Theorem cshfn 13536

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.)

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 substr ⟨0, (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 2626	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 = ∅ ↔ 𝑊 = ∅))
2	1	adantr 481	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 = ∅ ↔ 𝑊 = ∅))
3		simpl 473	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑤 = 𝑊)
4		simpr 477	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
5		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
6	5	adantr 481	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (#‘𝑤) = (#‘𝑊))
7	4, 6	oveq12d 6668	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑛 mod (#‘𝑤)) = (𝑁 mod (#‘𝑊)))
8	7, 6	opeq12d 4410	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩ = ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)
9	3, 8	oveq12d 6668	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) = (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩))
10	7	opeq2d 4409	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ⟨0, (𝑛 mod (#‘𝑤))⟩ = ⟨0, (𝑁 mod (#‘𝑊))⟩)
11	3, 10	oveq12d 6668	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩) = (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))
12	9, 11	oveq12d 6668	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩)) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
13	2, 12	ifbieq2d 4111	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
14		df-csh 13535	. 2 ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))))
15		0ex 4790	. . 3 ⊢ ∅ ∈ V
16		ovex 6678	. . 3 ⊢ ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) ∈ V
17	15, 16	ifex 4156	. 2 ⊢ if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) ∈ V
18	13, 14, 17	ovmpt2a 6791	1 ⊢ ((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  ∅c0 3915  ifcif 4086  ⟨cop 4183   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℕ₀cn0 11292  ℤcz 11377  ..^cfzo 12465   mod cmo 12668  #chash 13117   ++ cconcat 13293   substr csubstr 13295   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-csh 13535

This theorem is referenced by:  cshword  13537  cshword2  41437