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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmfv3 | Structured version Visualization version GIF version | ||
| Description: Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-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) | 
| cycpmfv3.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐷) | 
| cycpmfv3.2 | ⊢ (𝜑 → ¬ 𝑋 ∈ ran 𝑊) | 
| Ref | Expression | 
|---|---|
| cycpmfv3 | ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋) | 
| 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 33130 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) | 
| 6 | 5 | fveq1d 6907 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘𝑋)) | 
| 7 | f1oi 6885 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) | |
| 8 | f1ofn 6848 | . . . 4 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊)) | |
| 9 | 7, 8 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊)) | 
| 10 | 1zzd 12650 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 11 | cshwf 14839 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | |
| 12 | 3, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | 
| 13 | 12 | ffnd 6736 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) | 
| 14 | df-f1 6565 | . . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
| 15 | 4, 14 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | 
| 16 | 15 | simprd 495 | . . . . . 6 ⊢ (𝜑 → Fun ◡𝑊) | 
| 17 | 16 | funfnd 6596 | . . . . 5 ⊢ (𝜑 → ◡𝑊 Fn dom ◡𝑊) | 
| 18 | df-rn 5695 | . . . . . 6 ⊢ ran 𝑊 = dom ◡𝑊 | |
| 19 | 18 | fneq2i 6665 | . . . . 5 ⊢ (◡𝑊 Fn ran 𝑊 ↔ ◡𝑊 Fn dom ◡𝑊) | 
| 20 | 17, 19 | sylibr 234 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) | 
| 21 | dfdm4 5905 | . . . . . 6 ⊢ dom 𝑊 = ran ◡𝑊 | |
| 22 | 21 | eqimss2i 4044 | . . . . 5 ⊢ ran ◡𝑊 ⊆ dom 𝑊 | 
| 23 | wrdfn 14567 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
| 24 | 3, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) | 
| 25 | 24 | fndmd 6672 | . . . . 5 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) | 
| 26 | 22, 25 | sseqtrid 4025 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) | 
| 27 | fnco 6685 | . . . 4 ⊢ (((𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)) ∧ ◡𝑊 Fn ran 𝑊 ∧ ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) | |
| 28 | 13, 20, 26, 27 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) | 
| 29 | disjdifr 4472 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ | |
| 30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) | 
| 31 | cycpmfv3.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
| 32 | cycpmfv3.2 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ ran 𝑊) | |
| 33 | 31, 32 | eldifd 3961 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ ran 𝑊)) | 
| 34 | fvun1 6999 | . . 3 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊) ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊 ∧ (((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ ∧ 𝑋 ∈ (𝐷 ∖ ran 𝑊))) → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘𝑋) = (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋)) | |
| 35 | 9, 28, 30, 33, 34 | syl112anc 1375 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘𝑋) = (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋)) | 
| 36 | fvresi 7194 | . . 3 ⊢ (𝑋 ∈ (𝐷 ∖ ran 𝑊) → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋) | |
| 37 | 33, 36 | syl 17 | . 2 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋) | 
| 38 | 6, 35, 37 | 3eqtrd 2780 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 I cid 5576 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 ∘ ccom 5688 Fun wfun 6554 Fn wfn 6555 ⟶wf 6556 –1-1→wf1 6557 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 ℤcz 12615 ..^cfzo 13695 ♯chash 14370 Word cword 14553 cyclShift ccsh 14827 toCycctocyc 33127 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-hash 14371 df-word 14554 df-concat 14610 df-substr 14680 df-pfx 14710 df-csh 14828 df-tocyc 33128 | 
| This theorem is referenced by: cycpmco2 33154 cyc2fvx 33155 cyc3co2 33161 | 
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