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Mirrors > Home > MPE Home > Th. List > cshfn | Structured version Visualization version GIF version |
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.) |
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)))))) |
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)))))) |
Colors of variables: wff setvar class |
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 |
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