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Theorem wrdl3s3 13553
 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) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
Distinct variable groups:   𝑉,𝑎,𝑏,𝑐   𝑊,𝑎,𝑏,𝑐

Proof of Theorem wrdl3s3
StepHypRef Expression
1 c0ex 9913 . . . . . . . 8 0 ∈ V
21tpid1 4246 . . . . . . 7 0 ∈ {0, 1, 2}
3 fzo0to3tp 12421 . . . . . . 7 (0..^3) = {0, 1, 2}
42, 3eleqtrri 2687 . . . . . 6 0 ∈ (0..^3)
5 oveq2 6557 . . . . . 6 ((#‘𝑊) = 3 → (0..^(#‘𝑊)) = (0..^3))
64, 5syl5eleqr 2695 . . . . 5 ((#‘𝑊) = 3 → 0 ∈ (0..^(#‘𝑊)))
7 wrdsymbcl 13173 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
86, 7sylan2 490 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → (𝑊‘0) ∈ 𝑉)
9 1ex 9914 . . . . . . . 8 1 ∈ V
109tpid2 4247 . . . . . . 7 1 ∈ {0, 1, 2}
1110, 3eleqtrri 2687 . . . . . 6 1 ∈ (0..^3)
1211, 5syl5eleqr 2695 . . . . 5 ((#‘𝑊) = 3 → 1 ∈ (0..^(#‘𝑊)))
13 wrdsymbcl 13173 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ 𝑉)
1412, 13sylan2 490 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → (𝑊‘1) ∈ 𝑉)
15 2ex 10969 . . . . . . . 8 2 ∈ V
1615tpid3 4250 . . . . . . 7 2 ∈ {0, 1, 2}
1716, 3eleqtrri 2687 . . . . . 6 2 ∈ (0..^3)
1817, 5syl5eleqr 2695 . . . . 5 ((#‘𝑊) = 3 → 2 ∈ (0..^(#‘𝑊)))
19 wrdsymbcl 13173 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ 2 ∈ (0..^(#‘𝑊))) → (𝑊‘2) ∈ 𝑉)
2018, 19sylan2 490 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → (𝑊‘2) ∈ 𝑉)
21 simpr 476 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → (#‘𝑊) = 3)
22 eqid 2610 . . . . . 6 (𝑊‘0) = (𝑊‘0)
23 eqid 2610 . . . . . 6 (𝑊‘1) = (𝑊‘1)
24 eqid 2610 . . . . . 6 (𝑊‘2) = (𝑊‘2)
2522, 23, 243pm3.2i 1232 . . . . 5 ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))
2621, 25jctir 559 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → ((#‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2))))
27 eqeq2 2621 . . . . . . 7 (𝑎 = (𝑊‘0) → ((𝑊‘0) = 𝑎 ↔ (𝑊‘0) = (𝑊‘0)))
28273anbi1d 1395 . . . . . 6 (𝑎 = (𝑊‘0) → (((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐) ↔ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
2928anbi2d 736 . . . . 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 2621 . . . . . . 7 (𝑏 = (𝑊‘1) → ((𝑊‘1) = 𝑏 ↔ (𝑊‘1) = (𝑊‘1)))
31303anbi2d 1396 . . . . . 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)))
3231anbi2d 736 . . . . 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 2621 . . . . . . 7 (𝑐 = (𝑊‘2) → ((𝑊‘2) = 𝑐 ↔ (𝑊‘2) = (𝑊‘2)))
34333anbi3d 1397 . . . . . 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))))
3534anbi2d 736 . . . . 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)))))
3629, 32, 35rspc3ev 3297 . . . 4 ((((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘1) ∈ 𝑉 ∧ (𝑊‘2) ∈ 𝑉) ∧ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = (𝑊‘0) ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = (𝑊‘2)))) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
378, 14, 20, 26, 36syl31anc 1321 . . 3 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))
38 df-3an 1033 . . . . . . . . 9 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉))
39 eqwrds3 13552 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
4039ex 449 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → ((𝑎𝑉𝑏𝑉𝑐𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))))
4138, 40syl5bir 232 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐)))))
4241expd 451 . . . . . . 7 (𝑊 ∈ Word 𝑉 → ((𝑎𝑉𝑏𝑉) → (𝑐𝑉 → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))))
4342adantr 480 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → ((𝑎𝑉𝑏𝑉) → (𝑐𝑉 → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))))
4443imp31 447 . . . . 5 ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
4544rexbidva 3031 . . . 4 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ∃𝑐𝑉 ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
46452rexbidva 3038 . . 3 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → (∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝑎 ∧ (𝑊‘1) = 𝑏 ∧ (𝑊‘2) = 𝑐))))
4737, 46mpbird 246 . 2 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
48 s3cl 13474 . . . . . . . 8 ((𝑎𝑉𝑏𝑉𝑐𝑉) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
4948ad4ant123 1286 . . . . . . 7 ((((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉)
50 s3len 13489 . . . . . . 7 (#‘⟨“𝑎𝑏𝑐”⟩) = 3
5149, 50jctir 559 . . . . . 6 ((((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = 3))
52 eleq1 2676 . . . . . . . 8 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉))
53 fveq2 6103 . . . . . . . . 9 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (#‘𝑊) = (#‘⟨“𝑎𝑏𝑐”⟩))
5453eqeq1d 2612 . . . . . . . 8 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((#‘𝑊) = 3 ↔ (#‘⟨“𝑎𝑏𝑐”⟩) = 3))
5552, 54anbi12d 743 . . . . . . 7 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = 3)))
5655adantl 481 . . . . . 6 ((((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ Word 𝑉 ∧ (#‘⟨“𝑎𝑏𝑐”⟩) = 3)))
5751, 56mpbird 246 . . . . 5 ((((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3))
5857ex 449 . . . 4 (((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) → (𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3)))
5958rexlimdva 3013 . . 3 ((𝑎𝑉𝑏𝑉) → (∃𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3)))
6059rexlimivv 3018 . 2 (∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩ → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3))
6147, 60impbii 198 1 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {ctp 4129  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  3c3 10948  ..^cfzo 12334  #chash 12979  Word cword 13146  ⟨“cs3 13438 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-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-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445 This theorem is referenced by:  elwwlks2s3  41169
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