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Theorem cycpmfv3 30908
 Description: Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
Hypotheses
Ref Expression
tocycval.1 𝐶 = (toCyc‘𝐷)
tocycfv.d (𝜑𝐷𝑉)
tocycfv.w (𝜑𝑊 ∈ Word 𝐷)
tocycfv.1 (𝜑𝑊:dom 𝑊1-1𝐷)
cycpmfv3.1 (𝜑𝑋𝐷)
cycpmfv3.2 (𝜑 → ¬ 𝑋 ∈ ran 𝑊)
Assertion
Ref Expression
cycpmfv3 (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)

Proof of Theorem cycpmfv3
StepHypRef Expression
1 tocycval.1 . . . 4 𝐶 = (toCyc‘𝐷)
2 tocycfv.d . . . 4 (𝜑𝐷𝑉)
3 tocycfv.w . . . 4 (𝜑𝑊 ∈ Word 𝐷)
4 tocycfv.1 . . . 4 (𝜑𝑊:dom 𝑊1-1𝐷)
51, 2, 3, 4tocycfv 30902 . . 3 (𝜑 → (𝐶𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊)))
65fveq1d 6660 . 2 (𝜑 → ((𝐶𝑊)‘𝑋) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊))‘𝑋))
7 f1oi 6639 . . . 4 ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊)
8 f1ofn 6603 . . . 4 (( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊))
97, 8mp1i 13 . . 3 (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊))
10 1zzd 12052 . . . . . 6 (𝜑 → 1 ∈ ℤ)
11 cshwf 14209 . . . . . 6 ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
123, 10, 11syl2anc 587 . . . . 5 (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
1312ffnd 6499 . . . 4 (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)))
14 df-f1 6340 . . . . . . . 8 (𝑊:dom 𝑊1-1𝐷 ↔ (𝑊:dom 𝑊𝐷 ∧ Fun 𝑊))
154, 14sylib 221 . . . . . . 7 (𝜑 → (𝑊:dom 𝑊𝐷 ∧ Fun 𝑊))
1615simprd 499 . . . . . 6 (𝜑 → Fun 𝑊)
1716funfnd 6366 . . . . 5 (𝜑𝑊 Fn dom 𝑊)
18 df-rn 5535 . . . . . 6 ran 𝑊 = dom 𝑊
1918fneq2i 6432 . . . . 5 (𝑊 Fn ran 𝑊𝑊 Fn dom 𝑊)
2017, 19sylibr 237 . . . 4 (𝜑𝑊 Fn ran 𝑊)
21 dfdm4 5735 . . . . . 6 dom 𝑊 = ran 𝑊
2221eqimss2i 3951 . . . . 5 ran 𝑊 ⊆ dom 𝑊
23 wrdfn 13927 . . . . . . 7 (𝑊 ∈ Word 𝐷𝑊 Fn (0..^(♯‘𝑊)))
243, 23syl 17 . . . . . 6 (𝜑𝑊 Fn (0..^(♯‘𝑊)))
2524fndmd 6438 . . . . 5 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
2622, 25sseqtrid 3944 . . . 4 (𝜑 → ran 𝑊 ⊆ (0..^(♯‘𝑊)))
27 fnco 6448 . . . 4 (((𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)) ∧ 𝑊 Fn ran 𝑊 ∧ ran 𝑊 ⊆ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1) ∘ 𝑊) Fn ran 𝑊)
2813, 20, 26, 27syl3anc 1368 . . 3 (𝜑 → ((𝑊 cyclShift 1) ∘ 𝑊) Fn ran 𝑊)
29 incom 4106 . . . . 5 (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊)
30 disjdif 4368 . . . . 5 (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ∅
3129, 30eqtr3i 2783 . . . 4 ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅
3231a1i 11 . . 3 (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅)
33 cycpmfv3.1 . . . 4 (𝜑𝑋𝐷)
34 cycpmfv3.2 . . . 4 (𝜑 → ¬ 𝑋 ∈ ran 𝑊)
3533, 34eldifd 3869 . . 3 (𝜑𝑋 ∈ (𝐷 ∖ ran 𝑊))
36 fvun1 6743 . . 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))
379, 28, 32, 35, 36syl112anc 1371 . 2 (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊))‘𝑋) = (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋))
38 fvresi 6926 . . 3 (𝑋 ∈ (𝐷 ∖ ran 𝑊) → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋)
3935, 38syl 17 . 2 (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋)
406, 37, 393eqtrd 2797 1 (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225   I cid 5429  ◡ccnv 5523  dom cdm 5524  ran crn 5525   ↾ cres 5526   ∘ ccom 5528  Fun wfun 6329   Fn wfn 6330  ⟶wf 6331  –1-1→wf1 6332  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576  ℤcz 12020  ..^cfzo 13082  ♯chash 13740  Word cword 13913   cyclShift ccsh 14197  toCycctocyc 30899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fzo 13083  df-fl 13211  df-mod 13287  df-hash 13741  df-word 13914  df-concat 13970  df-substr 14050  df-pfx 14080  df-csh 14198  df-tocyc 30900 This theorem is referenced by:  cycpmco2  30926  cyc2fvx  30927  cyc3co2  30933
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