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Theorem cshwlen 13396
 Description: The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.)
Assertion
Ref Expression
cshwlen ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))

Proof of Theorem cshwlen
StepHypRef Expression
1 oveq1 6556 . . . . 5 (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (∅ cyclShift 𝑁))
2 0csh0 13390 . . . . . 6 (∅ cyclShift 𝑁) = ∅
32a1i 11 . . . . 5 (𝑊 = ∅ → (∅ cyclShift 𝑁) = ∅)
4 eqcom 2617 . . . . . 6 (𝑊 = ∅ ↔ ∅ = 𝑊)
54biimpi 205 . . . . 5 (𝑊 = ∅ → ∅ = 𝑊)
61, 3, 53eqtrd 2648 . . . 4 (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = 𝑊)
76fveq2d 6107 . . 3 (𝑊 = ∅ → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
87a1d 25 . 2 (𝑊 = ∅ → ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
9 cshword 13388 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
109fveq2d 6107 . . . . 5 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
1110adantr 480 . . . 4 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
12 swrdcl 13271 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ∈ Word 𝑉)
13 swrdcl 13271 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) ∈ Word 𝑉)
14 ccatlen 13213 . . . . . . 7 (((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ∈ Word 𝑉 ∧ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) ∈ Word 𝑉) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
1512, 13, 14syl2anc 691 . . . . . 6 (𝑊 ∈ Word 𝑉 → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
1615adantr 480 . . . . 5 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
1716adantr 480 . . . 4 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
18 lennncl 13180 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
19 pm3.21 463 . . . . . . . . . . 11 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))
2019ex 449 . . . . . . . . . 10 ((#‘𝑊) ∈ ℕ → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
2118, 20syl 17 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
2221ex 449 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
2322com24 93 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
2423pm2.43i 50 . . . . . 6 (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
2524imp31 447 . . . . 5 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))
26 simpl 472 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
27 pm3.22 464 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
2827adantl 481 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
29 zmodfzp1 12556 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)))
3028, 29syl 17 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)))
31 lencl 13179 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
32 nn0fz0 12306 . . . . . . . . . 10 ((#‘𝑊) ∈ ℕ0 ↔ (#‘𝑊) ∈ (0...(#‘𝑊)))
3331, 32sylib 207 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ (0...(#‘𝑊)))
3433adantr 480 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ (0...(#‘𝑊)))
35 swrdlen 13275 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)) ∧ (#‘𝑊) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) = ((#‘𝑊) − (𝑁 mod (#‘𝑊))))
3626, 30, 34, 35syl3anc 1318 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) = ((#‘𝑊) − (𝑁 mod (#‘𝑊))))
37 zmodcl 12552 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
3837ancoms 468 . . . . . . . . . 10 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
3938adantl 481 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ0)
40 0elfz 12305 . . . . . . . . 9 ((𝑁 mod (#‘𝑊)) ∈ ℕ0 → 0 ∈ (0...(𝑁 mod (#‘𝑊))))
4139, 40syl 17 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → 0 ∈ (0...(𝑁 mod (#‘𝑊))))
42 swrdlen 13275 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0...(𝑁 mod (#‘𝑊))) ∧ (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑁 mod (#‘𝑊)) − 0))
4326, 41, 30, 42syl3anc 1318 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑁 mod (#‘𝑊)) − 0))
4436, 43oveq12d 6567 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + ((𝑁 mod (#‘𝑊)) − 0)))
4537nn0cnd 11230 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
4645ancoms 468 . . . . . . . . 9 (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
4746adantl 481 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
4847subid1d 10260 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((𝑁 mod (#‘𝑊)) − 0) = (𝑁 mod (#‘𝑊)))
4948oveq2d 6565 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + ((𝑁 mod (#‘𝑊)) − 0)) = (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))))
5031nn0cnd 11230 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℂ)
51 npcan 10169 . . . . . . 7 (((#‘𝑊) ∈ ℂ ∧ (𝑁 mod (#‘𝑊)) ∈ ℂ) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
5250, 46, 51syl2an 493 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
5344, 49, 523eqtrd 2648 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (#‘𝑊))
5425, 53syl 17 . . . 4 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (#‘𝑊))
5511, 17, 543eqtrd 2648 . . 3 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
5655expcom 450 . 2 (𝑊 ≠ ∅ → ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
578, 56pm2.61ine 2865 1 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197   mod cmo 12530  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   cyclShift ccsh 13385 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  ax-pre-sup 9893 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-rmo 2904  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-sup 8231  df-inf 8232  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-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  cshwf  13397  2cshw  13410  lswcshw  13412  cshwleneq  13414  clwwisshclwwlem  26334  clwwnisshclwwn  26337  erclwwlkeqlen  26340  erclwwlkneqlen  26352  crctcshlem2  41021  clwwisshclwwslem  41234  clwwisshclwws  41235  clwwnisshclwwsn  41237  erclwwlkseqlen  41240  erclwwlksneqlen  41252  eucrct2eupth  41413
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