| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tocycfv | Structured version Visualization version GIF version | ||
| Description: Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-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) |
| Ref | Expression |
|---|---|
| tocycfv | ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tocycfv.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
| 2 | tocycval.1 | . . . 4 ⊢ 𝐶 = (toCyc‘𝐷) | |
| 3 | 2 | tocycval 33341 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) |
| 4 | 1, 3 | syl 18 | . 2 ⊢ (𝜑 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) |
| 5 | simpr 489 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊) | |
| 6 | 5 | rneqd 5919 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ran 𝑤 = ran 𝑊) |
| 7 | 6 | difeq2d 4083 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝐷 ∖ ran 𝑤) = (𝐷 ∖ ran 𝑊)) |
| 8 | 7 | reseq2d 5969 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑤)) = ( I ↾ (𝐷 ∖ ran 𝑊))) |
| 9 | 5 | oveq1d 7415 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝑤 cyclShift 1) = (𝑊 cyclShift 1)) |
| 10 | 5 | cnveqd 5852 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ◡𝑤 = ◡𝑊) |
| 11 | 9, 10 | coeq12d 5841 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ((𝑤 cyclShift 1) ∘ ◡𝑤) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
| 12 | 8, 11 | uneq12d 4125 | . 2 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
| 13 | id 23 | . . . 4 ⊢ (𝑢 = 𝑊 → 𝑢 = 𝑊) | |
| 14 | dmeq 5884 | . . . 4 ⊢ (𝑢 = 𝑊 → dom 𝑢 = dom 𝑊) | |
| 15 | eqidd 2766 | . . . 4 ⊢ (𝑢 = 𝑊 → 𝐷 = 𝐷) | |
| 16 | 13, 14, 15 | f1eq123d 6802 | . . 3 ⊢ (𝑢 = 𝑊 → (𝑢:dom 𝑢–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
| 17 | tocycfv.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | |
| 18 | tocycfv.1 | . . 3 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | |
| 19 | 16, 17, 18 | elrabd 3655 | . 2 ⊢ (𝜑 → 𝑊 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷}) |
| 20 | 1 | difexd 5292 | . . . 4 ⊢ (𝜑 → (𝐷 ∖ ran 𝑊) ∈ V) |
| 21 | 20 | resiexd 7204 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) ∈ V) |
| 22 | cshwcl 14825 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 cyclShift 1) ∈ Word 𝐷) | |
| 23 | 17, 22 | syl 18 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) ∈ Word 𝐷) |
| 24 | cnvexg 7909 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → ◡𝑊 ∈ V) | |
| 25 | 17, 24 | syl 18 | . . . 4 ⊢ (𝜑 → ◡𝑊 ∈ V) |
| 26 | coexg 7914 | . . . 4 ⊢ (((𝑊 cyclShift 1) ∈ Word 𝐷 ∧ ◡𝑊 ∈ V) → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) | |
| 27 | 23, 25, 26 | syl2anc 595 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) |
| 28 | unexg 7730 | . . 3 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∈ V ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ∈ V) | |
| 29 | 21, 27, 28 | syl2anc 595 | . 2 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ∈ V) |
| 30 | 4, 12, 19, 29 | fvmptd 6987 | 1 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 {crab 3417 Vcvv 3457 ∖ cdif 3904 ∪ cun 3905 ↦ cmpt 5186 I cid 5546 ◡ccnv 5651 dom cdm 5652 ran crn 5653 ↾ cres 5654 ∘ ccom 5656 –1-1→wf1 6522 ‘cfv 6525 (class class class)co 7400 1c1 11089 Word cword 14540 cyclShift ccsh 14815 toCycctocyc 33339 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-hash 14358 df-word 14541 df-concat 14598 df-substr 14669 df-pfx 14699 df-csh 14816 df-tocyc 33340 |
| This theorem is referenced by: tocycfvres1 33343 tocycfvres2 33344 cycpmfvlem 33345 cycpmfv3 33348 cycpmcl 33349 tocyc01 33351 cycpm2tr 33352 cycpmconjv 33375 cycpmrn 33376 |
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