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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cyc2fv1 | Structured version Visualization version GIF version |
Description: Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
Ref | Expression |
---|---|
cycpm2.c | β’ πΆ = (toCycβπ·) |
cycpm2.d | β’ (π β π· β π) |
cycpm2.i | β’ (π β πΌ β π·) |
cycpm2.j | β’ (π β π½ β π·) |
cycpm2.1 | β’ (π β πΌ β π½) |
cycpm2cl.s | β’ π = (SymGrpβπ·) |
Ref | Expression |
---|---|
cyc2fv1 | β’ (π β ((πΆββ¨βπΌπ½ββ©)βπΌ) = π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm2.c | . . 3 β’ πΆ = (toCycβπ·) | |
2 | cycpm2.d | . . 3 β’ (π β π· β π) | |
3 | cycpm2.i | . . . 4 β’ (π β πΌ β π·) | |
4 | cycpm2.j | . . . 4 β’ (π β π½ β π·) | |
5 | 3, 4 | s2cld 14862 | . . 3 β’ (π β β¨βπΌπ½ββ© β Word π·) |
6 | cycpm2.1 | . . . 4 β’ (π β πΌ β π½) | |
7 | 3, 4, 6 | s2f1 32689 | . . 3 β’ (π β β¨βπΌπ½ββ©:dom β¨βπΌπ½ββ©β1-1βπ·) |
8 | c0ex 11246 | . . . . . 6 β’ 0 β V | |
9 | 8 | snid 4669 | . . . . 5 β’ 0 β {0} |
10 | s2len 14880 | . . . . . . . . 9 β’ (β―ββ¨βπΌπ½ββ©) = 2 | |
11 | 10 | oveq1i 7436 | . . . . . . . 8 β’ ((β―ββ¨βπΌπ½ββ©) β 1) = (2 β 1) |
12 | 2m1e1 12376 | . . . . . . . 8 β’ (2 β 1) = 1 | |
13 | 11, 12 | eqtr2i 2757 | . . . . . . 7 β’ 1 = ((β―ββ¨βπΌπ½ββ©) β 1) |
14 | 13 | oveq2i 7437 | . . . . . 6 β’ (0..^1) = (0..^((β―ββ¨βπΌπ½ββ©) β 1)) |
15 | fzo01 13754 | . . . . . 6 β’ (0..^1) = {0} | |
16 | 14, 15 | eqtr3i 2758 | . . . . 5 β’ (0..^((β―ββ¨βπΌπ½ββ©) β 1)) = {0} |
17 | 9, 16 | eleqtrri 2828 | . . . 4 β’ 0 β (0..^((β―ββ¨βπΌπ½ββ©) β 1)) |
18 | 17 | a1i 11 | . . 3 β’ (π β 0 β (0..^((β―ββ¨βπΌπ½ββ©) β 1))) |
19 | 1, 2, 5, 7, 18 | cycpmfv1 32855 | . 2 β’ (π β ((πΆββ¨βπΌπ½ββ©)β(β¨βπΌπ½ββ©β0)) = (β¨βπΌπ½ββ©β(0 + 1))) |
20 | s2fv0 14878 | . . . 4 β’ (πΌ β π· β (β¨βπΌπ½ββ©β0) = πΌ) | |
21 | 3, 20 | syl 17 | . . 3 β’ (π β (β¨βπΌπ½ββ©β0) = πΌ) |
22 | 21 | fveq2d 6906 | . 2 β’ (π β ((πΆββ¨βπΌπ½ββ©)β(β¨βπΌπ½ββ©β0)) = ((πΆββ¨βπΌπ½ββ©)βπΌ)) |
23 | 0p1e1 12372 | . . . 4 β’ (0 + 1) = 1 | |
24 | 23 | fveq2i 6905 | . . 3 β’ (β¨βπΌπ½ββ©β(0 + 1)) = (β¨βπΌπ½ββ©β1) |
25 | s2fv1 14879 | . . . 4 β’ (π½ β π· β (β¨βπΌπ½ββ©β1) = π½) | |
26 | 4, 25 | syl 17 | . . 3 β’ (π β (β¨βπΌπ½ββ©β1) = π½) |
27 | 24, 26 | eqtrid 2780 | . 2 β’ (π β (β¨βπΌπ½ββ©β(0 + 1)) = π½) |
28 | 19, 22, 27 | 3eqtr3d 2776 | 1 β’ (π β ((πΆββ¨βπΌπ½ββ©)βπΌ) = π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2937 {csn 4632 βcfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 + caddc 11149 β cmin 11482 2c2 12305 ..^cfzo 13667 β―chash 14329 β¨βcs2 14832 SymGrpcsymg 19328 toCycctocyc 32848 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-csh 14779 df-s2 14839 df-tocyc 32849 |
This theorem is referenced by: cycpmco2lem1 32868 cyc3co2 32882 |
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