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Mirrors > Home > ILE Home > Th. List > abssinper | Unicode version |
Description: The absolute value of sine has period . (Contributed by NM, 17-Aug-2008.) |
Ref | Expression |
---|---|
abssinper |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9083 | . . . . . . . . . 10 | |
2 | halfcl 8970 | . . . . . . . . . . . 12 | |
3 | 2cn 8815 | . . . . . . . . . . . . 13 | |
4 | picn 12916 | . . . . . . . . . . . . 13 | |
5 | mulass 7775 | . . . . . . . . . . . . 13 | |
6 | 3, 4, 5 | mp3an23 1308 | . . . . . . . . . . . 12 |
7 | 2, 6 | syl 14 | . . . . . . . . . . 11 |
8 | 2ap0 8837 | . . . . . . . . . . . . 13 # | |
9 | divcanap1 8465 | . . . . . . . . . . . . 13 # | |
10 | 3, 8, 9 | mp3an23 1308 | . . . . . . . . . . . 12 |
11 | 10 | oveq1d 5797 | . . . . . . . . . . 11 |
12 | 7, 11 | eqtr3d 2175 | . . . . . . . . . 10 |
13 | 1, 12 | syl 14 | . . . . . . . . 9 |
14 | 13 | adantl 275 | . . . . . . . 8 |
15 | 14 | oveq2d 5798 | . . . . . . 7 |
16 | 15 | fveq2d 5433 | . . . . . 6 |
17 | 16 | eqcomd 2146 | . . . . 5 |
18 | 17 | adantr 274 | . . . 4 |
19 | sinper 12938 | . . . . 5 | |
20 | 19 | adantlr 469 | . . . 4 |
21 | 18, 20 | eqtrd 2173 | . . 3 |
22 | 21 | fveq2d 5433 | . 2 |
23 | peano2cn 7921 | . . . . . . . . . . . 12 | |
24 | halfcl 8970 | . . . . . . . . . . . 12 | |
25 | 23, 24 | syl 14 | . . . . . . . . . . 11 |
26 | 3, 4 | mulcli 7795 | . . . . . . . . . . 11 |
27 | mulcl 7771 | . . . . . . . . . . 11 | |
28 | 25, 26, 27 | sylancl 410 | . . . . . . . . . 10 |
29 | subadd23 7998 | . . . . . . . . . . 11 | |
30 | 4, 29 | mp3an2 1304 | . . . . . . . . . 10 |
31 | 28, 30 | sylan2 284 | . . . . . . . . 9 |
32 | divcanap1 8465 | . . . . . . . . . . . . . . . . . . 19 # | |
33 | 3, 8, 32 | mp3an23 1308 | . . . . . . . . . . . . . . . . . 18 |
34 | 23, 33 | syl 14 | . . . . . . . . . . . . . . . . 17 |
35 | 34 | oveq1d 5797 | . . . . . . . . . . . . . . . 16 |
36 | ax-1cn 7737 | . . . . . . . . . . . . . . . . 17 | |
37 | adddir 7781 | . . . . . . . . . . . . . . . . 17 | |
38 | 36, 4, 37 | mp3an23 1308 | . . . . . . . . . . . . . . . 16 |
39 | 35, 38 | eqtrd 2173 | . . . . . . . . . . . . . . 15 |
40 | 4 | mulid2i 7793 | . . . . . . . . . . . . . . . 16 |
41 | 40 | oveq2i 5793 | . . . . . . . . . . . . . . 15 |
42 | 39, 41 | eqtr2di 2190 | . . . . . . . . . . . . . 14 |
43 | mulass 7775 | . . . . . . . . . . . . . . . 16 | |
44 | 3, 4, 43 | mp3an23 1308 | . . . . . . . . . . . . . . 15 |
45 | 25, 44 | syl 14 | . . . . . . . . . . . . . 14 |
46 | 42, 45 | eqtr2d 2174 | . . . . . . . . . . . . 13 |
47 | 46 | oveq1d 5797 | . . . . . . . . . . . 12 |
48 | mulcl 7771 | . . . . . . . . . . . . . 14 | |
49 | 4, 48 | mpan2 422 | . . . . . . . . . . . . 13 |
50 | pncan 7992 | . . . . . . . . . . . . 13 | |
51 | 49, 4, 50 | sylancl 410 | . . . . . . . . . . . 12 |
52 | 47, 51 | eqtrd 2173 | . . . . . . . . . . 11 |
53 | 52 | adantl 275 | . . . . . . . . . 10 |
54 | 53 | oveq2d 5798 | . . . . . . . . 9 |
55 | 31, 54 | eqtr2d 2174 | . . . . . . . 8 |
56 | 1, 55 | sylan2 284 | . . . . . . 7 |
57 | 56 | fveq2d 5433 | . . . . . 6 |
58 | 57 | adantr 274 | . . . . 5 |
59 | subcl 7985 | . . . . . . . . 9 | |
60 | 4, 59 | mpan2 422 | . . . . . . . 8 |
61 | sinper 12938 | . . . . . . . 8 | |
62 | 60, 61 | sylan 281 | . . . . . . 7 |
63 | 62 | adantlr 469 | . . . . . 6 |
64 | sinmpi 12944 | . . . . . . 7 | |
65 | 64 | ad2antrr 480 | . . . . . 6 |
66 | 63, 65 | eqtrd 2173 | . . . . 5 |
67 | 58, 66 | eqtrd 2173 | . . . 4 |
68 | 67 | fveq2d 5433 | . . 3 |
69 | sincl 11449 | . . . . 5 | |
70 | 69 | absnegd 10993 | . . . 4 |
71 | 70 | ad2antrr 480 | . . 3 |
72 | 68, 71 | eqtrd 2173 | . 2 |
73 | zeo 9180 | . . 3 | |
74 | 73 | adantl 275 | . 2 |
75 | 22, 72, 74 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1332 wcel 1481 class class class wbr 3937 cfv 5131 (class class class)co 5782 cc 7642 cc0 7644 c1 7645 caddc 7647 cmul 7649 cmin 7957 cneg 7958 # cap 8367 cdiv 8456 c2 8795 cz 9078 cabs 10801 csin 11387 cpi 11390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834 |
This theorem is referenced by: sinkpi 12976 |
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