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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 2799 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 = ∅ ↔ 𝑊 = ∅)) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 = ∅ ↔ 𝑊 = ∅)) |
3 | simpl 483 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑤 = 𝑊) | |
4 | simpr 485 | . . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁) | |
5 | fveq2 6538 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
6 | 5 | adantr 481 | . . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (♯‘𝑤) = (♯‘𝑊)) |
7 | 4, 6 | oveq12d 7034 | . . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑛 mod (♯‘𝑤)) = (𝑁 mod (♯‘𝑊))) |
8 | 7, 6 | opeq12d 4718 | . . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉 = 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) |
9 | 3, 8 | oveq12d 7034 | . . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) = (𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉)) |
10 | 3, 7 | oveq12d 7034 | . . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (𝑤 prefix (𝑛 mod (♯‘𝑤))) = (𝑊 prefix (𝑁 mod (♯‘𝑊)))) |
11 | 9, 10 | oveq12d 7034 | . . 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 4406 | . 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 13987 | . 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 5102 | . . 3 ⊢ ∅ ∈ V | |
15 | ovex 7048 | . . 3 ⊢ ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ V | |
16 | 14, 15 | ifex 4429 | . 2 ⊢ if(𝑊 = ∅, ∅, ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) ∈ V |
17 | 12, 13, 16 | ovmpoa 7161 | 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 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {cab 2775 ∃wrex 3106 ∅c0 4211 ifcif 4381 〈cop 4478 Fn wfn 6220 ‘cfv 6225 (class class class)co 7016 0cc0 10383 ℕ0cn0 11745 ℤcz 11829 ..^cfzo 12883 mod cmo 13087 ♯chash 13540 ++ cconcat 13768 substr csubstr 13838 prefix cpfx 13868 cyclShift ccsh 13986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-iota 6189 df-fun 6227 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-csh 13987 |
This theorem is referenced by: cshword 13989 |
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