Theorem 2swrd2eqwrdeq 13696

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 13324	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2		1z 11407	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
3		nn0z 11400	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
4		zltp1le 11427	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ) → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
5	2, 3, 4	sylancr 695	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 < (#‘𝑊) ↔ (1 + 1) ≤ (#‘𝑊)))
6		1p1e2 11134	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ0 → (1 + 1) = 2)
8	7	breq1d 4663	. . . . . . . . . 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 445	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → 2 ≤ (#‘𝑊))
12		2nn0 11309	. . . . . . . 8 ⊢ 2 ∈ ℕ0
13		simpl 473	. . . . . . . 8 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ0)
14		nn0sub 11343	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) → (2 ≤ (#‘𝑊) ↔ ((#‘𝑊) − 2) ∈ ℕ0))
15	12, 13, 14	sylancr 695	. . . . . . 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 481	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℤ)
18		0red 10041	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 0 ∈ ℝ)
19		1red 10055	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → 1 ∈ ℝ)
20		nn0re 11301	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℝ)
21	18, 19, 20	3jca 1242	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
22		0lt1 10550	. . . . . . . . 9 ⊢ 0 < 1
23		lttr 10114	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((0 < 1 ∧ 1 < (#‘𝑊)) → 0 < (#‘𝑊)))
24	23	expd 452	. . . . . . . . 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 445	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → 0 < (#‘𝑊))
27		elnnz 11387	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
28	17, 26, 27	sylanbrc 698	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
29		2pos 11112	. . . . . . . 8 ⊢ 0 < 2
30		2re 11090	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ0 → 2 ∈ ℝ)
32	31, 20	ltsubposd 10613	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (0 < 2 ↔ ((#‘𝑊) − 2) < (#‘𝑊)))
33	29, 32	mpbii 223	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ0 → ((#‘𝑊) − 2) < (#‘𝑊))
34	33	adantr 481	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) < (#‘𝑊))
35		elfzo0 12508	. . . . . 6 ⊢ (((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)) ↔ (((#‘𝑊) − 2) ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ ((#‘𝑊) − 2) < (#‘𝑊)))
36	16, 28, 34, 35	syl3anbrc 1246	. . . . 5 ⊢ (((#‘𝑊) ∈ ℕ0 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
37	1, 36	sylan 488	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
38	37	3adant2 1080	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → ((#‘𝑊) − 2) ∈ (0..^(#‘𝑊)))
39		2swrdeqwrdeq 13453	. . 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 1371	. 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 13695	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
42	41	3adant2 1080	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
43	42	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
44		breq2 4657	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → (1 < (#‘𝑊) ↔ 1 < (#‘𝑈)))
45	44	3anbi3d 1405	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈))))
46		swrd2lsw 13695	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
47	46	3adant1 1079	. . . . . . . . . 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 446	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩) = ⟨“(𝑈‘((#‘𝑈) − 2))( lastS ‘𝑈)”⟩)
50		oveq1 6657	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 2) = ((#‘𝑈) − 2))
51		id 22	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
52	50, 51	opeq12d 4410	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 2), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 2), (#‘𝑈)⟩)
53	52	oveq2d 6666	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 2), (#‘𝑈)⟩))
54	53	eqeq1d 2624	. . . . . . . . 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 482	. . . . . . . 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 2637	. . . . . 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 6203	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊‘((#‘𝑊) − 2)) ∈ V)
59		fvexd 6203	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ V)
60		fvexd 6203	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) ∈ V)
61		fvexd 6203	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ V)
62		s2eq2s1eq 13681	. . . . . . 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 1327	. . . . . 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 6201	. . . . . . . . 9 ⊢ (𝑊‘((#‘𝑊) − 2)) ∈ V
65		s111 13395	. . . . . . . . 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 695	. . . . . . . 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 6657	. . . . . . . . . . . 12 ⊢ ((#‘𝑈) = (#‘𝑊) → ((#‘𝑈) − 2) = ((#‘𝑊) − 2))
68	67	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((#‘𝑈) = (#‘𝑊) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
69	68	eqcoms 2630	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
70	69	adantl 482	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈‘((#‘𝑈) − 2)) = (𝑈‘((#‘𝑊) − 2)))
71	70	eqeq2d 2632	. . . . . . . 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 6201	. . . . . . . 8 ⊢ ( lastS ‘𝑊) ∈ V
74		s111 13395	. . . . . . . 8 ⊢ ((( lastS ‘𝑊) ∈ V ∧ ( lastS ‘𝑈) ∈ V) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
75	73, 61, 74	sylancr 695	. . . . . . 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 1042	. . . 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 673	. 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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292  ⟨“cs1 13294   substr csubstr 13295  ⟨“cs2 13586

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-fal 1489  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  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-s2 13593

This theorem is referenced by:  numclwlk1lem2f1  27227