pfx2 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > pfx2		Structured version Visualization version GIF version

Theorem pfx2 41412

Description: A prefix of length 2. (Contributed by AV, 15-May-2020.)

**Assertion**
Ref	Expression
pfx2	⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Proof of Theorem pfx2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11309	. . . 4 ⊢ 2 ∈ ℕ₀
2	1	a1i 11	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ ℕ₀)
3		lencl 13324	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
4	3	adantr 481	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ₀)
5		simpr 477	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ≤ (#‘𝑊))
6		elfz2nn0 12431	. . 3 ⊢ (2 ∈ (0...(#‘𝑊)) ↔ (2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)))
7	2, 4, 5, 6	syl3anbrc 1246	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → 2 ∈ (0...(#‘𝑊)))
8		pfxlen 41391	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (#‘(𝑊 prefix 2)) = 2)
9		s2len 13634	. . . . . . 7 ⊢ (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) = 2
10	9	eqcomi 2631	. . . . . 6 ⊢ 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩)
11	10	a1i 11	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩))
12		simpl 473	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
13		simpr 477	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 2 ∈ (0...(#‘𝑊)))
14		2nn 11185	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
15		lbfzo0 12507	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
16	14, 15	mpbir 221	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^2)
17	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → 0 ∈ (0..^2))
18	12, 13, 17	3jca 1242	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
19	18	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)))
20		pfxfv 41399	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 0 ∈ (0..^2)) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
21	19, 20	syl 17	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (𝑊‘0))
22		fvex 6201	. . . . . . . 8 ⊢ (𝑊‘0) ∈ V
23		s2fv0 13632	. . . . . . . 8 ⊢ ((𝑊‘0) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) = (𝑊‘0)
25	21, 24	syl6eqr 2674	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
26		1nn0 11308	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
27		1lt2 11194	. . . . . . . . . 10 ⊢ 1 < 2
28		elfzo0 12508	. . . . . . . . . 10 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
29	26, 14, 27, 28	mpbir3an 1244	. . . . . . . . 9 ⊢ 1 ∈ (0..^2)
30		pfxfv 41399	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊)) ∧ 1 ∈ (0..^2)) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
31	29, 30	mp3an3 1413	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (𝑊‘1))
32		fvex 6201	. . . . . . . . 9 ⊢ (𝑊‘1) ∈ V
33		s2fv1 13633	. . . . . . . . 9 ⊢ ((𝑊‘1) ∈ V → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1))
34	32, 33	ax-mp 5	. . . . . . . 8 ⊢ (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1) = (𝑊‘1)
35	31, 34	syl6eqr 2674	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
36	35	adantr 481	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
37		0nn0 11307	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
38	37, 26	pm3.2i 471	. . . . . . . 8 ⊢ (0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀)
39	38	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀))
40		fveq2 6191	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘0))
41		fveq2 6191	. . . . . . . . 9 ⊢ (𝑖 = 0 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0))
42	40, 41	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑖 = 0 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0)))
43		fveq2 6191	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑊 prefix 2)‘𝑖) = ((𝑊 prefix 2)‘1))
44		fveq2 6191	. . . . . . . . 9 ⊢ (𝑖 = 1 → (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))
45	43, 44	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑖 = 1 → (((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1)))
46	42, 45	ralprg 4234	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))))
47	39, 46	syl 17	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ (((𝑊 prefix 2)‘0) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘0) ∧ ((𝑊 prefix 2)‘1) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘1))))
48	25, 36, 47	mpbir2and 957	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))
49		eqeq1 2626	. . . . . . 7 ⊢ ((#‘(𝑊 prefix 2)) = 2 → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ↔ 2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩)))
50		oveq2 6658	. . . . . . . . 9 ⊢ ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = (0..^2))
