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Theorem cshnzOLD 13876
 Description: Obsolete version of cshnz 13875 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
cshnzOLD 𝑁 ∈ ℤ → (𝑊 cyclShiftOLD 𝑁) = ∅)

Proof of Theorem cshnzOLD
Dummy variables 𝑓 𝑙 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cshOLD 13871 . . 3 cyclShiftOLD = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩))))
2 0ex 4984 . . . 4 ∅ ∈ V
3 ovex 6910 . . . 4 ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩)) ∈ V
42, 3ifex 4325 . . 3 if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩))) ∈ V
51, 4dmmpt2 7476 . 2 dom cyclShiftOLD = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ)
6 id 22 . . 3 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ)
76intnand 483 . 2 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ))
8 ndmovg 7051 . 2 ((dom cyclShiftOLD = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) ∧ ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) → (𝑊 cyclShiftOLD 𝑁) = ∅)
95, 7, 8sylancr 582 1 𝑁 ∈ ℤ → (𝑊 cyclShiftOLD 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {cab 2785  ∃wrex 3090  ∅c0 4115  ifcif 4277  ⟨cop 4374   × cxp 5310  dom cdm 5312   Fn wfn 6096  ‘cfv 6101  (class class class)co 6878  0cc0 10224  ℕ0cn0 11580  ℤcz 11666  ..^cfzo 12720   mod cmo 12923  ♯chash 13370   ++ cconcat 13590   substr csubstr 13664   cyclShiftOLD ccshOLD 13869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-cshOLD 13871 This theorem is referenced by:  0csh0OLD  13878
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