Theorem cshfn 14972

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.

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 = ∅ ↔ 𝑊 = ∅))
2	1	adantr 486	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 = ∅ ↔ 𝑊 = ∅))
3		simpl 488	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑤 = 𝑊)
4		simpr 490	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
5		fveq2 6911	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
6	5	adantr 486	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (♯‘𝑤) = (♯‘𝑊))
7	4, 6	oveq12d 7464	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑛 mod (♯‘𝑤)) = (𝑁 mod (♯‘𝑊)))
8	7, 6	opeq12d 4851	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩ = ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩)
9	3, 8	oveq12d 7464	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) = (𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩))
10	3, 7	oveq12d 7464	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 prefix (𝑛 mod (♯‘𝑤))) = (𝑊 prefix (𝑁 mod (♯‘𝑊))))
11	9, 10	oveq12d 7464	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))) = ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))))
12	2, 11	ifbieq2d 4512	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 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 14971	. 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 5265	. . 3 ⊢ ∅ ∈ V
15		ovex 7479	. . 3 ⊢ ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ V
16	14, 15	ifex 4536	. 2 ⊢ if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) ∈ V
17	12, 13, 16	ovmpoa 7602	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  ∅c0 4281  ifcif 4485  ⟨cop 4838   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  0cc0 11230  ℕ₀cn0 12636  ℤcz 12723  ..^cfzo 13818   mod cmo 14039  ♯chash 14504   ++ cconcat 14745   substr csubstr 14819   prefix cpfx 14851   cyclShift ccsh 14970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-csh 14971

This theorem is used by:  cshword  14973