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Mirrors > Home > MPE Home > Th. List > 0csh0 | Structured version Visualization version GIF version |
Description: Cyclically shifting an empty set/word always results in the empty word/set. (Contributed by AV, 25-Oct-2018.) (Revised by AV, 17-Nov-2018.) |
Ref | Expression |
---|---|
0csh0 | ⊢ (∅ cyclShift 𝑁) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csh 14683 | . . . 4 ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))))) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))) |
3 | iftrue 4493 | . . . 4 ⊢ (𝑤 = ∅ → if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) | |
4 | 3 | ad2antrl 727 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑤 = ∅ ∧ 𝑛 = 𝑁)) → if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) |
5 | 0nn0 12433 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | f0 6724 | . . . . . . 7 ⊢ ∅:∅⟶V | |
7 | ffn 6669 | . . . . . . . 8 ⊢ (∅:∅⟶V → ∅ Fn ∅) | |
8 | fzo0 13602 | . . . . . . . . . 10 ⊢ (0..^0) = ∅ | |
9 | 8 | eqcomi 2742 | . . . . . . . . 9 ⊢ ∅ = (0..^0) |
10 | 9 | fneq2i 6601 | . . . . . . . 8 ⊢ (∅ Fn ∅ ↔ ∅ Fn (0..^0)) |
11 | 7, 10 | sylib 217 | . . . . . . 7 ⊢ (∅:∅⟶V → ∅ Fn (0..^0)) |
12 | 6, 11 | ax-mp 5 | . . . . . 6 ⊢ ∅ Fn (0..^0) |
13 | id 22 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 0 ∈ ℕ0) | |
14 | oveq2 7366 | . . . . . . . . 9 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
15 | 14 | fneq2d 6597 | . . . . . . . 8 ⊢ (𝑙 = 0 → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
16 | 15 | adantl 483 | . . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑙 = 0) → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
17 | 13, 16 | rspcedv 3573 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (∅ Fn (0..^0) → ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
18 | 5, 12, 17 | mp2 9 | . . . . 5 ⊢ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙) |
19 | 0ex 5265 | . . . . . 6 ⊢ ∅ ∈ V | |
20 | fneq1 6594 | . . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓 Fn (0..^𝑙) ↔ ∅ Fn (0..^𝑙))) | |
21 | 20 | rexbidv 3172 | . . . . . 6 ⊢ (𝑓 = ∅ → (∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
22 | 19, 21 | elab 3631 | . . . . 5 ⊢ (∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙)) |
23 | 18, 22 | mpbir 230 | . . . 4 ⊢ ∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} |
24 | 23 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → ∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}) |
25 | id 22 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
26 | 19 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → ∅ ∈ V) |
27 | 2, 4, 24, 25, 26 | ovmpod 7508 | . 2 ⊢ (𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) |
28 | cshnz 14686 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) | |
29 | 27, 28 | pm2.61i 182 | 1 ⊢ (∅ cyclShift 𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3444 ∅c0 4283 ifcif 4487 ⟨cop 4593 Fn wfn 6492 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 0cc0 11056 ℕ0cn0 12418 ℤcz 12504 ..^cfzo 13573 mod cmo 13780 ♯chash 14236 ++ cconcat 14464 substr csubstr 14534 prefix cpfx 14564 cyclShift ccsh 14682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-csh 14683 |
This theorem is referenced by: cshw0 14688 cshwmodn 14689 cshwn 14691 cshwlen 14693 repswcshw 14706 |
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