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 30750 | . . 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 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊) | |
6 | 5 | rneqd 5808 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ran 𝑤 = ran 𝑊) |
7 | 6 | difeq2d 4099 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝐷 ∖ ran 𝑤) = (𝐷 ∖ ran 𝑊)) |
8 | 7 | reseq2d 5853 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑤)) = ( I ↾ (𝐷 ∖ ran 𝑊))) |
9 | 5 | oveq1d 7171 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → (𝑤 cyclShift 1) = (𝑊 cyclShift 1)) |
10 | 5 | cnveqd 5746 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ◡𝑤 = ◡𝑊) |
11 | 9, 10 | coeq12d 5735 | . . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ((𝑤 cyclShift 1) ∘ ◡𝑤) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) |
12 | 8, 11 | uneq12d 4140 | . 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 5772 | . . . 4 ⊢ (𝑢 = 𝑊 → dom 𝑢 = dom 𝑊) | |
15 | eqidd 2822 | . . . 4 ⊢ (𝑢 = 𝑊 → 𝐷 = 𝐷) | |
16 | 13, 14, 15 | f1eq123d 6608 | . . 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 3682 | . 2 ⊢ (𝜑 → 𝑊 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷}) |
20 | difexg 5231 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∖ ran 𝑊) ∈ V) | |
21 | 1, 20 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷 ∖ ran 𝑊) ∈ V) |
22 | 21 | resiexd 6979 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) ∈ V) |
23 | cshwcl 14160 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 cyclShift 1) ∈ Word 𝐷) | |
24 | 17, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) ∈ Word 𝐷) |
25 | cnvexg 7629 | . . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → ◡𝑊 ∈ V) | |
26 | 17, 25 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝑊 ∈ V) |
27 | coexg 7634 | . . . 4 ⊢ (((𝑊 cyclShift 1) ∈ Word 𝐷 ∧ ◡𝑊 ∈ V) → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) | |
28 | 24, 26, 27 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) |
29 | unexg 7472 | . . 3 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∈ V ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊) ∈ V) → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ∈ V) | |
30 | 22, 28, 29 | syl2anc 586 | . 2 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)) ∈ V) |
31 | 4, 12, 19, 30 | fvmptd 6775 | 1 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ↦ cmpt 5146 I cid 5459 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 1c1 10538 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 |
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-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-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 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: tocycfvres1 30752 tocycfvres2 30753 cycpmfvlem 30754 cycpmfv3 30757 cycpmcl 30758 tocyc01 30760 cycpm2tr 30761 cycpmconjv 30784 cycpmrn 30785 |
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