Mathbox for Thierry Arnoux |
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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 30751 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | fveq1d 6672 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘𝑋)) |
7 | f1oi 6652 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) | |
8 | f1ofn 6616 | . . . 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 12014 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | cshwf 14162 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | |
12 | 3, 10, 11 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) |
13 | 12 | ffnd 6515 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
14 | df-f1 6360 | . . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
15 | 4, 14 | sylib 220 | . . . . . . 7 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) |
16 | 15 | simprd 498 | . . . . . 6 ⊢ (𝜑 → Fun ◡𝑊) |
17 | 16 | funfnd 6386 | . . . . 5 ⊢ (𝜑 → ◡𝑊 Fn dom ◡𝑊) |
18 | df-rn 5566 | . . . . . 6 ⊢ ran 𝑊 = dom ◡𝑊 | |
19 | 18 | fneq2i 6451 | . . . . 5 ⊢ (◡𝑊 Fn ran 𝑊 ↔ ◡𝑊 Fn dom ◡𝑊) |
20 | 17, 19 | sylibr 236 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) |
21 | dfdm4 5764 | . . . . . 6 ⊢ dom 𝑊 = ran ◡𝑊 | |
22 | 21 | eqimss2i 4026 | . . . . 5 ⊢ ran ◡𝑊 ⊆ dom 𝑊 |
23 | wrdfn 13877 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
24 | 3, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
25 | 24 | fndmd 6456 | . . . . 5 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
26 | 22, 25 | sseqtrid 4019 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
27 | fnco 6465 | . . . 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 1367 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
29 | incom 4178 | . . . . 5 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) | |
30 | disjdif 4421 | . . . . 5 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ∅ | |
31 | 29, 30 | eqtr3i 2846 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ |
32 | 31 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) |
33 | cycpmfv3.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
34 | cycpmfv3.2 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ ran 𝑊) | |
35 | 33, 34 | eldifd 3947 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ ran 𝑊)) |
36 | fvun1 6754 | . . 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)) | |
37 | 9, 28, 32, 35, 36 | syl112anc 1370 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘𝑋) = (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋)) |
38 | fvresi 6935 | . . 3 ⊢ (𝑋 ∈ (𝐷 ∖ ran 𝑊) → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋) | |
39 | 35, 38 | syl 17 | . 2 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊))‘𝑋) = 𝑋) |
40 | 6, 37, 39 | 3eqtrd 2860 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 I cid 5459 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 ℤcz 11982 ..^cfzo 13034 ♯chash 13691 Word cword 13862 cyclShift ccsh 14150 toCycctocyc 30748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-hash 13692 df-word 13863 df-concat 13923 df-substr 14003 df-pfx 14033 df-csh 14151 df-tocyc 30749 |
This theorem is referenced by: cycpmco2 30775 cyc2fvx 30776 cyc3co2 30782 |
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