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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tocycfvres1 | Structured version Visualization version GIF version |
Description: A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
Ref | Expression |
---|---|
tocycval.1 | ⊢ 𝐶 = (toCyc‘𝐷) |
tocycfv.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
tocycfv.w | ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
tocycfv.1 | ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) |
Ref | Expression |
---|---|
tocycfvres1 | ⊢ (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tocycval.1 | . . . 4 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | tocycfv.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | tocycfv.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | |
4 | tocycfv.1 | . . . 4 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | |
5 | 1, 2, 3, 4 | tocycfv 32739 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | reseq1d 5971 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ↾ ran 𝑊)) |
7 | fnresi 6670 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊) | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊)) |
9 | 1zzd 12591 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
10 | cshwfn 14749 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) | |
11 | 3, 9, 10 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
12 | f1f1orn 6835 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊) | |
13 | f1ocnv 6836 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1-onto→ran 𝑊 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊) | |
14 | f1ofn 6825 | . . . . 5 ⊢ (◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ◡𝑊 Fn ran 𝑊) | |
15 | 4, 12, 13, 14 | 4syl 19 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) |
16 | dfdm4 5886 | . . . . 5 ⊢ dom 𝑊 = ran ◡𝑊 | |
17 | wrddm 14469 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊))) | |
18 | 3, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
19 | ssidd 3998 | . . . . . 6 ⊢ (𝜑 → (0..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))) | |
20 | 18, 19 | eqsstrd 4013 | . . . . 5 ⊢ (𝜑 → dom 𝑊 ⊆ (0..^(♯‘𝑊))) |
21 | 16, 20 | eqsstrrid 4024 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
22 | fnco 6658 | . . . 4 ⊢ (((𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)) ∧ ◡𝑊 Fn ran 𝑊 ∧ ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) | |
23 | 11, 15, 21, 22 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
24 | disjdifr 4465 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ | |
25 | 24 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) |
26 | fnunres2 6653 | . . 3 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊) ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊 ∧ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) | |
27 | 8, 23, 25, 26 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
28 | 6, 27 | eqtrd 2764 | 1 ⊢ (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3938 ∪ cun 3939 ∩ cin 3940 ⊆ wss 3941 ∅c0 4315 I cid 5564 ◡ccnv 5666 dom cdm 5667 ran crn 5668 ↾ cres 5669 ∘ ccom 5671 Fn wfn 6529 –1-1→wf1 6531 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 ℤcz 12556 ..^cfzo 13625 ♯chash 14288 Word cword 14462 cyclShift ccsh 14736 toCycctocyc 32736 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-fz 13483 df-fzo 13626 df-fl 13755 df-mod 13833 df-hash 14289 df-word 14463 df-concat 14519 df-substr 14589 df-pfx 14619 df-csh 14737 df-tocyc 32737 |
This theorem is referenced by: cycpmconjslem1 32784 cycpmconjslem2 32785 |
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