Theorem cshfn 14780

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 2732	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 = ∅ ↔ 𝑊 = ∅))
2	1	adantr 479	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 = ∅ ↔ 𝑊 = ∅))
3		simpl 481	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑤 = 𝑊)
4		simpr 483	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
5		fveq2 6902	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
6	5	adantr 479	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (♯‘𝑤) = (♯‘𝑊))
7	4, 6	oveq12d 7444	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑛 mod (♯‘𝑤)) = (𝑁 mod (♯‘𝑊)))
8	7, 6	opeq12d 4886	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩ = ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩)
9	3, 8	oveq12d 7444	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) = (𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩))
10	3, 7	oveq12d 7444	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 prefix (𝑛 mod (♯‘𝑤))) = (𝑊 prefix (𝑁 mod (♯‘𝑊))))
11	9, 10	oveq12d 7444	. . 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 4558	. 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 14779	. 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 5311	. . 3 ⊢ ∅ ∈ V
15		ovex 7459	. . 3 ⊢ ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ V
16	14, 15	ifex 4582	. 2 ⊢ if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) ∈ V
17	12, 13, 16	ovmpoa 7582	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))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  ∃wrex 3067  ∅c0 4326  ifcif 4532  ⟨cop 4638   Fn wfn 6548  ‘cfv 6553  (class class class)co 7426  0cc0 11146  ℕ₀cn0 12510  ℤcz 12596  ..^cfzo 13667   mod cmo 13874  ♯chash 14329   ++ cconcat 14560   substr csubstr 14630   prefix cpfx 14660   cyclShift ccsh 14778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-csh 14779

This theorem is referenced by:  cshword  14781