Theorem cshnz 14802

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.

Step	Hyp	Ref	Expression
1		df-csh 14799	. . 3 ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))
2		0ex 5256	. . . 4 ⊢ ∅ ∈ V
3		ovex 7425	. . . 4 ⊢ ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))) ∈ V
4	2, 3	ifex 4530	. . 3 ⊢ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) ∈ V
5	1, 4	dmmpo 8048	. 2 ⊢ dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ)
6		id 22	. . 3 ⊢ (¬ 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ)
7	6	intnand 492	. 2 ⊢ (¬ 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ))
8		ndmovg 7575	. 2 ⊢ ((dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) ∧ ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = ∅)
9	5, 7, 8	sylancr 596	1 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085  ∅c0 4285  ifcif 4479  ⟨cop 4587   × cxp 5643  dom cdm 5645   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  0cc0 11070  ℕ₀cn0 12478  ℤcz 12565  ..^cfzo 13656   mod cmo 13876  ♯chash 14340   ++ cconcat 14580   substr csubstr 14651   prefix cpfx 14681   cyclShift ccsh 14798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-csh 14799

This theorem is referenced by:  0csh0  14803  cshwcl  14808