![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14772 | . . . 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 4535 | . . . 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 12518 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | f0 6778 | . . . . . . 7 ⊢ ∅:∅⟶V | |
7 | ffn 6722 | . . . . . . . 8 ⊢ (∅:∅⟶V → ∅ Fn ∅) | |
8 | fzo0 13689 | . . . . . . . . . 10 ⊢ (0..^0) = ∅ | |
9 | 8 | eqcomi 2737 | . . . . . . . . 9 ⊢ ∅ = (0..^0) |
10 | 9 | fneq2i 6652 | . . . . . . . 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 7428 | . . . . . . . . 9 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
15 | 14 | fneq2d 6648 | . . . . . . . 8 ⊢ (𝑙 = 0 → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
16 | 15 | adantl 481 | . . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑙 = 0) → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
17 | 13, 16 | rspcedv 3602 | . . . . . 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 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
20 | fneq1 6645 | . . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓 Fn (0..^𝑙) ↔ ∅ Fn (0..^𝑙))) | |
21 | 20 | rexbidv 3175 | . . . . . 6 ⊢ (𝑓 = ∅ → (∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
22 | 19, 21 | elab 3667 | . . . . 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 7573 | . 2 ⊢ (𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) |
28 | cshnz 14775 | . 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 1534 ∈ wcel 2099 {cab 2705 ∃wrex 3067 Vcvv 3471 ∅c0 4323 ifcif 4529 ⟨cop 4635 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 0cc0 11139 ℕ0cn0 12503 ℤcz 12589 ..^cfzo 13660 mod cmo 13867 ♯chash 14322 ++ cconcat 14553 substr csubstr 14623 prefix cpfx 14653 cyclShift ccsh 14771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-csh 14772 |
This theorem is referenced by: cshw0 14777 cshwmodn 14778 cshwn 14780 cshwlen 14782 repswcshw 14795 |
Copyright terms: Public domain | W3C validator |