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Theorem cshwcshid 13793
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 27165 and erclwwlknsym 27222. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
Hypotheses
Ref Expression
cshwcshid.1 (𝜑𝑦 ∈ Word 𝑉)
cshwcshid.2 (𝜑 → (♯‘𝑥) = (♯‘𝑦))
Assertion
Ref Expression
cshwcshid (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
Distinct variable group:   𝑚,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚,𝑛)   𝑉(𝑥,𝑦,𝑚,𝑛)

Proof of Theorem cshwcshid
StepHypRef Expression
1 fznn0sub2 12660 . . . . . . 7 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2 oveq2 6822 . . . . . . . 8 ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
32eleq2d 2825 . . . . . . 7 ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
41, 3syl5ibr 236 . . . . . 6 ((♯‘𝑥) = (♯‘𝑦) → (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
5 cshwcshid.2 . . . . . 6 (𝜑 → (♯‘𝑥) = (♯‘𝑦))
64, 5syl11 33 . . . . 5 (𝑚 ∈ (0...(♯‘𝑦)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
76adantr 472 . . . 4 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
87impcom 445 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9 cshwcshid.1 . . . . . . . 8 (𝜑𝑦 ∈ Word 𝑉)
10 simpl 474 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11 elfzelz 12555 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
1211adantl 473 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13 elfz2nn0 12644 . . . . . . . . . . 11 (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)))
14 nn0z 11612 . . . . . . . . . . . . 13 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15 nn0z 11612 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
16 zsubcl 11631 . . . . . . . . . . . . 13 (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1714, 15, 16syl2anr 496 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18173adant3 1127 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1913, 18sylbi 207 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2019adantl 473 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2110, 12, 203jca 1123 . . . . . . . 8 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
229, 21sylan 489 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23 2cshw 13779 . . . . . . 7 ((𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
2422, 23syl 17 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
25 nn0cn 11514 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
26 nn0cn 11514 . . . . . . . . . . . 12 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
2725, 26anim12i 591 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28273adant3 1127 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
2913, 28sylbi 207 . . . . . . . . 9 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30 pncan3 10501 . . . . . . . . 9 ((𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3129, 30syl 17 . . . . . . . 8 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3231adantl 473 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3332oveq2d 6830 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34 cshwn 13763 . . . . . . . 8 (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
359, 34syl 17 . . . . . . 7 (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3635adantr 472 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3724, 33, 363eqtrrd 2799 . . . . 5 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
3837adantrr 755 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
39 oveq1 6821 . . . . . . 7 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
4039eqeq2d 2770 . . . . . 6 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4140adantl 473 . . . . 5 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4241adantl 473 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4338, 42mpbird 247 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
44 oveq2 6822 . . . . 5 (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
4544eqeq2d 2770 . . . 4 (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))))
4645rspcev 3449 . . 3 ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
478, 43, 46syl2anc 696 . 2 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
4847ex 449 1 (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6814  cc 10146  0cc0 10148   + caddc 10151  cle 10287  cmin 10478  0cn0 11504  cz 11589  ...cfz 12539  chash 13331  Word cword 13497   cyclShift ccsh 13754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-fl 12807  df-mod 12883  df-hash 13332  df-word 13505  df-concat 13507  df-substr 13509  df-csh 13755
This theorem is referenced by:  erclwwlksym  27165  erclwwlknsym  27222
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