Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpm3cl2 | Structured version Visualization version GIF version |
Description: Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
cycpm3.c | ⊢ 𝐶 = (toCyc‘𝐷) |
cycpm3.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
cycpm3.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
cycpm3.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
cycpm3.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
cycpm3.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
cycpm3.1 | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
cycpm3.2 | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
cycpm3.3 | ⊢ (𝜑 → 𝐾 ≠ 𝐼) |
Ref | Expression |
---|---|
cycpm3cl2 | ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
2 | cycpm3.c | . . . . 5 ⊢ 𝐶 = (toCyc‘𝐷) | |
3 | cycpm3.s | . . . . 5 ⊢ 𝑆 = (SymGrp‘𝐷) | |
4 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
5 | 2, 3, 4 | tocycf 30780 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
7 | 6 | ffnd 6512 | . 2 ⊢ (𝜑 → 𝐶 Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
8 | id 22 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝑤 = 〈“𝐼𝐽𝐾”〉) | |
9 | dmeq 5769 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → dom 𝑤 = dom 〈“𝐼𝐽𝐾”〉) | |
10 | eqidd 2821 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝐷 = 𝐷) | |
11 | 8, 9, 10 | f1eq123d 6605 | . . 3 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷)) |
12 | cycpm3.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
13 | cycpm3.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
14 | cycpm3.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
15 | 12, 13, 14 | s3cld 14230 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷) |
16 | cycpm3.1 | . . . 4 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
17 | cycpm3.2 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
18 | cycpm3.3 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ 𝐼) | |
19 | 12, 13, 14, 16, 17, 18 | s3f1 30623 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷) |
20 | 11, 15, 19 | elrabd 3680 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
21 | s3clhash 30624 | . . 3 ⊢ 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3}) | |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3})) |
23 | 7, 20, 22 | fnfvimad 6993 | 1 ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {crab 3141 {csn 4564 ◡ccnv 5551 dom cdm 5552 “ cima 5555 ⟶wf 6348 –1-1→wf1 6349 ‘cfv 6352 3c3 11691 ♯chash 13688 Word cword 13859 〈“cs3 14200 Basecbs 16479 SymGrpcsymg 18491 toCycctocyc 30769 |
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 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 |
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 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-map 8405 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-sup 8903 df-inf 8904 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-xnn0 11966 df-z 11980 df-uz 12242 df-rp 12388 df-fz 12891 df-fzo 13032 df-fl 13160 df-mod 13236 df-hash 13689 df-word 13860 df-concat 13919 df-s1 13946 df-substr 13999 df-pfx 14029 df-csh 14147 df-s2 14206 df-s3 14207 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-tset 16580 df-efmnd 18030 df-symg 18492 df-tocyc 30770 |
This theorem is referenced by: cyc3genpmlem 30814 |
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