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Theorem cycpmfv3 30409
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 30403 . . 3 (𝜑 → (𝐶𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊)))
65fveq1d 6545 . 2 (𝜑 → ((𝐶𝑊)‘𝑋) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊))‘𝑋))
7 f1oi 6525 . . . 4 ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊)
8 f1ofn 6489 . . . 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 11867 . . . . . 6 (𝜑 → 1 ∈ ℤ)
11 cshwf 14003 . . . . . 6 ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
123, 10, 11syl2anc 584 . . . . 5 (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
1312ffnd 6388 . . . 4 (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)))
14 df-f1 6235 . . . . . . . 8 (𝑊:dom 𝑊1-1𝐷 ↔ (𝑊:dom 𝑊𝐷 ∧ Fun 𝑊))
154, 14sylib 219 . . . . . . 7 (𝜑 → (𝑊:dom 𝑊𝐷 ∧ Fun 𝑊))
1615simprd 496 . . . . . 6 (𝜑 → Fun 𝑊)
1716funfnd 6261 . . . . 5 (𝜑𝑊 Fn dom 𝑊)
18 df-rn 5459 . . . . . 6 ran 𝑊 = dom 𝑊
1918fneq2i 6326 . . . . 5 (𝑊 Fn ran 𝑊𝑊 Fn dom 𝑊)
2017, 19sylibr 235 . . . 4 (𝜑𝑊 Fn ran 𝑊)
21 dfdm4 5655 . . . . . 6 dom 𝑊 = ran 𝑊
2221eqimss2i 3951 . . . . 5 ran 𝑊 ⊆ dom 𝑊
23 wrdfn 13727 . . . . . . 7 (𝑊 ∈ Word 𝐷𝑊 Fn (0..^(♯‘𝑊)))
243, 23syl 17 . . . . . 6 (𝜑𝑊 Fn (0..^(♯‘𝑊)))
2524fndmd 6331 . . . . 5 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
2622, 25sseqtrid 3944 . . . 4 (𝜑 → ran 𝑊 ⊆ (0..^(♯‘𝑊)))
27 fnco 6340 . . . 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 1364 . . 3 (𝜑 → ((𝑊 cyclShift 1) ∘ 𝑊) Fn ran 𝑊)
29 incom 4103 . . . . 5 (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊)
30 disjdif 4339 . . . . 5 (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ∅
3129, 30eqtr3i 2821 . . . 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 3874 . . 3 (𝜑𝑋 ∈ (𝐷 ∖ ran 𝑊))
36 fvun1 6626 . . 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 1367 . 2 (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊))‘𝑋) = (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋))
38 fvresi 6803 . . 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 2835 1 (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  cdif 3860  cun 3861  cin 3862  wss 3863  c0 4215   I cid 5352  ccnv 5447  dom cdm 5448  ran crn 5449  cres 5450  ccom 5452  Fun wfun 6224   Fn wfn 6225  wf 6226  1-1wf1 6227  1-1-ontowf1o 6229  cfv 6230  (class class class)co 7021  0cc0 10388  1c1 10389  cz 11834  ..^cfzo 12888  chash 13545  Word cword 13712   cyclShift ccsh 13991  toCycctocyc 30400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-oadd 7962  df-er 8144  df-map 8263  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-sup 8757  df-inf 8758  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-n0 11751  df-z 11835  df-uz 12099  df-rp 12245  df-fz 12748  df-fzo 12889  df-fl 13017  df-mod 13093  df-hash 13546  df-word 13713  df-concat 13774  df-substr 13844  df-pfx 13874  df-csh 13992  df-tocyc 30401
This theorem is referenced by:  cyc2fvx  30418  cyc3co2  30424
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