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Theorem cshwcshid 14866
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 30040 and erclwwlknsym 30089. (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 13675 . . . . . . 7 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2 oveq2 7439 . . . . . . . 8 ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
32eleq2d 2827 . . . . . . 7 ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
41, 3imbitrrid 246 . . . . . 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 480 . . . 4 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
87impcom 407 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9 cshwcshid.1 . . . . . . . 8 (𝜑𝑦 ∈ Word 𝑉)
10 simpl 482 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11 elfzelz 13564 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
1211adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13 elfz2nn0 13658 . . . . . . . . . . 11 (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)))
14 nn0z 12638 . . . . . . . . . . . . 13 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15 nn0z 12638 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
16 zsubcl 12659 . . . . . . . . . . . . 13 (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1714, 15, 16syl2anr 597 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18173adant3 1133 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1913, 18sylbi 217 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2019adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2110, 12, 203jca 1129 . . . . . . . 8 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
229, 21sylan 580 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23 2cshw 14851 . . . . . . 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 12536 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
26 nn0cn 12536 . . . . . . . . . . . 12 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
2725, 26anim12i 613 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28273adant3 1133 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
2913, 28sylbi 217 . . . . . . . . 9 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30 pncan3 11516 . . . . . . . . 9 ((𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3129, 30syl 17 . . . . . . . 8 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3231adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3332oveq2d 7447 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34 cshwn 14835 . . . . . . . 8 (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
359, 34syl 17 . . . . . . 7 (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3635adantr 480 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3724, 33, 363eqtrrd 2782 . . . . 5 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
3837adantrr 717 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
39 oveq1 7438 . . . . . . 7 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
4039eqeq2d 2748 . . . . . 6 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4140adantl 481 . . . . 5 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4241adantl 481 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4338, 42mpbird 257 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
44 oveq2 7439 . . . 4 (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
4544rspceeqv 3645 . . 3 ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
468, 43, 45syl2anc 584 . 2 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
4746ex 412 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153  0cc0 11155   + caddc 11158  cle 11296  cmin 11492  0cn0 12526  cz 12613  ...cfz 13547  chash 14369  Word cword 14552   cyclShift ccsh 14826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-hash 14370  df-word 14553  df-concat 14609  df-substr 14679  df-pfx 14709  df-csh 14827
This theorem is referenced by:  erclwwlksym  30040  erclwwlknsym  30089
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