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Mirrors > Home > MPE Home > Th. List > 3cshw | Structured version Visualization version GIF version |
Description: Cyclically shifting a word three times results in a once cyclically shifted word under certain circumstances. (Contributed by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.) |
Ref | Expression |
---|---|
3cshw | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cshwid 14178 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) = 𝑊) | |
2 | 1 | 3adant2 1127 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) = 𝑊) |
3 | 2 | eqcomd 2829 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑊 = ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀))) |
4 | 3 | oveq1d 7173 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) cyclShift 𝑁)) |
5 | cshwcl 14162 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑀) ∈ Word 𝑉) | |
6 | 5 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑀) ∈ Word 𝑉) |
7 | lencl 13885 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
8 | 7 | nn0zd 12088 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ) |
9 | zsubcl 12027 | . . . . 5 ⊢ (((♯‘𝑊) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((♯‘𝑊) − 𝑀) ∈ ℤ) | |
10 | 8, 9 | sylan 582 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ) → ((♯‘𝑊) − 𝑀) ∈ ℤ) |
11 | 10 | 3adant2 1127 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((♯‘𝑊) − 𝑀) ∈ ℤ) |
12 | simp2 1133 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℤ) | |
13 | 2cshwcom 14180 | . . 3 ⊢ (((𝑊 cyclShift 𝑀) ∈ Word 𝑉 ∧ ((♯‘𝑊) − 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀))) | |
14 | 6, 11, 12, 13 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀))) |
15 | 4, 14 | eqtrd 2858 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 − cmin 10872 ℤcz 11984 ♯chash 13693 Word cword 13864 cyclShift ccsh 14152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-hash 13694 df-word 13865 df-concat 13925 df-substr 14005 df-pfx 14035 df-csh 14153 |
This theorem is referenced by: cshweqdif2 14183 |
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