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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cyc3fv2 | Structured version Visualization version GIF version |
Description: Function value of a 3-cycle at the second point. (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 |
---|---|
cyc3fv2 | β’ (π β ((πΆββ¨βπΌπ½πΎββ©)βπ½) = πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.c | . . 3 β’ πΆ = (toCycβπ·) | |
2 | cycpm3.d | . . 3 β’ (π β π· β π) | |
3 | cycpm3.i | . . . 4 β’ (π β πΌ β π·) | |
4 | cycpm3.j | . . . 4 β’ (π β π½ β π·) | |
5 | cycpm3.k | . . . 4 β’ (π β πΎ β π·) | |
6 | 3, 4, 5 | s3cld 14826 | . . 3 β’ (π β β¨βπΌπ½πΎββ© β Word π·) |
7 | cycpm3.1 | . . . 4 β’ (π β πΌ β π½) | |
8 | cycpm3.2 | . . . 4 β’ (π β π½ β πΎ) | |
9 | cycpm3.3 | . . . 4 β’ (π β πΎ β πΌ) | |
10 | 3, 4, 5, 7, 8, 9 | s3f1 32615 | . . 3 β’ (π β β¨βπΌπ½πΎββ©:dom β¨βπΌπ½πΎββ©β1-1βπ·) |
11 | 1ex 11211 | . . . . . 6 β’ 1 β V | |
12 | 11 | prid2 4762 | . . . . 5 β’ 1 β {0, 1} |
13 | s3len 14848 | . . . . . . . . 9 β’ (β―ββ¨βπΌπ½πΎββ©) = 3 | |
14 | 13 | oveq1i 7414 | . . . . . . . 8 β’ ((β―ββ¨βπΌπ½πΎββ©) β 1) = (3 β 1) |
15 | 3m1e2 12341 | . . . . . . . 8 β’ (3 β 1) = 2 | |
16 | 14, 15 | eqtri 2754 | . . . . . . 7 β’ ((β―ββ¨βπΌπ½πΎββ©) β 1) = 2 |
17 | 16 | oveq2i 7415 | . . . . . 6 β’ (0..^((β―ββ¨βπΌπ½πΎββ©) β 1)) = (0..^2) |
18 | fzo0to2pr 13720 | . . . . . 6 β’ (0..^2) = {0, 1} | |
19 | 17, 18 | eqtri 2754 | . . . . 5 β’ (0..^((β―ββ¨βπΌπ½πΎββ©) β 1)) = {0, 1} |
20 | 12, 19 | eleqtrri 2826 | . . . 4 β’ 1 β (0..^((β―ββ¨βπΌπ½πΎββ©) β 1)) |
21 | 20 | a1i 11 | . . 3 β’ (π β 1 β (0..^((β―ββ¨βπΌπ½πΎββ©) β 1))) |
22 | 1, 2, 6, 10, 21 | cycpmfv1 32775 | . 2 β’ (π β ((πΆββ¨βπΌπ½πΎββ©)β(β¨βπΌπ½πΎββ©β1)) = (β¨βπΌπ½πΎββ©β(1 + 1))) |
23 | s3fv1 14846 | . . . 4 β’ (π½ β π· β (β¨βπΌπ½πΎββ©β1) = π½) | |
24 | 4, 23 | syl 17 | . . 3 β’ (π β (β¨βπΌπ½πΎββ©β1) = π½) |
25 | 24 | fveq2d 6888 | . 2 β’ (π β ((πΆββ¨βπΌπ½πΎββ©)β(β¨βπΌπ½πΎββ©β1)) = ((πΆββ¨βπΌπ½πΎββ©)βπ½)) |
26 | 1p1e2 12338 | . . . 4 β’ (1 + 1) = 2 | |
27 | 26 | fveq2i 6887 | . . 3 β’ (β¨βπΌπ½πΎββ©β(1 + 1)) = (β¨βπΌπ½πΎββ©β2) |
28 | s3fv2 14847 | . . . 4 β’ (πΎ β π· β (β¨βπΌπ½πΎββ©β2) = πΎ) | |
29 | 5, 28 | syl 17 | . . 3 β’ (π β (β¨βπΌπ½πΎββ©β2) = πΎ) |
30 | 27, 29 | eqtrid 2778 | . 2 β’ (π β (β¨βπΌπ½πΎββ©β(1 + 1)) = πΎ) |
31 | 22, 25, 30 | 3eqtr3d 2774 | 1 β’ (π β ((πΆββ¨βπΌπ½πΎββ©)βπ½) = πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2934 {cpr 4625 βcfv 6536 (class class class)co 7404 0cc0 11109 1c1 11110 + caddc 11112 β cmin 11445 2c2 12268 3c3 12269 ..^cfzo 13630 β―chash 14292 β¨βcs3 14796 SymGrpcsymg 19283 toCycctocyc 32768 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-hash 14293 df-word 14468 df-concat 14524 df-s1 14549 df-substr 14594 df-pfx 14624 df-csh 14742 df-s2 14802 df-s3 14803 df-tocyc 32769 |
This theorem is referenced by: cyc3co2 32802 |
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