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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 14572 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0) | |
| 6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0) | 
| 7 | 6 | nn0zd 12641 | . . . . 5 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ) | 
| 8 | fzossrbm1 13729 | . . . . 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 3983 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) | 
| 12 | 1, 2, 3, 4, 11 | cycpmfvlem 33133 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) | 
| 13 | df-f1 6565 | . . . . 5 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
| 14 | 4, 13 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | 
| 15 | 14 | simprd 495 | . . 3 ⊢ (𝜑 → Fun ◡𝑊) | 
| 16 | wrdfn 14567 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
| 17 | 3, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) | 
| 18 | fnfvelrn 7099 | . . . . 5 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑁) ∈ ran 𝑊) | |
| 19 | 17, 11, 18 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑊‘𝑁) ∈ ran 𝑊) | 
| 20 | df-rn 5695 | . . . 4 ⊢ ran 𝑊 = dom ◡𝑊 | |
| 21 | 19, 20 | eleqtrdi 2850 | . . 3 ⊢ (𝜑 → (𝑊‘𝑁) ∈ dom ◡𝑊) | 
| 22 | fvco 7006 | . . 3 ⊢ ((Fun ◡𝑊 ∧ (𝑊‘𝑁) ∈ dom ◡𝑊) → (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁)) = ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁)))) | |
| 23 | 15, 21, 22 | syl2anc 584 | . 2 ⊢ (𝜑 → (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁)) = ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁)))) | 
| 24 | f1f1orn 6858 | . . . . . 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 6672 | . . . . . 6 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) | 
| 27 | 11, 26 | eleqtrrd 2843 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom 𝑊) | 
| 28 | f1ocnvfv1 7297 | . . . . 5 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝑁 ∈ dom 𝑊) → (◡𝑊‘(𝑊‘𝑁)) = 𝑁) | |
| 29 | 25, 27, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (◡𝑊‘(𝑊‘𝑁)) = 𝑁) | 
| 30 | 29 | fveq2d 6909 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁))) = ((𝑊 cyclShift 1)‘𝑁)) | 
| 31 | 1zzd 12650 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 32 | cshwidxmod 14842 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1)‘𝑁) = (𝑊‘((𝑁 + 1) mod (♯‘𝑊)))) | |
| 33 | 3, 31, 11, 32 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘𝑁) = (𝑊‘((𝑁 + 1) mod (♯‘𝑊)))) | 
| 34 | fzo0ss1 13730 | . . . . . 6 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊)) | |
| 35 | fzoaddel2 13760 | . . . . . . 7 ⊢ ((𝑁 ∈ (0..^((♯‘𝑊) − 1)) ∧ (♯‘𝑊) ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 + 1) ∈ (1..^(♯‘𝑊))) | |
| 36 | 10, 7, 31, 35 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ (1..^(♯‘𝑊))) | 
| 37 | 34, 36 | sselid 3980 | . . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (0..^(♯‘𝑊))) | 
| 38 | zmodidfzoimp 13942 | . . . . 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 6909 | . . 3 ⊢ (𝜑 → (𝑊‘((𝑁 + 1) mod (♯‘𝑊))) = (𝑊‘(𝑁 + 1))) | 
| 41 | 30, 33, 40 | 3eqtrd 2780 | . 2 ⊢ (𝜑 → ((𝑊 cyclShift 1)‘(◡𝑊‘(𝑊‘𝑁))) = (𝑊‘(𝑁 + 1))) | 
| 42 | 12, 23, 41 | 3eqtrd 2780 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ∘ 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 + caddc 11159 − cmin 11493 ℕ0cn0 12528 ℤcz 12615 ..^cfzo 13695 mod cmo 13910 ♯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-2 12330 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: cyc2fv1 33142 cycpmco2lem4 33150 cycpmco2lem6 33152 cycpmco2lem7 33153 cycpmco2 33154 cyc3fv1 33158 cyc3fv2 33159 cycpmrn 33164 | 
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