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Theorem cshfn 13988
Description: Perform a cyclical shift for a function over a half-open range of nonnegative integers. (Contributed by AV, 20-May-2018.) (Revised by AV, 17-Nov-2018.) (Revised by AV, 4-Nov-2022.)
Assertion
Ref Expression
cshfn ((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))))
Distinct variable group:   𝑓,𝑙
Allowed substitution hints:   𝑁(𝑓,𝑙)   𝑊(𝑓,𝑙)

Proof of Theorem cshfn
Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2799 . . . 4 (𝑤 = 𝑊 → (𝑤 = ∅ ↔ 𝑊 = ∅))
21adantr 481 . . 3 ((𝑤 = 𝑊𝑛 = 𝑁) → (𝑤 = ∅ ↔ 𝑊 = ∅))
3 simpl 483 . . . . 5 ((𝑤 = 𝑊𝑛 = 𝑁) → 𝑤 = 𝑊)
4 simpr 485 . . . . . . 7 ((𝑤 = 𝑊𝑛 = 𝑁) → 𝑛 = 𝑁)
5 fveq2 6538 . . . . . . . 8 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
65adantr 481 . . . . . . 7 ((𝑤 = 𝑊𝑛 = 𝑁) → (♯‘𝑤) = (♯‘𝑊))
74, 6oveq12d 7034 . . . . . 6 ((𝑤 = 𝑊𝑛 = 𝑁) → (𝑛 mod (♯‘𝑤)) = (𝑁 mod (♯‘𝑊)))
87, 6opeq12d 4718 . . . . 5 ((𝑤 = 𝑊𝑛 = 𝑁) → ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩ = ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩)
93, 8oveq12d 7034 . . . 4 ((𝑤 = 𝑊𝑛 = 𝑁) → (𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) = (𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩))
103, 7oveq12d 7034 . . . 4 ((𝑤 = 𝑊𝑛 = 𝑁) → (𝑤 prefix (𝑛 mod (♯‘𝑤))) = (𝑊 prefix (𝑁 mod (♯‘𝑊))))
119, 10oveq12d 7034 . . 3 ((𝑤 = 𝑊𝑛 = 𝑁) → ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))) = ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))))
122, 11ifbieq2d 4406 . 2 ((𝑤 = 𝑊𝑛 = 𝑁) → if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))))
13 df-csh 13987 . 2 cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))
14 0ex 5102 . . 3 ∅ ∈ V
15 ovex 7048 . . 3 ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ V
1614, 15ifex 4429 . 2 if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) ∈ V
1712, 13, 16ovmpoa 7161 1 ((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  {cab 2775  wrex 3106  c0 4211  ifcif 4381  cop 4478   Fn wfn 6220  cfv 6225  (class class class)co 7016  0cc0 10383  0cn0 11745  cz 11829  ..^cfzo 12883   mod cmo 13087  chash 13540   ++ cconcat 13768   substr csubstr 13838   prefix cpfx 13868   cyclShift ccsh 13986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-csh 13987
This theorem is referenced by:  cshword  13989
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