Theorem wrdl3s3 14895

Description: A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.)

Assertion

Ref	Expression
wrdl3s3	⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Distinct variable groups:   𝑉,𝑎,𝑏,𝑐   𝑊,𝑎,𝑏,𝑐

Proof of Theorem wrdl3s3

Step	Hyp	Ref	Expression
1		c0ex 11190	. . . . . . . 8 ⊢ 0 ∈ V
2	1	tpid1 4765	. . . . . . 7 ⊢ 0 ∈ {0, 1, 2}
3		fzo0to3tp 13700	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
4	2, 3	eleqtrri 2831	. . . . . 6 ⊢ 0 ∈ (0..^3)
5		oveq2 7401	. . . . . 6 ⊢ ((♯‘𝑊) = 3 → (0..^(♯‘𝑊)) = (0..^3))
6	4, 5	eleqtrrid 2839	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 0 ∈ (0..^(♯‘𝑊)))
7		wrdsymbcl 14459	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → (𝑊‘0) ∈ 𝑉)
8	6, 7	sylan2 593	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘0) ∈ 𝑉)
9		1ex 11192	. . . . . . . 8 ⊢ 1 ∈ V
10	9	tpid2 4767	. . . . . . 7 ⊢ 1 ∈ {0, 1, 2}
11	10, 3	eleqtrri 2831	. . . . . 6 ⊢ 1 ∈ (0..^3)
12	11, 5	eleqtrrid 2839	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 1 ∈ (0..^(♯‘𝑊)))
13		wrdsymbcl 14459	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑊))) → (𝑊‘1) ∈ 𝑉)
14	12, 13	sylan2 593	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘1) ∈ 𝑉)
15		2ex 12271	. . . . . . . 8 ⊢ 2 ∈ V
16	15	tpid3 4770	. . . . . . 7 ⊢ 2 ∈ {0, 1, 2}
17	16, 3	eleqtrri 2831	. . . . . 6 ⊢ 2 ∈ (0..^3)
18	17, 5	eleqtrrid 2839	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 2 ∈ (0..^(♯‘𝑊)))
19		wrdsymbcl 14459	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑊))) → (𝑊‘2) ∈ 𝑉)
20	18, 19	sylan2 593	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘2) ∈ 𝑉)
21		simpr 485	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (♯‘𝑊) = 3)
22		eqid 2731	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
23		eqid 2731	. . . . . 6 ⊢ (𝑊‘1) = (𝑊‘1)
24		eqid 2731	. . . . . 6 ⊢ (𝑊‘2) = (𝑊‘2)
25	22, 23, 24	3pm3.2i 1339	. . . . 5 ⊢ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))
26	21, 25	jctir 521	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))))
27		eqeq2 2743	. . . . . . 7 ⊢ (𝑎 = (𝑊‘0) → ((𝑊‘0) = 𝑎 ↔ (𝑊‘0) = (𝑊‘0)))
28	27	3anbi1d 1440	. . . . . 6 ⊢ (𝑎 = (𝑊‘0) → (((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐) ↔ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
29	28	anbi2d 629	. . . . 5 ⊢ (𝑎 = (𝑊‘0) → (((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)) ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
30		eqeq2 2743	. . . . . . 7 ⊢ (𝑏 = (𝑊‘1) → ((𝑊‘1) = 𝑏 ↔ (𝑊‘1) = (𝑊‘1)))
31	30	3anbi2d 1441	. . . . . 6 ⊢ (𝑏 = (𝑊‘1) → (((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐) ↔ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝑐)))
32	31	anbi2d 629	. . . . 5 ⊢ (𝑏 = (𝑊‘1) → (((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)) ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝑐))))
33		eqeq2 2743	. . . . . . 7 ⊢ (𝑐 = (𝑊‘2) → ((𝑊‘2) = 𝑐 ↔ (𝑊‘2) = (𝑊‘2)))
34	33	3anbi3d 1442	. . . . . 6 ⊢ (𝑐 = (𝑊‘2) → (((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝑐) ↔ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))))
35	34	anbi2d 629	. . . . 5 ⊢ (𝑐 = (𝑊‘2) → (((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝑐)) ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2)))))
36	29, 32, 35	rspc3ev 3624	. . . 4 ⊢ ((((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘1) ∈ 𝑉 ∧ (𝑊‘2) ∈ 𝑉) ∧ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2)))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
37	8, 14, 20, 26, 36	syl31anc 1373	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
38		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉))
39		eqwrds3 14894	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
40	39	ex 413	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))))
41	38, 40	biimtrrid 242	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))))
42	41	expd 416	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑐 ∈ 𝑉 → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))))
43	42	adantr 481	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑐 ∈ 𝑉 → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))))
44	43	imp31 418	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
45	44	rexbidva 3175	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ∃𝑐 ∈ 𝑉 ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
46	45	2rexbidva 3216	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
47	37, 46	mpbird 256	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
48		s3cl 14812	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
49	48	ad4ant123 1172	. . . . . 6 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
50		s3len 14827	. . . . . 6 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
51	49, 50	jctir 521	. . . . 5 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3))
52		eleq1 2820	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉))
53		fveqeq2 6887	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((♯‘𝑊) = 3 ↔ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3))
54	52, 53	anbi12d 631	. . . . . 6 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3)))
55	54	adantl 482	. . . . 5 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3)))
56	51, 55	mpbird 256	. . . 4 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3))
57	56	rexlimdva2 3156	. . 3 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3)))
58	57	rexlimivv 3198	. 2 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3))
59	47, 58	impbii 208	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3069  {ctp 4626  ‘cfv 6532  (class class class)co 7393  0cc0 11092  1c1 11093  2c2 12249  3c3 12250  ..^cfzo 13609  ♯chash 14272  Word cword 14446  ⟨“cs3 14775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-fzo 13610  df-hash 14273  df-word 14447  df-concat 14503  df-s1 14528  df-s2 14781  df-s3 14782

This theorem is referenced by:  elwwlks2s3  29070