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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 14766 | . . 3 β’ (π β β¨βπΌπ½ββ© β Word π·) |
6 | cycpm2.1 | . . . 4 β’ (π β πΌ β π½) | |
7 | 3, 4, 6 | s2f1 31850 | . . 3 β’ (π β β¨βπΌπ½ββ©:dom β¨βπΌπ½ββ©β1-1βπ·) |
8 | c0ex 11154 | . . . . . 6 β’ 0 β V | |
9 | 8 | snid 4623 | . . . . 5 β’ 0 β {0} |
10 | s2len 14784 | . . . . . . . . 9 β’ (β―ββ¨βπΌπ½ββ©) = 2 | |
11 | 10 | oveq1i 7368 | . . . . . . . 8 β’ ((β―ββ¨βπΌπ½ββ©) β 1) = (2 β 1) |
12 | 2m1e1 12284 | . . . . . . . 8 β’ (2 β 1) = 1 | |
13 | 11, 12 | eqtr2i 2762 | . . . . . . 7 β’ 1 = ((β―ββ¨βπΌπ½ββ©) β 1) |
14 | 13 | oveq2i 7369 | . . . . . 6 β’ (0..^1) = (0..^((β―ββ¨βπΌπ½ββ©) β 1)) |
15 | fzo01 13660 | . . . . . 6 β’ (0..^1) = {0} | |
16 | 14, 15 | eqtr3i 2763 | . . . . 5 β’ (0..^((β―ββ¨βπΌπ½ββ©) β 1)) = {0} |
17 | 9, 16 | eleqtrri 2833 | . . . 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 32011 | . 2 β’ (π β ((πΆββ¨βπΌπ½ββ©)β(β¨βπΌπ½ββ©β0)) = (β¨βπΌπ½ββ©β(0 + 1))) |
20 | s2fv0 14782 | . . . 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 6847 | . 2 β’ (π β ((πΆββ¨βπΌπ½ββ©)β(β¨βπΌπ½ββ©β0)) = ((πΆββ¨βπΌπ½ββ©)βπΌ)) |
23 | 0p1e1 12280 | . . . 4 β’ (0 + 1) = 1 | |
24 | 23 | fveq2i 6846 | . . 3 β’ (β¨βπΌπ½ββ©β(0 + 1)) = (β¨βπΌπ½ββ©β1) |
25 | s2fv1 14783 | . . . 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 2785 | . 2 β’ (π β (β¨βπΌπ½ββ©β(0 + 1)) = π½) |
28 | 19, 22, 27 | 3eqtr3d 2781 | 1 β’ (π β ((πΆββ¨βπΌπ½ββ©)βπΌ) = π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2940 {csn 4587 βcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 β cmin 11390 2c2 12213 ..^cfzo 13573 β―chash 14236 β¨βcs2 14736 SymGrpcsymg 19153 toCycctocyc 32004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-csh 14683 df-s2 14743 df-tocyc 32005 |
This theorem is referenced by: cycpmco2lem1 32024 cyc3co2 32038 |
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