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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 2732 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | 2, 3, 4 | tocycf 32263 | . . . 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 6715 | . 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 5901 | . . . 4 β’ (π€ = β¨βπΌπ½πΎββ© β dom π€ = dom β¨βπΌπ½πΎββ©) | |
10 | eqidd 2733 | . . . 4 β’ (π€ = β¨βπΌπ½πΎββ© β π· = π·) | |
11 | 8, 9, 10 | f1eq123d 6822 | . . 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 14819 | . . 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 32100 | . . 3 β’ (π β β¨βπΌπ½πΎββ©:dom β¨βπΌπ½πΎββ©β1-1βπ·) |
20 | 11, 15, 19 | elrabd 3684 | . 2 β’ (π β β¨βπΌπ½πΎββ© β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
21 | s3clhash 32101 | . . 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 7232 | 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 1541 β wcel 2106 β wne 2940 {crab 3432 {csn 4627 β‘ccnv 5674 dom cdm 5675 β cima 5678 βΆwf 6536 β1-1βwf1 6537 βcfv 6540 3c3 12264 β―chash 14286 Word cword 14460 β¨βcs3 14789 Basecbs 17140 SymGrpcsymg 19228 toCycctocyc 32252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-csh 14735 df-s2 14795 df-s3 14796 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-tset 17212 df-efmnd 18746 df-symg 19229 df-tocyc 32253 |
This theorem is referenced by: cyc3genpmlem 32297 |
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