Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmfv1 | Structured version Visualization version GIF version |
Description: Value of a cycle function for any element but the last. (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) |
cycpmfv1.1 | ⊢ (𝜑 → 𝑁 ∈ (0..^((♯‘𝑊) − 1))) |
Ref | Expression |
---|---|
cycpmfv1 | ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tocycval.1 | . . 3 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | tocycfv.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | tocycfv.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | |
4 | tocycfv.1 | . . 3 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | |
5 | lencl 13877 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0) | |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0) |
7 | 6 | nn0zd 12079 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ) |
8 | fzossrbm1 13063 | . . . . 5 ⊢ ((♯‘𝑊) ∈ ℤ → (0..^((♯‘𝑊) − 1)) ⊆ (0..^(♯‘𝑊))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝑊) − 1)) ⊆ (0..^(♯‘𝑊))) |
10 | cycpmfv1.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^((♯‘𝑊) − 1))) | |
11 | 9, 10 | sseldd 3961 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) |
12 | 1, 2, 3, 4, 11 | cycpmfvlem 30773 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
13 | df-f1 6353 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
14 | 4, 13 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) |
15 | 14 | simprd 498 | . . 3 ⊢ (𝜑 → Fun ◡𝑊) |
16 | wrdfn 13872 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
17 | 3, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
18 | fnfvelrn 6841 | . . . . 5 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑁) ∈ ran 𝑊) | |
19 | 17, 11, 18 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑊‘𝑁) ∈ ran 𝑊) |
20 | df-rn 5559 | . . . 4 ⊢ ran 𝑊 = dom ◡𝑊 | |
21 | 19, 20 | eleqtrdi 2922 | . . 3 ⊢ (𝜑 → (𝑊‘𝑁) ∈ dom ◡𝑊) |
22 | fvco 6752 | . . 3 ⊢ ((Fun ◡𝑊 ∧ (𝑊‘𝑁) ∈ dom ◡𝑊) → (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁)) = ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁)))) | |
23 | 15, 21, 22 | syl2anc 586 | . 2 ⊢ (𝜑 → (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁)) = ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁)))) |
24 | f1f1orn 6619 | . . . . . 6 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊) | |
25 | 4, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊) |
26 | 17 | fndmd 6449 | . . . . . 6 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
27 | 11, 26 | eleqtrrd 2915 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom 𝑊) |
28 | f1ocnvfv1 7026 | . . . . 5 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝑁 ∈ dom 𝑊) → (◡𝑊‘(𝑊‘𝑁)) = 𝑁) | |
29 | 25, 27, 28 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (◡𝑊‘(𝑊‘𝑁)) = 𝑁) |
30 | 29 | fveq2d 6667 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁))) = ((𝑊 cyclShift 1)‘𝑁)) |
31 | 1zzd 12007 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
32 | cshwidxmod 14158 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1)‘𝑁) = (𝑊‘((𝑁 + 1) mod (♯‘𝑊)))) | |
33 | 3, 31, 11, 32 | syl3anc 1366 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘𝑁) = (𝑊‘((𝑁 + 1) mod (♯‘𝑊)))) |
34 | fzo0ss1 13064 | . . . . . 6 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊)) | |
35 | fzoaddel2 13090 | . . . . . . 7 ⊢ ((𝑁 ∈ (0..^((♯‘𝑊) − 1)) ∧ (♯‘𝑊) ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 + 1) ∈ (1..^(♯‘𝑊))) | |
36 | 10, 7, 31, 35 | syl3anc 1366 | . . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ (1..^(♯‘𝑊))) |
37 | 34, 36 | sseldi 3958 | . . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (0..^(♯‘𝑊))) |
38 | zmodidfzoimp 13266 | . . . . 5 ⊢ ((𝑁 + 1) ∈ (0..^(♯‘𝑊)) → ((𝑁 + 1) mod (♯‘𝑊)) = (𝑁 + 1)) | |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑁 + 1) mod (♯‘𝑊)) = (𝑁 + 1)) |
40 | 39 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (𝑊‘((𝑁 + 1) mod (♯‘𝑊))) = (𝑊‘(𝑁 + 1))) |
41 | 30, 33, 40 | 3eqtrd 2859 | . 2 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁))) = (𝑊‘(𝑁 + 1))) |
42 | 12, 23, 41 | 3eqtrd 2859 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ∘ ccom 5552 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7149 0cc0 10530 1c1 10531 + caddc 10533 − cmin 10863 ℕ0cn0 11891 ℤcz 11975 ..^cfzo 13030 mod cmo 13234 ♯chash 13687 Word cword 13858 cyclShift ccsh 14143 toCycctocyc 30767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-hash 13688 df-word 13859 df-concat 13916 df-substr 13996 df-pfx 14026 df-csh 14144 df-tocyc 30768 |
This theorem is referenced by: cyc2fv1 30782 cycpmco2lem4 30790 cycpmco2lem6 30792 cycpmco2lem7 30793 cycpmco2 30794 cyc3fv1 30798 cyc3fv2 30799 cycpmrn 30804 |
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