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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 | β’ πΆ = (toCycβπ·) |
cycpm3.s | β’ π = (SymGrpβπ·) |
cycpm3.d | β’ (π β π· β π) |
cycpm3.i | β’ (π β πΌ β π·) |
cycpm3.j | β’ (π β π½ β π·) |
cycpm3.k | β’ (π β πΎ β π·) |
cycpm3.1 | β’ (π β πΌ β π½) |
cycpm3.2 | β’ (π β π½ β πΎ) |
cycpm3.3 | β’ (π β πΎ β πΌ) |
Ref | Expression |
---|---|
cycpm3cl2 | β’ (π β (πΆββ¨βπΌπ½πΎββ©) β (πΆ β (β‘β― β {3}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.d | . . . 4 β’ (π β π· β π) | |
2 | cycpm3.c | . . . . 5 β’ πΆ = (toCycβπ·) | |
3 | cycpm3.s | . . . . 5 β’ π = (SymGrpβπ·) | |
4 | eqid 2725 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | 2, 3, 4 | tocycf 32883 | . . . 4 β’ (π· β π β πΆ:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
6 | 1, 5 | syl 17 | . . 3 β’ (π β πΆ:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
7 | 6 | ffnd 6718 | . 2 β’ (π β πΆ Fn {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
8 | id 22 | . . . 4 β’ (π€ = β¨βπΌπ½πΎββ© β π€ = β¨βπΌπ½πΎββ©) | |
9 | dmeq 5900 | . . . 4 β’ (π€ = β¨βπΌπ½πΎββ© β dom π€ = dom β¨βπΌπ½πΎββ©) | |
10 | eqidd 2726 | . . . 4 β’ (π€ = β¨βπΌπ½πΎββ© β π· = π·) | |
11 | 8, 9, 10 | f1eq123d 6826 | . . 3 β’ (π€ = β¨βπΌπ½πΎββ© β (π€:dom π€β1-1βπ· β β¨βπΌπ½πΎββ©:dom β¨βπΌπ½πΎββ©β1-1βπ·)) |
12 | cycpm3.i | . . . 4 β’ (π β πΌ β π·) | |
13 | cycpm3.j | . . . 4 β’ (π β π½ β π·) | |
14 | cycpm3.k | . . . 4 β’ (π β πΎ β π·) | |
15 | 12, 13, 14 | s3cld 14855 | . . 3 β’ (π β β¨βπΌπ½πΎββ© β Word π·) |
16 | cycpm3.1 | . . . 4 β’ (π β πΌ β π½) | |
17 | cycpm3.2 | . . . 4 β’ (π β π½ β πΎ) | |
18 | cycpm3.3 | . . . 4 β’ (π β πΎ β πΌ) | |
19 | 12, 13, 14, 16, 17, 18 | s3f1 32715 | . . 3 β’ (π β β¨βπΌπ½πΎββ©:dom β¨βπΌπ½πΎββ©β1-1βπ·) |
20 | 11, 15, 19 | elrabd 3676 | . 2 β’ (π β β¨βπΌπ½πΎββ© β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
21 | s3clhash 32716 | . . 3 β’ β¨βπΌπ½πΎββ© β (β‘β― β {3}) | |
22 | 21 | a1i 11 | . 2 β’ (π β β¨βπΌπ½πΎββ© β (β‘β― β {3})) |
23 | 7, 20, 22 | fnfvimad 7242 | 1 β’ (π β (πΆββ¨βπΌπ½πΎββ©) β (πΆ β (β‘β― β {3}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2930 {crab 3419 {csn 4624 β‘ccnv 5671 dom cdm 5672 β cima 5675 βΆwf 6539 β1-1βwf1 6540 βcfv 6543 3c3 12298 β―chash 14321 Word cword 14496 β¨βcs3 14825 Basecbs 17179 SymGrpcsymg 19325 toCycctocyc 32872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-csh 14771 df-s2 14831 df-s3 14832 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-efmnd 18825 df-symg 19326 df-tocyc 32873 |
This theorem is referenced by: cyc3genpmlem 32917 |
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