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Theorem cshnz 13389
 Description: A cyclical shift is the empty set if the number of shifts is not an integer. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.)
Assertion
Ref Expression
cshnz 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅)

Proof of Theorem cshnz
Dummy variables 𝑓 𝑙 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-csh 13386 . . 3 cyclShift = (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 4718 . . . 4 ∅ ∈ V
3 ovex 6577 . . . 4 ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩)) ∈ V
42, 3ifex 4106 . . 3 if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))) ∈ V
51, 4dmmpt2 7129 . 2 dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ)
6 id 22 . . 3 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ)
76intnand 953 . 2 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ))
8 ndmovg 6715 . 2 ((dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) ∧ ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = ∅)
95, 7, 8sylancr 694 1 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ∅c0 3874  ifcif 4036  ⟨cop 4131   × cxp 5036  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ℤcz 11254  ..^cfzo 12334   mod cmo 12530  #chash 12979   ++ 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-csh 13386 This theorem is referenced by:  0csh0  13390  cshwcl  13395
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