
30		2re 11128	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → 2 ∈ ℝ)
32	31, 20	ltsubposd 10651	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 < 2 ↔ ((#‘𝑊) − 2) < (#‘𝑊)))
33	29, 32	mpbii 223	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) − 2) < (#‘𝑊))
34	33	adantr 480	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) < (#‘𝑊))
35		elfzo0 12548	. . . . . 6 ⊢ (((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)) ↔ (((#‘𝑊) − 2) ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ ((#‘𝑊) − 2) < (#‘𝑊)))
36	16, 28, 34, 35	syl3anbrc 1265	. . . . 5 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
37	1, 36	sylan 487	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
38	37	3adant2 1100	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
39		2swrdeqwrdeq 13499	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩)))))
40	38, 39	syld3an3 1411	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩)))))
41		swrd2lsw 13741	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
42	41	3adant2 1100	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
43	42	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
44		breq2 4689	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → (1 < (#‘𝑊) ↔ 1 < (#‘𝑈)))
45	44	3anbi3d 1445	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈))))
46		swrd2lsw 13741	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
47	46	3adant1 1099	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
48	45, 47	syl6bi 243	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩))
49	48	impcom 445	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
50		oveq1 6697	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 2) = ((#‘𝑈) − 2))
51		id 22	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
52	50, 51	opeq12d 4441	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 2), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 2), (#‘𝑈)⟩)
53	52	oveq2d 6706	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩))
54	53	eqeq1d 2653	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩))
55	54	adantl 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩))
56	49, 55	mpbird 247	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
57	43, 56	eqeq12d 2666	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) ↔ ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩))
58		fvexd 6241	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊‘((#‘𝑊) − 2)) ∈ V)
59		fvexd 6241	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ V)
60		fvexd 6241	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) ∈ V)
61		fvexd 6241	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ V)
62		s2eq2s1eq 13727	. . . . . . 7 ⊢ ((((𝑊‘((#‘𝑊) − 2)) ∈ V ∧ ( lastS ‘𝑊) ∈ V) ∧ ((𝑈‘((#‘𝑈) − 2)) ∈ V ∧ ( lastS ‘𝑈) ∈ V)) → (⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩ ↔ (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ∧ ⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩)))
63	58, 59, 60, 61, 62	syl22anc 1367	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩ ↔ (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ∧ ⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩)))
64		fvex 6239	. . . . . . . . 9 ⊢ (𝑊‘((#‘𝑊) − 2)) ∈ V
65		s111 13432	. . . . . . . . 9 ⊢ (((𝑊‘((#‘𝑊) − 2)) ∈ V ∧ (𝑈‘((#‘𝑈) − 2)) ∈ V) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2))))
66	64, 60, 65	sylancr 696	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2))))
67		oveq1 6697	. . . . . . . . . . . 12 ⊢ ((#‘𝑈) = (#‘𝑊) → ((#‘𝑈) − 2) = ((#‘𝑊) − 2))
68	67	fveq2d 6233	. . . . . . . . . . 11 ⊢ ((#‘𝑈) = (#‘𝑊) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
69	68	eqcoms 2659	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
70	69	adantl 481	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
71	70	eqeq2d 2661	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2)) ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2))))
72	66, 71	bitrd 268	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2))))
73		fvex 6239	. . . . . . . 8 ⊢ ( lastS ‘𝑊) ∈ V
74		s111 13432	. . . . . . . 8 ⊢ ((( lastS ‘𝑊) ∈ V ∧ ( lastS ‘𝑈) ∈ V) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
75	73, 61, 74	sylancr 696	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
76	72, 75	anbi12d 747	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ∧ ⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩) ↔ ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
77	57, 63, 76	3bitrd 294	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) ↔ ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
78	77	anbi2d 740	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩)) ↔ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
79		3anass 1059	. . . 4 ⊢ (((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)) ↔ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
80	78, 79	syl6bbr 278	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩)) ↔ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
81	80	pm5.32da 674	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩))) ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
82	40, 81	bitrd 268	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  2c2 11108  ℕ₀cn0 11330  ℤcz 11415  ..^cfzo 12504  #chash 13157  Word cword 13323   lastS clsw 13324  ⟨“cs1 13326   substr csubstr 13327  ⟨“cs2 13632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-s2 13639

This theorem is referenced by:  numclwlk1lem2f1  27347