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 2737 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
5 | 2, 3, 4 | tocycf 31103 | . . . 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 6546 | . 2 ⊢ (𝜑 → 𝐶 Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
8 | id 22 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝑤 = 〈“𝐼𝐽𝐾”〉) | |
9 | dmeq 5772 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → dom 𝑤 = dom 〈“𝐼𝐽𝐾”〉) | |
10 | eqidd 2738 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝐷 = 𝐷) | |
11 | 8, 9, 10 | f1eq123d 6653 | . . 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 14437 | . . 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 30941 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷) |
20 | 11, 15, 19 | elrabd 3604 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
21 | s3clhash 30942 | . . 3 ⊢ 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3}) | |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3})) |
23 | 7, 20, 22 | fnfvimad 7050 | 1 ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {crab 3065 {csn 4541 ◡ccnv 5550 dom cdm 5551 “ cima 5554 ⟶wf 6376 –1-1→wf1 6377 ‘cfv 6380 3c3 11886 ♯chash 13896 Word cword 14069 〈“cs3 14407 Basecbs 16760 SymGrpcsymg 18759 toCycctocyc 31092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-substr 14206 df-pfx 14236 df-csh 14354 df-s2 14413 df-s3 14414 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-tset 16821 df-efmnd 18296 df-symg 18760 df-tocyc 31093 |
This theorem is referenced by: cyc3genpmlem 31137 |
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