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Definition df-cshOLD 13943
 Description: Obsolete version of df-csh 13942 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by Mario Carneiro/ Alexander van der Vekens/Gerard Lang, 17-Nov-2018.) (New usage is discouraged.)
Assertion
Ref Expression
df-cshOLD 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))⟩))))
Distinct variable group:   𝑓,𝑙,𝑛,𝑤

Detailed syntax breakdown of Definition df-cshOLD
StepHypRef Expression
1 ccshOLD 13941 . 2 class cyclShiftOLD
2 vw . . 3 setvar 𝑤
3 vn . . 3 setvar 𝑛
4 vf . . . . . . 7 setvar 𝑓
54cv 1600 . . . . . 6 class 𝑓
6 cc0 10274 . . . . . . 7 class 0
7 vl . . . . . . . 8 setvar 𝑙
87cv 1600 . . . . . . 7 class 𝑙
9 cfzo 12789 . . . . . . 7 class ..^
106, 8, 9co 6924 . . . . . 6 class (0..^𝑙)
115, 10wfn 6132 . . . . 5 wff 𝑓 Fn (0..^𝑙)
12 cn0 11647 . . . . 5 class 0
1311, 7, 12wrex 3091 . . . 4 wff 𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)
1413, 4cab 2763 . . 3 class {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}
15 cz 11733 . . 3 class
162cv 1600 . . . . 5 class 𝑤
17 c0 4141 . . . . 5 class
1816, 17wceq 1601 . . . 4 wff 𝑤 = ∅
193cv 1600 . . . . . . . 8 class 𝑛
20 chash 13441 . . . . . . . . 9 class
2116, 20cfv 6137 . . . . . . . 8 class (♯‘𝑤)
22 cmo 12992 . . . . . . . 8 class mod
2319, 21, 22co 6924 . . . . . . 7 class (𝑛 mod (♯‘𝑤))
2423, 21cop 4404 . . . . . 6 class ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩
25 csubstr 13736 . . . . . 6 class substr
2616, 24, 25co 6924 . . . . 5 class (𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩)
276, 23cop 4404 . . . . . 6 class ⟨0, (𝑛 mod (♯‘𝑤))⟩
2816, 27, 25co 6924 . . . . 5 class (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩)
29 cconcat 13666 . . . . 5 class ++
3026, 28, 29co 6924 . . . 4 class ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩))
3118, 17, 30cif 4307 . . 3 class if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩)))
322, 3, 14, 15, 31cmpt2 6926 . 2 class (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (♯‘𝑤))⟩))))
331, 32wceq 1601 1 wff 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))⟩))))
 Colors of variables: wff setvar class This definition is referenced by:  cshfnOLD  13945  cshnzOLD  13948  0csh0OLD  13950
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