51		fzo0to2pr 12553	. . . . . . . . 9 ⊢ (0..^2) = {0, 1}
52	50, 51	syl6eq 2672	. . . . . . . 8 ⊢ ((#‘(𝑊 prefix 2)) = 2 → (0..^(#‘(𝑊 prefix 2))) = {0, 1})
53	52	raleqdv 3144	. . . . . . 7 ⊢ ((#‘(𝑊 prefix 2)) = 2 → (∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖) ↔ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
54	49, 53	anbi12d 747	. . . . . 6 ⊢ ((#‘(𝑊 prefix 2)) = 2 → (((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
55	54	adantl 482	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → (((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)) ↔ (2 = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ {0, 1} ((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
56	11, 48, 55	mpbir2and 957	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) ∧ (#‘(𝑊 prefix 2)) = 2) → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
57	8, 56	mpdan 702	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖)))
58		pfxcl 41386	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix 2) ∈ Word 𝑉)
59	58	adantr 481	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) ∈ Word 𝑉)
60		simp2 1062	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ₀)
61		1red 10055	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ₀ → 1 ∈ ℝ)
62		2re 11090	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
63	62	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ₀ → 2 ∈ ℝ)
64		nn0re 11301	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℝ)
65	61, 63, 64	3jca 1242	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) ∈ ℕ₀ → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
66		ltleletr 10130	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊)))
67	65, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊)))
68	27, 67	mpani 712	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) ∈ ℕ₀ → (2 ≤ (#‘𝑊) → 1 ≤ (#‘𝑊)))
69	68	imp 445	. . . . . . . . . . 11 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
70	69	3adant1 1079	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → 1 ≤ (#‘𝑊))
71	60, 70	jca 554	. . . . . . . . 9 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → ((#‘𝑊) ∈ ℕ₀ ∧ 1 ≤ (#‘𝑊)))
72		elnnnn0c 11338	. . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ₀ ∧ 1 ≤ (#‘𝑊)))
73	71, 72	sylibr 224	. . . . . . . 8 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
74	6, 73	sylbi 207	. . . . . . 7 ⊢ (2 ∈ (0...(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
75		lbfzo0 12507	. . . . . . 7 ⊢ (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
76	74, 75	sylibr 224	. . . . . 6 ⊢ (2 ∈ (0...(#‘𝑊)) → 0 ∈ (0..^(#‘𝑊)))
77		wrdsymbcl 13318	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
78	76, 77	sylan2 491	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
79	26	a1i 11	. . . . . . 7 ⊢ (2 ∈ (0...(#‘𝑊)) → 1 ∈ ℕ₀)
80	65	adantl 482	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
81		ltletr 10129	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊)))
82	80, 81	syl 17	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀) → ((1 < 2 ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊)))
83	27, 82	mpani 712	. . . . . . . . 9 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀) → (2 ≤ (#‘𝑊) → 1 < (#‘𝑊)))
84	83	3impia 1261	. . . . . . . 8 ⊢ ((2 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 2 ≤ (#‘𝑊)) → 1 < (#‘𝑊))
85	6, 84	sylbi 207	. . . . . . 7 ⊢ (2 ∈ (0...(#‘𝑊)) → 1 < (#‘𝑊))
86		elfzo0 12508	. . . . . . 7 ⊢ (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
87	79, 74, 85, 86	syl3anbrc 1246	. . . . . 6 ⊢ (2 ∈ (0...(#‘𝑊)) → 1 ∈ (0..^(#‘𝑊)))
88		wrdsymbcl 13318	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
89	87, 88	sylan2 491	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
90	78, 89	s2cld 13616	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word 𝑉)
91		eqwrd 13346	. . . 4 ⊢ (((𝑊 prefix 2) ∈ Word 𝑉 ∧ ⟨“(𝑊‘0)(𝑊‘1)”⟩ ∈ Word 𝑉) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
92	59, 90, 91	syl2anc 693	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → ((𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩ ↔ ((#‘(𝑊 prefix 2)) = (#‘⟨“(𝑊‘0)(𝑊‘1)”⟩) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 prefix 2)))((𝑊 prefix 2)‘𝑖) = (⟨“(𝑊‘0)(𝑊‘1)”⟩‘𝑖))))
93	57, 92	mpbird 247	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0...(#‘𝑊))) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)
94	7, 93	syldan 487	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {cpr 4179 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ⟨“cs2 13586 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-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-s2 13593 df-pfx 41382

This theorem is referenced by: (None)

W3C validator