| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| 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 33128 | . . 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 17 | . 2 ⊢ (𝜑 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) |
| 5 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊) | |
| 6 | 5 | rneqd 5949 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ran 𝑤 = ran 𝑊) |
| 7 | 6 | difeq2d 4126 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝐷 ∖ ran 𝑤) = (𝐷 ∖ ran 𝑊)) |
| 8 | 7 | reseq2d 5997 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑤)) = ( I ↾ (𝐷 ∖ ran 𝑊))) |
| 9 | 5 | oveq1d 7446 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝑤 cyclShift 1) = (𝑊 cyclShift 1)) |
| 10 | 5 | cnveqd 5886 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ◡𝑤 = ◡𝑊) |
| 11 | 9, 10 | coeq12d 5875 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ((𝑤 cyclShift 1) ∘ ◡𝑤) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
| 12 | 8, 11 | uneq12d 4169 | . 2 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
| 13 | id 22 | . . . 4 ⊢ (𝑢 = 𝑊 → 𝑢 = 𝑊) | |
| 14 | dmeq 5914 | . . . 4 ⊢ (𝑢 = 𝑊 → dom 𝑢 = dom 𝑊) | |
| 15 | eqidd 2738 | . . . 4 ⊢ (𝑢 = 𝑊 → 𝐷 = 𝐷) | |
| 16 | 13, 14, 15 | f1eq123d 6840 | . . 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 3694 | . 2 ⊢ (𝜑 → 𝑊 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷}) |
| 20 | 1 | difexd 5331 | . . . 4 ⊢ (𝜑 → (𝐷 ∖ ran 𝑊) ∈ V) |
| 21 | 20 | resiexd 7236 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) ∈ V) |
| 22 | cshwcl 14836 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 cyclShift 1) ∈ Word 𝐷) | |
| 23 | 17, 22 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) ∈ Word 𝐷) |
| 24 | cnvexg 7946 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → ◡𝑊 ∈ V) | |
| 25 | 17, 24 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑊 ∈ V) |
| 26 | coexg 7951 | . . . 4 ⊢ (((𝑊 cyclShift 1) ∈ Word 𝐷 ∧ ◡𝑊 ∈ V) → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) | |
| 27 | 23, 25, 26 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) |
| 28 | unexg 7763 | . . 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 584 | . 2 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ∈ V) |
| 30 | 4, 12, 19, 29 | fvmptd 7023 | 1 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ↦ cmpt 5225 I cid 5577 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 1c1 11156 Word cword 14552 cyclShift ccsh 14826 toCycctocyc 33126 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-substr 14679 df-pfx 14709 df-csh 14827 df-tocyc 33127 |
| This theorem is referenced by: tocycfvres1 33130 tocycfvres2 33131 cycpmfvlem 33132 cycpmfv3 33135 cycpmcl 33136 tocyc01 33138 cycpm2tr 33139 cycpmconjv 33162 cycpmrn 33163 |
| Copyright terms: Public domain | W3C validator |