Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tocycfvres2 | Structured version Visualization version GIF version |
Description: A cyclic permutation is the identity outside of 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 |
---|---|
tocycfvres2 | ⊢ (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 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 30953 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | reseq1d 5824 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ↾ (𝐷 ∖ ran 𝑊))) |
7 | fnresi 6465 | . . . 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 12094 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
10 | cshwfn 14252 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) | |
11 | 3, 9, 10 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
12 | f1f1orn 6629 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊) | |
13 | f1ocnv 6630 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1-onto→ran 𝑊 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊) | |
14 | f1ofn 6619 | . . . . 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 5738 | . . . . 5 ⊢ dom 𝑊 = ran ◡𝑊 | |
17 | wrddm 13962 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊))) | |
18 | 3, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
19 | ssidd 3900 | . . . . . 6 ⊢ (𝜑 → (0..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))) | |
20 | 18, 19 | eqsstrd 3915 | . . . . 5 ⊢ (𝜑 → dom 𝑊 ⊆ (0..^(♯‘𝑊))) |
21 | 16, 20 | eqsstrrid 3926 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
22 | fnco 6453 | . . . 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 1372 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
24 | disjdifr 4362 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ | |
25 | 24 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) |
26 | fnunres1 30519 | . . 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)) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊))) | |
27 | 8, 23, 25, 26 | syl3anc 1372 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊))) |
28 | 6, 27 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 ∪ cun 3841 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 I cid 5428 ◡ccnv 5524 dom cdm 5525 ran crn 5526 ↾ cres 5527 ∘ ccom 5529 Fn wfn 6334 –1-1→wf1 6336 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7170 0cc0 10615 1c1 10616 ℤcz 12062 ..^cfzo 13124 ♯chash 13782 Word cword 13955 cyclShift ccsh 14239 toCycctocyc 30950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-hash 13783 df-word 13956 df-concat 14012 df-substr 14092 df-pfx 14122 df-csh 14240 df-tocyc 30951 |
This theorem is referenced by: cycpmconjslem2 30999 cyc3conja 31001 |
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