Theorem cshfn 14862

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 2764	. . . 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 6879	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
6	5	adantr 486	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (♯‘𝑤) = (♯‘𝑊))
7	4, 6	oveq12d 7432	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑛 mod (♯‘𝑤)) = (𝑁 mod (♯‘𝑊)))
8	7, 6	opeq12d 4841	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩ = ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩)
9	3, 8	oveq12d 7432	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) = (𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩))
10	3, 7	oveq12d 7432	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 prefix (𝑛 mod (♯‘𝑤))) = (𝑊 prefix (𝑁 mod (♯‘𝑊))))
11	9, 10	oveq12d 7432	. . 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 4509	. 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 14861	. 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 5264	. . 3 ⊢ ∅ ∈ V
15		ovex 7447	. . 3 ⊢ ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ V
16	14, 15	ifex 4533	. 2 ⊢ if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) ∈ V
17	12, 13, 16	ovmpoa 7569	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 2145  {cab 2738  ∃wrex 3086  ∅c0 4279  ifcif 4482  ⟨cop 4590   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414  0cc0 11125  ℕ₀cn0 12529  ℤcz 12616  ..^cfzo 13710   mod cmo 13931  ♯chash 14395   ++ cconcat 14636   substr csubstr 14709   prefix cpfx 14741   cyclShift ccsh 14860

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-csh 14861

This theorem is used by:  cshword  14863