Theorem 2swrd2eqwrdeq 13544

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 13179	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2		1z 11284	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
3		nn0z 11277	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
4		zltp1le 11304	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ) → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
5	2, 3, 4	sylancr 694	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
6		1p1e2 11011	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 + 1) = 2)
8	7	breq1d 4593	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → ((1 + 1) ≤ (#‘𝑊) ↔ 2 ≤ (#‘𝑊)))
9	8	biimpd 218	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → ((1 + 1) ≤ (#‘𝑊) → 2 ≤ (#‘𝑊)))
10	5, 9	sylbid 229	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) → 2 ≤ (#‘𝑊)))
11	10	imp 444	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → 2 ≤ (#‘𝑊))
12		2nn0 11186	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13		simpl 472	. . . . . . . 8 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
14		nn0sub 11220	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (2 ≤ (#‘𝑊) ↔ ((#‘𝑊) − 2) ∈ ℕ0))
15	12, 13, 14	sylancr 694	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (2 ≤ (#‘𝑊) ↔ ((#‘𝑊) − 2) ∈ ℕ0))
16	11, 15	mpbid 221	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ ℕ0)
17	3	adantr 480	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℤ)
18		0red 9920	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 0 ∈ ℝ)
19		1red 9934	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 1 ∈ ℝ)
20		nn0re 11178	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ)
21	18, 19, 20	3jca 1235	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
22		0lt1 10429	. . . . . . . . 9 ⊢ 0 < 1
23		lttr 9993	. . . . . . . . . 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 11264	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
28	17, 26, 27	sylanbrc 695	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
29		2pos 10989	. . . . . . . 8 ⊢ 0 < 2
30		2re 10967	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → 2 ∈ ℝ)
32	31, 20	ltsubposd 10492	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 < 2 ↔ ((#‘𝑊) − 2) < (#‘𝑊)))
33	29, 32	mpbii 222	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) − 2) < (#‘𝑊))
34	33	adantr 480	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) < (#‘𝑊))
35		elfzo0 12376	. . . . . 6 ⊢ (((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)) ↔ (((#‘𝑊) − 2) ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ ((#‘𝑊) − 2) < (#‘𝑊)))
36	16, 28, 34, 35	syl3anbrc 1239	. . . . 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 1073	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
39		2swrdeqwrdeq 13305	. . 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 1363	. 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 13543	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
42	41	3adant2 1073	. . . . . . . 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 4587	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → (1 < (#‘𝑊) ↔ 1 < (#‘𝑈)))
45	44	3anbi3d 1397	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈))))
46		swrd2lsw 13543	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
47	46	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
48	45, 47	syl6bi 242	. . . . . . . . 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 6556	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 2) = ((#‘𝑈) − 2))
51		id 22	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
52	50, 51	opeq12d 4348	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 2), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 2), (#‘𝑈)⟩)
53	52	oveq2d 6565	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩))
54	53	eqeq1d 2612	. . . . . . . . 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 246	. . . . . . 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 2625	. . . . . 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		fvex 6113	. . . . . . . 8 ⊢ (𝑊‘((#‘𝑊) − 2)) ∈ V
59	58	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊‘((#‘𝑊) − 2)) ∈ V)
60		fvex 6113	. . . . . . . 8 ⊢ ( lastS ‘𝑊) ∈ V
61	60	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ V)
62		fvex 6113	. . . . . . . 8 ⊢ (𝑈‘((#‘𝑈) − 2)) ∈ V
63	62	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) ∈ V)
64		fvex 6113	. . . . . . . 8 ⊢ ( lastS ‘𝑈) ∈ V
65	64	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ V)
66		s2eq2s1eq 13531	. . . . . . 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)”⟩)))
67	59, 61, 63, 65, 66	syl22anc 1319	. . . . . 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)”⟩)))
68		s111 13248	. . . . . . . . 9 ⊢ (((𝑊‘((#‘𝑊) − 2)) ∈ V ∧ (𝑈‘((#‘𝑈) − 2)) ∈ V) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2))))
69	58, 63, 68	sylancr 694	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2))))
70		oveq1 6556	. . . . . . . . . . . 12 ⊢ ((#‘𝑈) = (#‘𝑊) → ((#‘𝑈) − 2) = ((#‘𝑊) − 2))
71	70	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((#‘𝑈) = (#‘𝑊) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
72	71	eqcoms 2618	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
73	72	adantl 481	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
74	73	eqeq2d 2620	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑈) − 2)) ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2))))
75	69, 74	bitrd 267	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“(𝑊‘((#‘𝑊) − 2))”⟩ = ⟨“(𝑈‘((#‘𝑈) − 2))”⟩ ↔ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2))))
76		s111 13248	. . . . . . . 8 ⊢ ((( lastS ‘𝑊) ∈ V ∧ ( lastS ‘𝑈) ∈ V) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
77	60, 65, 76	sylancr 694	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
78	75, 77	anbi12d 743	. . . . . 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))))
79	57, 67, 78	3bitrd 293	. . . . 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))))
80	79	anbi2d 736	. . . 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)))))
81		3anass 1035	. . . 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))))
82	80, 81	syl6bbr 277	. . 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))))
83	82	pm5.32da 671	. 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)))))
84	40, 83	bitrd 267	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147  ⟨“cs1 13149   substr csubstr 13150  ⟨“cs2 13437

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-fal 1481  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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  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-s2 13444

This theorem is referenced by:  numclwlk1lem2f1  26621  av-numclwlk1lem2f1  41524