Theorem wrdl3s3 14953

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 11246	. . . . . . . 8 ⊢ 0 ∈ V
2	1	tpid1 4777	. . . . . . 7 ⊢ 0 ∈ {0, 1, 2}
3		fzo0to3tp 13758	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
4	2, 3	eleqtrri 2828	. . . . . 6 ⊢ 0 ∈ (0..^3)
5		oveq2 7434	. . . . . 6 ⊢ ((♯‘𝑊) = 3 → (0..^(♯‘𝑊)) = (0..^3))
6	4, 5	eleqtrrid 2836	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 0 ∈ (0..^(♯‘𝑊)))
7		wrdsymbcl 14517	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → (𝑊‘0) ∈ 𝑉)
8	6, 7	sylan2 591	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘0) ∈ 𝑉)
9		1ex 11248	. . . . . . . 8 ⊢ 1 ∈ V
10	9	tpid2 4779	. . . . . . 7 ⊢ 1 ∈ {0, 1, 2}
11	10, 3	eleqtrri 2828	. . . . . 6 ⊢ 1 ∈ (0..^3)
12	11, 5	eleqtrrid 2836	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 1 ∈ (0..^(♯‘𝑊)))
13		wrdsymbcl 14517	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑊))) → (𝑊‘1) ∈ 𝑉)
14	12, 13	sylan2 591	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘1) ∈ 𝑉)
15		2ex 12327	. . . . . . . 8 ⊢ 2 ∈ V
16	15	tpid3 4782	. . . . . . 7 ⊢ 2 ∈ {0, 1, 2}
17	16, 3	eleqtrri 2828	. . . . . 6 ⊢ 2 ∈ (0..^3)
18	17, 5	eleqtrrid 2836	. . . . 5 ⊢ ((♯‘𝑊) = 3 → 2 ∈ (0..^(♯‘𝑊)))
19		wrdsymbcl 14517	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑊))) → (𝑊‘2) ∈ 𝑉)
20	18, 19	sylan2 591	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (𝑊‘2) ∈ 𝑉)
21		simpr 483	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → (♯‘𝑊) = 3)
22		eqid 2728	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
23		eqid 2728	. . . . . 6 ⊢ (𝑊‘1) = (𝑊‘1)
24		eqid 2728	. . . . . 6 ⊢ (𝑊‘2) = (𝑊‘2)
25	22, 23, 24	3pm3.2i 1336	. . . . 5 ⊢ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))
26	21, 25	jctir 519	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))))
27		eqeq2 2740	. . . . . . 7 ⊢ (𝑎 = (𝑊‘0) → ((𝑊‘0) = 𝑎 ↔ (𝑊‘0) = (𝑊‘0)))
28	27	3anbi1d 1436	. . . . . 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 628	. . . . 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 2740	. . . . . . 7 ⊢ (𝑏 = (𝑊‘1) → ((𝑊‘1) = 𝑏 ↔ (𝑊‘1) = (𝑊‘1)))
31	30	3anbi2d 1437	. . . . . 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 628	. . . . 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 2740	. . . . . . 7 ⊢ (𝑐 = (𝑊‘2) → ((𝑊‘2) = 𝑐 ↔ (𝑊‘2) = (𝑊‘2)))
34	33	3anbi3d 1438	. . . . . 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 628	. . . . 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 3628	. . . 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 1370	. . 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 1086	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉))
39		eqwrds3 14952	. . . . . . . . . 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 411	. . . . . . . . 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 414	. . . . . . 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 479	. . . . . 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 416	. . . . 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 3174	. . . 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 3215	. . 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 14870	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
49	48	ad4ant123 1169	. . . . . 6 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
50		s3len 14885	. . . . . 6 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
51	49, 50	jctir 519	. . . . 5 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3))
52		eleq1 2817	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉))
53		fveqeq2 6911	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((♯‘𝑊) = 3 ↔ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3))
54	52, 53	anbi12d 630	. . . . . 6 ⊢ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3)))
55	54	adantl 480	. . . . 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 3154	. . 3 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3)))
58	57	rexlimivv 3197	. 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∃wrex 3067  {ctp 4636  ‘cfv 6553  (class class class)co 7426  0cc0 11146  1c1 11147  2c2 12305  3c3 12306  ..^cfzo 13667  ♯chash 14329  Word cword 14504  ⟨“cs3 14833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-s2 14839  df-s3 14840

This theorem is referenced by:  elwwlks2s3  29782